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TRANSCRIPT
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A. Ravindran, D. Phillips, J. Solberg, “Operations Research:Principles and Practice”. 2nd Ed. John Wiley, 1987.
Hamdy A. Taha, “Operations Research: An Introduction”. 8th Ed.Pearson Prentice Hall, 2007.
Frederick S. Hillier, “Introduction to Operations Research”.McGraw-Hill Education, 2012.
Jorge Nocedal, Stephen J. Wright, “Numerical Optimization”.Second Edition, Springer, 2006.
Wikipedia (the free encyclopedia), “Operations research”.https://en.wikipedia.org/wiki/Operations_research
...
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Mathematical programming
Network optimization
Decision analysis
Queuing theory
Game theory
Industrial engineering
Logistics
Supply chain management
Financial engineering
...
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实际问题 数学模型
实际问题的解释
数学结论
简化抽象
求解
结果分析
检验
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......
“Optimization lies at the heart of machine learning.”
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Supervised learning (Regression, Classification)
Unsupervised learning (Clustering, K-means, PCA)
Deep learning
Reinforcement learning
Collaborative filtering
Nonparametric Bayesian inference
Social activity modeling for networks
......
“Optimization lies at the heart of machine learning.”
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ùpci (x)´å¼ê§EÚI©O´ªåI8ÚتåI8"
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`z¯K©a
— 8I¼êÚå¼êþ5¼ê§¯K¡55y(LinearProgramming)¶
— 8I¼êå¼ê¥k´Cþx5¼ê§¯K¡55y(Nonlinear Programming)"
— d§âûüCþ!8I¼êÚ¦ØÓ§`z©ê5y!Ä5y!ä5y!1w5y!Å5y!õ8I5y"
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`z¯K)
÷vå^x ∈ Sx¡¯K1)(feasible solution), XJ1)x∗ ∈ S?Ú÷v
f (x∗) 6 f (x),∀x ∈ S . (4)
K¡x∗¯K(1)Û`)(global optimal solution). ,§3¹1)x∗ ∈ S·U(x∗)p§¤á
f (x∗) 6 f (x), ∀x ∈ S ∩ U(x∗). (5)
d¡x∗¯K(1)ÛÜ`)(local optimal solution).
دK8I¼ê½å^UéE,§éÑÛ`)~(J§ù·8IÒ¬´¦ÑÛÜ`)"
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`5^
`5^µ¯K`)¤÷v7½ö¿©^"§`z¯K¦)O!©ÛJø7ØnØÄ:"
ÃÃÃååå¯KKK444^
7^µ8I¼êf (x)3:x?§ex ´ÛÜ4:§K∇f (x) = 0.
7^µ8I¼êf (x)3:x?g§ex´ÛÜ4:§K∇f (x) = 0, ¿HesseÝ∇2f (x) ≥ 0.
¿©^µ8I¼êf (x)3:x?g§e∇f (x) = 0∇2f (x) > 0, Kx´ÛÜ4:"
¿^µf (x)´½Â3Rnþà¼ê§Kx´N4:£Û`)¤¿^´∇f (x) = 0.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 27 / 422
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`5^
ååå¯KKK`555^
Kuhn-Tucker7^µxå¯K(3)1:§PIe = i ∈ I | ci (x) = 0, fÚci (i ∈ Ie)3:x§ci (i ∈ I \ Ie)3:xëY§ci (i ∈ E)3:xëY§þ8∇ci (x) | i ∈ E ∪ Ie5Ã'"XJx´ÛÜ`)§K3êλi ≥ 0(i ∈ Ie)Úµi (i ∈ E)¦
∇f (x)−∑i∈Ie
λi∇ci (x)−∑i∈E
µi∇ci (x) = 0. (6)
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`5^
½ÂLagrange¼êL(x, λ, µ) = f (x)−∑i∈I
λici (x)−∑i∈E
µici (x). ex¯K
ÛÜ`)§K3¦fþλ ≥ 0, µ¦∇xL(x, λ, µ) = 0.
d§7^±L
(K− T)
∇xL(x, λ, µ) = 0,ci (x) = 0, i ∈ E ,ci (x) ≥ 0, i ∈ I,λici (x) = 0, i ∈ I,λi ≥ 0, i ∈ I.
(7)
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1 XØ
2 55y
3 ä`z
4 Ä5y
5 55yÄ:nØ
6 Ãå`z
7 g5y
8 5å`z
9 o(
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 30 / 422
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55y.
min cTxs.t. Aex = be
Ax ≤ b.(8)
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~fµr
,rúi)ü«a.ÄÔrµ®GrÚâr")ùr¤Idÿð!Ú0U|¤"ÄkIòù£Ø0U¤§,ò¤k·Ü/¤r¤¬"3)óS¥§ò¤¬¤âG½®"G§l ªr¬"
磨碎 混合
筛粉
结粒
糖渣
玉米
燕麦
颗粒饲料
粉状饲料
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 32 / 422
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~fµr
r¬Ñ÷vE¤©I¦"L1Ñ«x§Ún¹þz©'§±9ª¬A¹þ¦"
Table: E¤©¹þz©'
x n
ÿð 13.6 7.1 7.0 4.1 2.4 3.70U 5.0 0.3 25.0
¹þ¦ >= 9.5 >= 2.0 <= 6.0
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 33 / 422
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~fµr
«^þk"L2Ñ«^þ±9éAd"
Table: ^þd
^þ£Z¤ d£/Z¤
ÿð 11900 1.3 23500 1.70U 750 1.2
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 34 / 422
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~fµr
L3ÑóS¤"
Table: \ó¤
·Ü (â ç®
2.5 0.5 4.2 1.7
XJzUI¦þ9ëâr§12ë®Gr§K«A©O¦^õ?1·ÜâU¦o¤$º
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 35 / 422
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.L
3ùp§ÿð!!0U¦^þûüCþ§¿©Ox1Z§x2Z§x3Z"
r`z.µ
min (1.3x1 + 1.7x2 + 1.2x3) + 2.5(x1 + x2) + 0.5(x1 + x2 + x3)s.t. 13.6x1 + 4.1x2 + 5.0x3 ≥ 9.5(x1 + x2 + x3),
7.1x1 + 2.4x2 + 0.3x3 ≥ 2.0(x1 + x2 + x3),7.0x1 + 3.7x2 + 25.0x3 ≤ 6.0(x1 + x2 + x3),x1 + x2 + x3 ≥ 9000 + 12000,0 ≤ x1 ≤ 11900,0 ≤ x2 ≤ 23500,0 ≤ x3 ≤ 750.
(9)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 36 / 422
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linprog
Solve a linear programming problem
minx
f T x
s.t. A x ≤ bAeq x = beqlb ≤ x ≤ ub
(10)
where f , x , b, beq, lb, and ub are vectors and A and Aeq are matrices.
[x,fval,exitflag,output,lambda] = linprog(f,A,b,Aeq,beq,lb,ub,x0,options)
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linprog
f = [4.3; 4.7; 1.7];A = [−4.1, 5.4, 4.5;−5.1,−0.4, 1.7; 1,−2.3, 19;−1,−1,−1];b = [0; 0; 0;−21000];lb = zeros(3, 1);ub = [11900; 23500; 750];
[x,fval,exitflag,output,lambda] = linprog(f,A,b,[],[],lb,ub);
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.(J
)ÑI¦9ëârÚ12ë®Gr§$¤150868"[(JXe¤«µ
Table: rZY
ÿð£Z¤ £Z¤ 0U£Z¤
¦^þ 11897 8678.9 424.47
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 39 / 422
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~fµ)Oy
,à|3ykè/þ«ØÓàÔ)OySü
½5©ÛµÜn©è/] §3å^9½be¼Ð¤"
Table: àÔ)'êâ
àÔ è/ ^ó ³ z |d
A 1 450 35 350 1500B 1 600 25 400 1200C 1 900 30 300 1300
o] 100 63000 3300 33000
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~fµ)Oy
©Û¿ïá.àÔA, B, C«¡È©Ox1, x2, x3ú§Ko|d´
1500x1 + 1200x2 + 1800x3
] åµè/ x1 + x2 + x3 ≤ 100,Nå 450x1 + 600x2 + 900x3 ≤ 63000,³ 35x1 + 25x2 + 30x3 ≤ 3300,z 350x1 + 400x2 + 300x3 ≤ 33000,d§¢S«¡ÈK5§=x1, x2, x3 ≥ 0.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 41 / 422
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~fµ)Oy
nþ¤ã§à)OySü¯KêÆ.£55y¤µ
max 1500x1 + 1200x2 + 1800x3
s.t. x1 + x2 + x3 ≤ 100,450x1 + 600x2 + 900x3 ≤ 63000,35x1 + 25x2 + 30x3 ≤ 3300,350x1 + 400x2 + 300x3 ≤ 33000,x1, x2, x3 ≥ 0.
(11)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 42 / 422
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55y.9ÙIOz
55y/ªµ
min(max) z = c1x1 + · · ·+ cnxns.t. a11x1 + · · ·+ a1nxn ≤ (=,≥)b1
...am1x1 + · · ·+ amnxn ≤ (=,≥)bm
(12)
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55yIO.
55y¯Ko±¤XeIO/ªµ
(LP)
minn∑
j=1cjxj
s.t.n∑
j=1aijxj = bi , i = 1, · · · ,m
xj ≥ 0, j = 1, · · · , n.
(13)
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55yIO.
½ö^ÝL«µ
(LP)min cTxs.t. Ax = b
x ≥ 0.(14)
Ù¥ÝA ∈ Rm×n, c´nþ§b´mþ"¡üX/OI§·±bb ≥ 0.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 45 / 422
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IOzÚ½
UIOzÚ½kµ
8I¼êmax f (x) −→ min−f (x)
تåªz£Ú\tµCþ½öCþ¤
gdCþKzxj = x′j − x
′′j , x
′j , x
′′j ≥ 0
Cþþe.?n
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55yÄnØ
(((ØØØ1µµµ 355y¥§å^þ5ª95ت§¤±1S´à8"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 47 / 422
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55yÄnØ
444:::µµµ S´à8§ex = λx(1) + (1− λ)x(2), λ ∈ (0, 1),x(1), x(2) ∈ S, 7kx = x(1) = x(2), K¡x´à8S4:"
(((ØØØ2µµµ éuk.4à8§?Û:ÑUL«¤4:à|Ü"
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55yÄnØ
(Ø2éÃ.8¿Ø¤á§dIÚ\4Vg"
S ⊂ Rn´4à8£UÃ.¤§d"þ"XJéS¥zxÑkx + λd | λ ≥ 0 ⊂ S, K¡þdS"
qd(1)Úd(2)´Sü§eé?Ûêλkd(1) 6= λd(2), K¡d(1)Úd(2)´üØÓ"
444µµµ eSdØUL«¤T8ÜüØÓ5|ܧK¡dS4
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55yÄnØ
LLL«««½½½nnnµµµ S = x | Ax = b, x ≥ 0õ¡8§Kk
(1) 4:8§3k4:x(1), · · · , x(k);
(2) 48Ü8¿^´Sk.¶(3) eSÃ.§K3k4d(1), · · · ,d(`);
(4) x ∈ S¿^´
x =k∑
i=1
λix(i) +
∑j=1
µjd(j)
Ù¥λi ≥ 0, i = 1, · · · , k ,k∑
i=1λi = 1, µj ≥ 0, j = 1, · · · , `.
[SK1.1µy²þãà8L«½n...]
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55yÄnØ
ÄIO/ª
(LP)min cTxs.t. Ax = b
x ≥ 0.(15)
1S = x | Ax = b, x ≥ 04:x(1), · · · , x(k), 4d(1), · · · ,d(`). âL«½n§?Û1:xL«
x =k∑
i=1
λix(i) +
∑j=1
µjd(j).
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55yÄnØ
òÙ\IO.¥µ
min cTx =k∑
i=1λi (cTx(i)) +
∑j=1
µj(cTd(j))
s.t. λi ≥ 0, i = 1, · · · , k,k∑
i=1λi = 1, µj ≥ 0, j = 1, · · · , `.
(16)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 52 / 422
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55yÄnØ
`)½
1=⇒ Ã)1k.=⇒ ) ½ö áõ)1Ã.=⇒ ) ½ö áõ) ½ö −∞
·r/)0Ú/áõ)0¡.3`)§ r/Ã.) −∞08\Ø3`)/"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 53 / 422
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55yÄnØ
(((ØØØ3µµµ IO/ª55y.1§3k.`)=¤kcTd(j) Kê£Ù¥d(j)14¤"?Úk§e3k.`)§K8I¼ê`½U3,4:£`4:¤?"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 54 / 422
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55yÄnØ
55y*ã)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 55 / 422
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55yÄnØ
55y*ã)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 56 / 422
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55yÄnØ
55y*ã)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 57 / 422
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55yÄnØ
55yIO/ª
(LP)min cTxs.t. Ax = b
x ≥ 0.(17)
bA = (B,N), Ù¥B´m_Ý£Ø5¤"ÓPx = (xTB , x
TN )T , Ù¥xB©þB¥éA§xN©þN¥
éA"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 58 / 422
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55yÄnØ
ùAx = b=¤
(B,N)
(xBxN
)= b,
=BxB + NxN = b =⇒ xB = B−1b− B−1NxN .
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 59 / 422
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55yÄnØ
ÄÄÄ)))/ÄÄÄÝÝݵµµ
3þª¥§xN©þÒ´5§|Ax = bgdCþ"AO/-xN = 0, K)
x =
(xBxN
)=
(B−1b
0
)¡§|Ä)§éAB¡ÄÝ"
xB©þ¡ÄCþ§xN©þ¡ÄCþ"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 60 / 422
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55yÄnØ
111ÄÄÄ)))/111ÄÄÄÝÝݵµµ
qeB−1b ≥ 0, K¡x =
(B−1b
0
)(LP)1Ä)§A¡B
1ÄݧxB =
xB1
...xBm
|1ÄCþ"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 61 / 422
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55yÄnØ
~fµ
min −x1 + 3x2
s.t. x1 + 2x2 ≤ 8x2 ≤ 2x1, x2 ≥ 0.
(18)
[SK1.2µÁÑþã~f¤k4:Ú1Ä)...]
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 62 / 422
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55yÄnØ
444:::888 -vs- 111ÄÄÄ)))888
55y.¥§14:1Ä)m´déA"
½½½nnnµµµ 55yIO.(LP)1S = x | Ax = b, x ≥ 0,rank(Am×n) = m, K1S4:8(LP)1Ä)8´d"
[SK1.3µy²4:8–1Ä)8d½n...]
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 63 / 422
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55yÄnØ
dààà888LLL«««½½½nnnÚ444:::–111ÄÄÄ)))ddd½½½nnn§·´µ
55y¯K3`)§8I¼ê`½U31,4:?§=(LP)3`)§K½31Ä)´`)"
ù§55y.¦)£`)¤8(¦`1Ä)"ùg´üX/ÄÑu:"1Ä)ê éõ§Ø¨qÞ"TæÛ«üѺ ù´üX/¢...
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 64 / 422
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55yÄnØ
,§·3c¡Ñààà888LLL«««½½½nnn§Ù(Ø´/4:3k50§3ùpò±,«ªJÑ"
111ÄÄÄ)))333555½½½nnnµµµ (LP)k1)§K½31Ä)"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 65 / 422
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üX/
ÄÄÄgggµµµ l1Ä)Ñu§¦e¦8I¼êk¤Uõ1Ä)¶ÏLØäSU?1Ä)åã`1Ä)"
ÌÌÌÚÚÚ½½½µµµ
`½(optimality)
=¶$(pivoting)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 66 / 422
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üX/
Äkb½é55yIO/¯K
min cTx | Ax = b, x ≥ 0
®1Äy©A = (B,N), x =
(xBxN
), c =
(cBcN
).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 67 / 422
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üX/
@odxB = B−1b − B−1NxN§8I¼êL«
z = cTB xB + cTNxN= cTBB
−1b + (cTN − cTBB−1N)xN
= z0 +∑j∈N
(cj − zj)xj ,(19)
ùp§z0 = cTBB
−1b, (20)
zj = cTBB−1aj , (21)
ajL«ÝA1jþ"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 68 / 422
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üX/
éAu,y©x =
(xBxN
), edª(21)¤û½zj÷v£`½½½^
¤cj − zj ≥ 0, ∀j ∈ N. (22)
P?¿1)x£÷vxj ≥ 0, j ∈ N¤8I¼êz , Kd(19)
z = z0 +∑j∈N
(cj − zj)xj ≥ z0 = cTBB−1b.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 69 / 422
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üX/
=d1Ä)x =
(xBxN
)=
(B−1b
0
)¤éA8I¼êz0Ø'
?Û1)8I¼êz§¤±x =
(B−1b
0
)=¯K
`)"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 70 / 422
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üX/
§eª(22)ؤá§=3,p ∈ N¦
cp − zp < 0. (23)
Ïdrxp±ÄCþ½y30§òxpd0O\∆(≥ 0),8I¼êzA/dz0~z0 + (cp − zp)∆(≤ z0).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 71 / 422
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üX/
y)15§k7?Ú©Û∆9ÄCþxBCz¹"
exp = ∆, xj = 0(j ∈ N, j 6= p), Kd5§|Ax = bxB
xB = B−1b− B−1ap∆.
Pb = B−1b, yp = B−1ap, þª¤xB1
xB2
...xBm
=
b1
b2...bm
−
y1p
y2p...
ymp
∆. (24)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 72 / 422
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üX/
ÉuKå§3ª(24)m>ØKp∆, =
∆ =bryrp
= min biyip| yip > 0(i = 1, · · · ,m) (25)
þª∆xp#§Ó¿3yip > 0cJebiyip
, ¿ò
TeIiPr . ù½e
xBi= bi − yip∆(i = 1, · · · ,m)
xp = ∆xj = 0(j ∈ N, j 6= p)
(26)
¤¯K1)§AO/¤áxBr = 0.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 73 / 422
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üX/
d§xBrÚxj(j ∈ N, j 6= p)Üå5kn −mCþ0; U´ØdxB1 , · · · , xBr−1 , xp, xBr+1 , · · · , xBm. rùmCþw¤#ÄCþ§,n −mw¤ÄCþ§Ò#/y©0"
u´§·¼#1Ä)
x = (xB1 , · · · , xBr−1 , 0, xBr+1 , · · · , xBm , 0, · · · , xp, · · · , 0)T .
[SK1.4µy²þã#x´1Ä)...]
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 74 / 422
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üX/
±þö´ÏLOÄCþxBrÄCþxp5(½#1Ä)§·¡===¶¶¶$$$(pivoting).
,§¡ùxBrÑÄCþ§xp\ÄCþ"ù#1Ä)§|^ª(22)?1`½§¿Eþãö"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 75 / 422
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üX/
üüüXXX///(SIMPLEX)
1ڣЩz¤µ(½1Äy©A = (B,N), ¿OxB = b := B−1b.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 76 / 422
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üX/
üüüXXX///(SIMPLEX)
1Ú£`½¤µOþw = (BT )−1cB , é¤kÄ©þj ∈ N¦Ñzj = wTaj .
XJcj − zj ≥ 0(j ∈ N), Kc1Ä)x =
(xBxN
)=
(b0
)®
´`)[stop!]"ÄK§À÷vcp − zp < 0p ∈ N?\eÚ"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 77 / 422
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üX/
üüüXXX///(SIMPLEX)
1nÚ£=¶$¤µOþyp = B−1ap. eyp ≤ 0, K¯KÃk.)[stop!!]"
ÄK§éÑr¦∆ = bryrp
= min biyip| yip > 0(i = 1, · · · ,m), òÄ
ÝBþaBr^apO§#ÄÝ
B = (aB1 , · · · , aBr−1 , ap, aBr+1 , · · · , aBm).
? §P#ÄCþ
xBi= bi − yip∆(i = 1, · · · ,m, i 6= r), xp = ∆.
-ÄCþeI8ÜÄCþeI8Ü©O
B := B ∪ p − Br, N := N ∪ Br − p.
£1Ú"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 78 / 422
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üX/
üüüXXX///nnnØØØÂÂÂñññ555
Xc¤ã§²L=¶$§éAucp − zp < 0ÄCþxp\Ä#1Ä)§8I¼êò~|cp − zp|∆.
3ùp§∆ := bryrp
= min biyip| yip > 0(i = 1, · · · ,m).
Ïb = B−1b ≥ 0, ¤±∆ ≥ 0.
AOb > 0§k∆ > 0§d8I¼êî~"
,§1Ä)ê´k§SجÃEe§73kgS(åO"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 79 / 422
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üX/
üüüXXX///kkkÂÂÂñññ555½½½nnnµµµ£òz/¤e¤55y¯K±¦ÑЩ1Ä)§ =¶$L§¥¤k1Ä)Ñ´òz§K|^üX/3kgS§oé`)§o£OѯKÃk.)§l (åO"
555µµµ 1Ä)x =
(xBxN
)=
(b0
)÷vb > 0§¡§òz"
§3bi = 0, ¡T1Ä)òz"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 80 / 422
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üX/
ÐÐЩ©©111ÄÄÄ)))µµµ cãüX/I1Ä)Щ)"
¯K¥ÑЩ1Ä)§·3ùp0~^µ
üã
M
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 81 / 422
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üX/
¤¢üüüãã㧴1ã¦9ϯK`)§1ã2±§¯KЩ1Ä)?1O"
3üüüããã1ãp§é55y¯K
min cTx | Ax = b, x ≥ 0,
ÄXe9ϯKmin 1Tys.t. Ax + y = b
x ≥ 0, y ≥ 0.(27)
ùp§y#Ú?m<óCþþ§1´¤k©þþ1~êþ"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 82 / 422
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üX/
39ϯK(27)¥§
(xy
)=
(0b
)´1Ä)"rT1Ä)Ð
©)§Ò^þüX/?1S"
w,§9ϯK(27)8I¼ê1Ty31p´K"eÙ
`)´
(xy
), K¤á1T y ≥ 0
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 83 / 422
![Page 93: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/93.jpg)
üX/
XJ¯K31)§K?¿1)xÚy = 0|Ü
(x0
)Ñ÷v9
ϯK(27)å^§8I¼ê1Ty = 0.
ù¯¢¿X§9ϯK`)
(xy
)¤á1T y > 0§¯K
vk1)"
§XJ9ϯK`)
(xy
)¤á1T y = 0§=y = 0. d
§xw,´¯K1)£´1Ä)¤"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 84 / 422
![Page 94: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/94.jpg)
üX/
k¦ÑЩ1Ä)2?11ã`)OÜ¿¤Ú§¡M"
MÄXe55y¯K
min cTx + M · 1Tys.t. Ax + y = b
x ≥ 0, y ≥ 0.(28)
ùpM¿©~ê§<óCþyÚ~êþ1üã¥"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 85 / 422
![Page 95: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/95.jpg)
üX/
3M¯K(28)¥§8I¼êM · 1TyÓ|/ §¤±üX/SòÄk4<óCþyi~"Ù(J´§e¤k<óCþѤ0, K8I¼êcTx + M · 1Ty Ú¯K8I¼êcTxþ§@3düX/S¤?1Ò´¦¯K`)O"
¯¢þ§XJ¯Kk`)§3M¿©§M¯K(28)
`)
(xy
)¥y = 0§x ¤¯K`)"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 86 / 422
![Page 96: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/96.jpg)
üX/
^üX/¦)Xe¯Kµ
min −4x1 − x2
s.t. −x1 + 2x2 ≤ 42x1 + 3x2 ≤ 12x1 − x2 ≤ 3x1, x2 ≥ 0.
(29)
[SK1.5µkÑüX/OÚ½§,¦)þã55y¯K...]
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 87 / 422
![Page 97: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/97.jpg)
üX/
üX/L£üX/O(¤ï
min fs.t. f − cTx = 0
Ax = bx ≥ 0.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 88 / 422
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üX/
üX/L£üX/O(¤ï
min fs.t. xB +B−1NxN = B−1b
f +0xB +(cTBB−1N − cTN )xN = cTBB
−1bxB ≥ 0, xN ≥ 0.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 89 / 422
![Page 99: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/99.jpg)
üX/
Table: üX/L
f xB xN RHS
0 Im B−1N B−1b
1 0 cTBB−1N − cTN cTBB
−1b
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 90 / 422
![Page 100: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/100.jpg)
üX/
Table: üX/L
xB1 · · · xBr · · · xBm · · · xj · · · xp · · ·
xB1
...xBr
...xBm
1 · · · 0 · · · 0...
......
0 · · · 1 · · · 0...
......
0 · · · 0 · · · 1
· · · y1j · · · y1p · · ·...
...· · · yrj · · · yrp · · ·
......
· · · ymj · · · ymp · · ·
b1...br...bm
0 · · · 0 · · · 0 · · · zj − cj · · · zp − cp · · · cTB b
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 91 / 422
![Page 101: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/101.jpg)
üX/
^M¦)Xe¯Kµ
min x1 + x2 − 3x3
s.t. x1 − 2x2 + x3 ≤ 112x1 + x2 − 4x3 ≥ 3x1 − 2x3 = 1x1, x2, x3 ≥ 0.
(30)
[±üX/L/ª?1O...]
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 92 / 422
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üX/
IOz.£M¤µ
min x1 + x2 − 3x3 + M(x6 + x7)s.t. x1 − 2x2 + x3 + x4 = 11
2x1 + x2 − 4x3 − x5 + x6 = 3x1 − 2x3 + x7 = 1x1, x2, x3, x4, x5, x6, x7 ≥ 0.
(31)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 93 / 422
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üX/
Table: SüX/L-1
x1 x2 x3 x4 x5 x6 x7
x4 1 -2 1 1 0 0 0 11x6 2 1 -4 0 -1 1 0 3x7 1 0 -2 0 0 0 1 1
3M − 1 M − 1 −6M + 3 0 −M 0 0 4M
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 94 / 422
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üX/
Table: SüX/L-2
x1 x2 x3 x4 x5 x6 x7
x4 0 -2 3 1 0 0 -1 10x6 0 1 0 0 -1 1 -2 1x1 1 0 -2 0 0 0 1 1
0 M − 1 1 0 −M 0 1− 3M M + 1
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 95 / 422
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üX/
Table: SüX/L-3
x1 x2 x3 x4 x5 x6 x7
x4 0 0 3 1 -2 2 -5 12x2 0 1 0 0 -1 1 -2 1x1 1 0 -2 0 0 0 1 1
0 0 1 0 -1 1−M −1−M 2
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 96 / 422
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üX/
Table: SüX/L-4 (opt)
x1 x2 x3 x4 x5 x6 x7
x3 0 0 1 1/3 -2/3 2/3 -5/3 4x2 0 1 0 0 -1 1 -2 1x1 1 0 0 2/3 -4/3 4/3 -7/3 9
0 0 0 -1/3 -1/3 1/3−M 2/3−M −2
5µüX/¤kOÑéAuüX/L£Ý¤$£=1ÐC¤"
[SK1.6µy²31ÌcüØÓ1ÄeOXêÚ8I¼êkXe'X]
(zj − cj)new = (zj − cj)−
yrjyrp
(zp − cp);
(cTBB−1b)new = cTBB
−1b− bryrp
(zp − cp).(32)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 97 / 422
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éónØ
é?¿55y.§±½Â¡éó¯K(Dual Problem) ."
éééóóónnnØØØò«¯Kéó¯KmS3éX§?Ú\ïÄ55y¦)JønØâ"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 98 / 422
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éónØ
¯Kµ
(LP)min cTxs.t. Ax ≥ b
x ≥ 0.(33)
éó¯Kµ
(DP)max bTws.t. ATw ≤ c
w ≥ 0.(34)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 99 / 422
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éónØ
éó'X£ãµ
(1) 8I¼ê4(min) ←→ 4(max)
(2) ¯K8I¼êXêþ¤,¯Kå^m>~êþ
(3) å^ÝA ←→ AT , å^ØÒ
(4) –éó¯KCþÑkKå
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 100 / 422
![Page 110: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/110.jpg)
éónØ
éó¯K£d/ª¤µ
(DP)min (−b)Tws.t. (−A)Tw ≥ −c
w ≥ 0.(35)
éó¯Kéóµ
(DDP)max (−c)Txs.t. (−A)x ≤ −b
x ≥ 0.(36)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 101 / 422
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éónØ
IO.éóµ
min cTxs.t. Ax = b
x ≥ 0.⇒
min cTxs.t. Ax ≥ b
−Ax ≥ −bx ≥ 0.
(37)
max bTws.t. ATw ≤ c.
⇐max bTw
′ − bTw′′
s.t. ATw′ − ATw
′′ ≤ c
w′ ≥ 0,w
′′ ≥ 0.
(38)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 102 / 422
![Page 112: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/112.jpg)
éónØ
ªéóµ
min cTxs.t. AGx ≥ bG
AEx = bE
ALx ≤ bL
x ≥ 0.
(39)
max bTG wG + bT
E wE + bTL wL
s.t. ATG wG + AT
E wE + ATL wL ≤ c
wG ≥ 0, wE free, wL ≤ 0.(40)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 103 / 422
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éónØ
¯Kéó¯KmkXSS3'X§PxÚw©O(LP)Ú(DP)?¿1)"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 104 / 422
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éónØ
ffféééóóó½½½nnnµµµ ?¿55y¯K(LP)9Ùéó¯K(DP)m¤á±e'X
(a) x´(LP)¯K1)§w´(DP)¯K1)§⇒ cTx ≥ bTw.
(b) x´(LP)¯K1)⇒ cTx ≥ ¯K(DP)"w´(DP)¯K1)⇒ bTw ≤ ¯K(LP)"
(c) x´(LP)¯K1)§w´(DP)¯K1)§cTx = bTw⇒ x´(LP)¯K`)§w´(DP)¯K`)"
(d) ¯K(LP)Ã. ⇒ ¯K(DP)Ã1)"¯K(DP)Ã. ⇒ ¯K(LP)Ã1)"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 105 / 422
![Page 115: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/115.jpg)
éónØ
`)))333555½½½nnnµµµ e(LP)(DP)Ñk1)§K§þ3`)"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 106 / 422
![Page 116: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/116.jpg)
éónØ
éééóóó½½½nnnµµµ ¯K(LP)Úéó¯K(DP)k`)§K,k`)§dü`"
[SK1.7µy²þãéó½n...]
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 107 / 422
![Page 117: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/117.jpg)
éónØ
pppÖÖÖtttµµµ½½½nnnµµµ xÚw©O´(LP)Ú(DP)1)§@oxÚw´éA¯K`)¿^´µ
xT (AT w − c) = 0wT (Ax− b) = 0
(41)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 108 / 422
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éónØ
55yKuhn-Tucker^µ
(K− T)
Ax− b ≥ 0x ≥ 0ATw − c ≤ 0w ≥ 0xT (ATw − c) = 0wT (Ax− b) = 0
(42)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 109 / 422
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éónØ
|^pÖtµ½n§¯K`)§¦ÑÙéó¯K`)"
max w1 + 2w2
s.t. 3w1 + w2 ≤ 2−w1 + 2w2 ≤ 3w1 − 3w2 ≤ 1w1 ≥ 0,w2 ≥ 0.
`)w =
(w1
w2
)=
(1/7
11/7
).
min 2x1 + 3x2 + x3
s.t. 3x1 − x2 + x3 ≥ 1x1 + 2x2 − 3x3 ≥ 2x1 ≥ 0, x2 ≥ 0, x3 ≥ 0.
`)x =
x1
x2
x3
=
4/75/7
0
.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 110 / 422
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éónØ
|^pÖtµ½n§¯K`)§¦ÑÙéó¯K`)"
max w1 + 2w2
s.t. 3w1 + w2 ≤ 2−w1 + 2w2 ≤ 3w1 − 3w2 ≤ 1w1 ≥ 0,w2 ≥ 0.
`)w =
(w1
w2
)=
(1/7
11/7
).
min 2x1 + 3x2 + x3
s.t. 3x1 − x2 + x3 ≥ 1x1 + 2x2 − 3x3 ≥ 2x1 ≥ 0, x2 ≥ 0, x3 ≥ 0.
`)x =
x1
x2
x3
=
4/75/7
0
.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 110 / 422
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éónØ
|^pÖtµ½n§¯K`)§¦ÑÙéó¯K`)"
max w1 + 2w2
s.t. 3w1 + w2 ≤ 2−w1 + 2w2 ≤ 3w1 − 3w2 ≤ 1w1 ≥ 0,w2 ≥ 0.
`)w =
(w1
w2
)=
(1/7
11/7
).
min 2x1 + 3x2 + x3
s.t. 3x1 − x2 + x3 ≥ 1x1 + 2x2 − 3x3 ≥ 2x1 ≥ 0, x2 ≥ 0, x3 ≥ 0.
`)x =
x1
x2
x3
=
4/75/7
0
.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 110 / 422
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(¯Ý©Û
¢S¯K¥§éõÿ´Äu,æ8êâ5û½.Xê"3ù«¹e§³7¬ÑyXê6Ä9ÚåCz"
¤¢(¯Ý©ÛÒ´|^)¯K(J§ïÄXêkCz`)A"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 111 / 422
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(¯Ý©Û
üå§Ä¯Kmin cTxs.t. Ax = b
x ≥ 0(43)
å^m>~êþCz
b + ∆b = (b1 + ∆b1, · · · , bm + ∆bm)T ,
#55y¯K
min cTxs.t. Ax = b + ∆b
x ≥ 0.(44)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 112 / 422
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(¯Ý©Û
ybé¯K(43)|^üX/®Ù`
)x∗ =
(x∗Bx∗N
)=
(B−1b
0
).
ùéu`ÄB, â15^k
B−1b ≥ 0, (45)
â`5^kcTN − cTBB
−1N ≥ 0. (46)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 113 / 422
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(¯Ý©Û
bCb + ∆b§XJ¤á
B−1(b + ∆b) ≥ 0, (47)
K¯K(43)`ÄB´¯K(44)`Ä"
d¯K(44)`)¤x =
(xBxN
)=
(B−1(b + ∆b)
0
).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 114 / 422
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(¯Ý©Û
?Ú§P¯K(43)8I¼ê`z(b), ¯K(44)8I¼ê`z(b + ∆b), Kk
z(b) = cTBB−1b
z(b + ∆b) = cTBB−1(b + ∆b)
(48)
¤áz(b + ∆b)− z(b) = cTBB
−1∆b. (49)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 115 / 422
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(¯Ý©Û
¯¢þ§(BT )−1cB´éAuÄÝBüX/¦fþw. âéónا§=¯K(43)éó¯K`)"
Ïd§Pw∗ = (BT )−1cB , Kþª¤
z(b + ∆b)− z(b) = (w∗)T∆b =m∑i=1
w∗i ∆bi . (50)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 116 / 422
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(¯Ý©Û
Ä∆bi → 04§Ò±
∂z(b)
∂bi= w∗i , (i = 1, · · · ,m). (51)
ùp§¢SþÑéó¯K`)²L¿Â£Kfd¤
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 117 / 422
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55yõªm
lOþÝ5w§üX/Ø´õªm"'uüX/nØ9¢SþOÇnã©zµ
R. Shamir. The efficiency of the simplex method: A survey. ManagementScience, Vol. 33, 1987: 301-334.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 118 / 422
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55yõªm
é55y¯Kõªmµ
ý¥§k3nØþ²(55y¯Kð3õªmS)§Ï §´¿Â"L.G. Khachiyan. A polynomial algorithm in linear programming.Soviet Mathematics Doklady, Vol. 20, 1979: 191-194.
S:§OüX/k Ú\58§Ó§í2¿A^Øu55y¯K"N. Karmarkar. A new polynomial time algorithm for linearprogramming. Combinatorica, Vol. 4, 1984: 373-395.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 119 / 422
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Ù£55y¤
Exercise 1.1: y²55y1S = x | Ax = b, x ≥ 0à8L«½n"
Exercise 1.2: Ñ~f(18)¤k4:Ú1Ä)"
Exercise 1.3: y²55y.Ä1)8Ù14:8´d"
Exercise 1.4: y²üX/=¶$
x = (xB1 , · · · , xBr−1 , 0, xBr+1 , · · · , xBm , 0, · · · , xp, · · · , 0)T
´#1Ä)"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 120 / 422
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Ù£55y¤
Exercise 1.5: kÑüX/OÚ½§,¦)55y~f(29)"
Exercise 1.6: y²31ÌcüØÓ1ÄeOXêÚ8I¼ê÷v(32).
Exercise 1.7: y²55y¯Kéó½n§=¯K(LP)Úéó¯K(DP)k`)§K,k`)dü`"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 121 / 422
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1 XØ
2 55y
3 ä`z
4 Ä5y
5 55yÄ:nØ
6 Ãå`z
7 g5y
8 5å`z
9 o(
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 122 / 422
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ä`z0
$ѯK§á´¯K§6¯K§¤6¯Kk'ä`zÏ~A^u)û¢S¯K§¿äk²L¿Â"
ÒêÆ. 󧧴55yAA~"é55y.®kõªm§Ïä.AÏêÆ(§|^Ù(A5±OÑÇp¦)"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 123 / 422
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$Ñ.
$Ñ.´aAÏ55y§=lÀ /£X)[¤C$ÀÔ8/£X²û󥤧Ù8I´(½$ÑL¦o$Ѥ¿÷vøAÚI¦^"
$Ñ.*ÐA^uÙ¦+§)Ý]§óNݧ<"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 124 / 422
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$Ñ.
²;$ѯKµ
óiÀÔþsi , i = 1, 2, · · · ,m.
I¦:jI¦þdj , j = 1, 2, · · · , n.
lóiI¦:jü À$¤^cij9ÙuÀþxij .
ÀU¦$Ño¤^´»5y"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 125 / 422
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$Ñ.
…
S_2
S_1
S_m
…D_2
D_1
D_n
1111 : xc
mnmn xc :
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 126 / 422
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$Ñ.
$ѯKêÆ£ã
minm∑i=1
n∑j=1
cijxij
s.t.n∑
j=1xij ≤ si , i = 1, · · · ,m
m∑i=1
xij ≥ dj , j = 1, · · · , n
xij ≥ 0, i = 1, · · · ,m j = 1, · · · , n.
(52)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 127 / 422
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$Ñ.
Table: Mileage Chart
Denver Miami
Los Angeles 1000 2690Detroit 1250 1350New Orleans 1275 850
Table: Transportation Cost per Car
Denver(j = 1) Miami(j = 2)
Los Angeles(i = 1) $80 $215Detroit(i = 2) $100 $108New Orleans(i = 3) $102 $68
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 128 / 422
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$Ñ.
Table: MG Auto Transportation Model
Denver Miami SupplyLos Angeles 80 215
x11 x12 1000Detroit 100 108
x21 x22 1500New Orleans 102 68
x31 x32 1200Demand 2300 1400
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 129 / 422
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$Ñ.
min z = 80x11 + 215x12 + 100x21 + 108x22 + 102x31 + 68x32
s.t. x11 + x12 ≤ 1000 (LosAngeles)x21 + x22 ≤ 1500 (Detroit)x31 + x32 ≤ 1200 (NewOreleans)x11 + x21 + x31 ≥ 2300 (Denver)x12 + x22 + x32 ≥ 1400 (Miami)xij ≥ 0, i = 1, 2, 3, j = 1, 2
(53)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 130 / 422
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$Ñ.
IJï/
minm∑i=1
n∑j=1
cijxij
s.t.n∑
j=1xij = si , i = 1, · · · ,m
m∑i=1
xij = dj , j = 1, · · · , n
xij ≥ 0, i = 1, · · · ,m j = 1, · · · , n.
(54)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 131 / 422
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$Ñ.
$ѯKéó
maxm∑i=1
siui +n∑
j=1djvj
s.t. ui + vj ≤ cij , i = 1, · · · ,m, j = 1, · · · , nui , vj Ã
(55)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 132 / 422
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$Ñ.
|^éó'Xµ
éÄCþxij ó§ui + vj = cij , =σij = ui + vj − cij = 0.
éÄCþxij ó§eσij = ui + vj − cij ≤ 0, ®éó1¶eσij = ui + vj − cij > 0, éó1§KÚ?Ä"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 133 / 422
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$$$ÑÑÑ...üüüXXX///¦¦¦)))ÚÚÚ½½½µµµ
1. À|m + n − 1´»§Ð©1Ä)"
2. uc)´ÄU?§XJU?§KÚ?ÄCþ?1Ú3§ÄKÊ"
3. rÚ2¥]ÀCþÚ?§(½=´»AdÄ)¥òÑ"
4. N٦Ĵ»6þ£÷v15¤§£Ú2.
·ò±~f5`²$Ñ.üX/"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 134 / 422
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Table: ,úi$ÑL
D(1) D(2) D(3) D(4) SupplyS(1) 10 2 20 11
x11 x12 x13 x14 15S(2) 12 7 9 20
x21 x22 x23 x24 25S(3) 4 14 16 18
x31 x32 x33 x34 10Demand 5 15 15 15
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 135 / 422
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Table: Щ1Ä)(Northwest-Corner Starting Solution)
D(1) D(2) D(3) D(4) SupplyS(1) 10 2 20 11
5 10 15S(2) 12 7 9 20
5 15 5 25S(3) 4 14 16 18
10 10Demand 5 15 15 15
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 136 / 422
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Table: S-1
v1 = 10 v2 = 2 v3 = 4 v4 = 15 Supply
10 2 20 11
u1 ≡ 0 5 10 15[-16] [4]
12 7 9 20
u2 = 5 5 15 5 25[3]
4 14 16 18
u3 = 3 10 10[9] [-9] [-9]
Demand 5 15 15 15
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 137 / 422
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Table: S-1 ££´¤
v1 = 10 v2 = 2 v3 = 4 v4 = 15 Supply
10 2 20 11
u1 ≡ 0 5− θ 10 + θ 15[-16] [4]
12 7 9 20
u2 = 5 5− θ 15 5 + θ 25[3]
4 14 16 18
u3 = 3 θ 10− θ 10[9] [-9] [-9]
Demand 5 15 15 15
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 138 / 422
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$Ñ.
Table: S-2
v1 = 1 v2 = 2 v3 = 4 v4 = 15 Supply
10 2 20 11
u1 ≡ 0 15 15[-9] [-16] [4]
12 7 9 20
u2 = 5 0 15 10 25[-6]
4 14 16 18
u3 = 3 5 5 10[-9] [-9]
Demand 5 15 15 15
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 139 / 422
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$Ñ.
Table: S-2££´¤
v1 = 1 v2 = 2 v3 = 4 v4 = 15 Supply
10 2 20 11
u1 ≡ 0 15− θ θ 15[-9] [-16] [4]
12 7 9 20
u2 = 5 0 + θ 15 10− θ 25[-6]
4 14 16 18
u3 = 3 5 5 10[-9] [-9]
Demand 5 15 15 15
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 140 / 422
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$Ñ.
Table: S-3 £`)¤
v1 = −3 v2 = 2 v3 = 4 v4 = 11 Supply
10 2 20 11
u1 ≡ 0 5 10 15[-13] [-16]
12 7 9 20
u2 = 5 10 15 [-4] 25[-10]
4 14 16 18
u3 = 7 5 5 10[-5] [-5]
Demand 5 15 15 15
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 141 / 422
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ä.
G = (V ,E )kã§Ù¥V´!:8ܧE´>8Ü"
kr,!:Щ:s, ,ª:t AÏé"
>e ∈ EþDk¤c(e)±9Nþu(e), Ñ¢ê"
d§|¤|E |Aþ©OP
c = (c(e)|e ∈ E ), u = (u(e)|e ∈ E ).
Ú)ùN = (G , s, t, c,u)¡ä"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 142 / 422
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á´»¯K
á´»¯K¥§¤c(e))º>eÝ"á´»¯K;./ª´µ
3äN = (G , s, c)¥¦Ñ©:sÙ§:v ∈ Vá´»9ÙÝ"
,§½Âw ∈ Vv ∈ V´»(path)
π = vi0(= w), vi1 , · · · , vik (= v)
Ý
c(π) =k−1∑j=0
c(vij , vij+1).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 143 / 422
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á´»¯K
N¥vkÝK£´§3kls¤k:vá´»§§^±sááá´äää5L«"
¦svá´»§÷Xsvá´ä>reÒ1"
s 2 4 2
3
2
1
-1 1 1
1 1 2
1 2 0
-1
2
2
1
3
-2
2
1
3
2 1 0
1 -1 2
2 2 -1
XÛ¦Tá´äºá´»¯K.º
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 144 / 422
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á´»¯K
á´»¯KêÆ£ãµ
min∑
(i ,j)∈Ecijxij
s.t.∑
(s,j)∈Exsj = 1∑
(k,j)∈Exkj −
∑(i ,k)∈E
xik = 0, ∀k ∈ V − s, t
−∑
(i ,t)∈Exit = −1
xij ≥ 0,∀(i , j) ∈ E
(56)
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á´»¯K
á´»¯Kéóµ
max (ys − yt)s.t. yi − yj ≤ cij , ∀(i , j) ∈ E
(57)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 146 / 422
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á´»¯K
äá´»¯K±w¤´55y.£¯¢þ´AÏ$Ñ.¤§âéó5EÙ¦)"
XJÄuÄ5yg§Ñá´»¯Krõªm"±e=`²äkL5µ
Dijkstra’s algorithm.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 147 / 422
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á´»¯K
DIJKSTRA
Ñ\ kãG = (V ,E ), >Ýc : E → R+, ©:s ∈ V .
ÑÑ l©:¤k!:v ∈ Vá´»9ÙÝc∗(v).
Ú Ð©zµ-d(s) := 0, d(v) :=∞(v ∈ V − s), ±9P := ∅.Ú SµÀ÷vd(v∗) = mind(v) | v ∈ V − P!
:v∗ ∈ V − P.
Ún #µc∗(v∗) := d(v∗), P := P ∪ v∗.?Úéw ∈ V − P>e = (v∗,w) ∈ EXe#µ
d(w) := mind(w), d(v∗) + c(e).
Úo (å½µXJP = VK(åO¶ÄK£Ú"
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á´»¯K
OL§¥¦Ñd(v)ðlsvá´»Ýc∗(v)þ.§=d(v) ≥ c∗(v).
3®1!:8ÜP ⊂ Vp§äk5d(v) = c∗(v), v ∈ P. u´§P = V=¦Ñs¤k!:á´»9Ùݧ(åO"
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á´»¯K~
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graphshortestpath
[dist, path, pred] = graphshortestpath(G,S)
This procedure determines the single-source shortest paths from node S toall other nodes in the graph represented by matrix G.
Input G is an N-by-N sparse matrix that represents a graph. Nonzeroentries in matrix G represent the weights of the edges.
Output dist is the vector of N distances from the source to every node(using Infs for nonreachable nodes and 0 for the source node).
Output path contains the winning paths to every node.
Output pred contains the predecessor nodes of the winning paths.
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graphshortestpath
Create and view a directed graph with 6 nodes and 11 edges.
W = [.41 .99 .51 .32 .15 .45 .38 .32 .36 .29 .21];
DG = sparse([6 1 2 2 3 4 4 5 5 6 1],[2 6 3 5 4 1 6 3 4 3 5],W);
h = view(biograph(DG,[],’ShowWeights’,’on’))
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graphshortestpath
Find the shortest path in the graph from node 1 to node 6.
[dist,path,pred] = graphshortestpath(DG,1,6)
set(h.Nodes(path),’Color’,[1 0.5 0.5])
edges = getedgesbynodeid(h,get(h.Nodes(path),’ID’));
set(edges,’LineColor’,[1 0 0]); set(edges,’LineWidth’,2)
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á´»¯K¢~
á´»¯KA^¢~µ#
Table: Equipment replacement
Replacement costfor given years in operation
Equipment acquiredat start of year +1 +2 +3
1 4000 5400 98002 4300 6200 87003 4800 7100 –4 4900 – –
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 154 / 422
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á´»¯K¢~
á´»¯KA^¢~µ#
543214000 4300 4800 4900
8700
6200
7100
9800
5400
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 155 / 422
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6¯K
äN = (G , s, t,u)p§¤¢l :s®:t6(flow) ´¢êþx = (x(e) | e ∈ E ), ÷vXe666þþþÅÅÅððð^µ∑
e∈Out(v)
x(e)−∑
e∈In(v)
x(e) = 0, v ∈ V − s, t (58)
ùpOut(v)ÚIn(v)©OL«G¥6ÑÚ6\v!:>8Ü"
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6¯K
lsX6Ñþx(s) ,
∑e∈Out(s)
x(e)−∑
e∈In(s)
x(e)
tX6\þ
x(t) ,∑
e∈In(t)
x(e)−∑
e∈Out(t)
x(e)
§=¤áx(s) = x(t). (59)
ùx(s)¡x6"
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6¯K
6x?Ú÷vNNNþþþååå (capacity constraint)^
0 ≤ x(e) ≤ u(e), e ∈ E (60)
K¡§111666 (feasible flow)"
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6¯K
¤¢6¯K§Ò´¦Ñ¦616¯K§êÆþ£ãXeµ
max x(s)s.t.
∑e∈Out(v)
x(e)−∑
e∈In(v)
x(e) = 0, v ∈ V − s, t
x(s) =∑
e∈Out(s)
x(e)−∑
e∈In(s)
x(e)
0 ≤ x(e) ≤ u(e), e ∈ E .
(61)
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6¯K
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 160 / 422
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¯K
r6äº:8y©¤ü8ÜSÚT , ¦ :s ∈ S®:t ∈ T . ©(S ,T )Nþ
U(S ,T ) =∑
(i , j) ∈ Ei ∈ S , j ∈ T
u(i , j).
¤¢§Ò´¤k©¥Nþ"
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6½n
l*þw§8(S ,T )´l :s®:t7²´§XJT´×lK6lsÃt"u´·±e¡½n"
666½½½nnnµµµ ?¿6ä6þuTäNþ"
5µ½ny²±ëãØ¡©z]"
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6½n
l*þw§8(S ,T )´l :s®:t7²´§XJT´×lK6lsÃt"u´·±e¡½n"
666½½½nnnµµµ ?¿6ä6þuTäNþ"
5µ½ny²±ëãØ¡©z]"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 162 / 422
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6½n
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 163 / 422
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6¯K
äN = (G , s, t,u)¥½16x§ÏLEäääÚ666***¿¿¿´5S#6"
lN>e = (v ,w) ∈ EÑu§Uì±e5K)¤>(v ,w)½(w , v)£â¹ö=Ù¤§>8ÜPEx, Ó(½ÙNþu.
Rule 1 XJu(e)− x(e) > 0, K)¤(v ,w) ∈ Ex, ¿-ÙNþu(v ,w) = u(e)− x(e).
Rule 2 XJx(e) > 0, K)¤(w , v) ∈ Ex, ¿-ÙNþu(w , v) = x(e).
¤kãGx = (V ,Ex) ÚNþuÜå5B´äNx = (Gx, s, t, u). @o§Nx¥lst´¡6*¿´"
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6¯K
ÚÚÚnnn £6*¿´¤½äN = (G , s, t,u), Ù16x6¿^´§äNx¥Ø3k6*¿´"
XJ3k6*¿´§Kx?66x′.
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6¯K
b½3Nx¥3lst´π, K|^
∆ := minu(e) | e ∈ π, (62)
é>e = (v ,w) ∈ EXe?
x′(e) =
x(e) + ∆, (v ,w) ∈ π(dRule 1 )¤>)x(e)−∆, (w , v) ∈ π(dRule 2 )¤>)x(e), Ù§/
(63)
´yx′´N16§Ù6´
x′(s) = x(s) + ∆.
[göSK]
yþãx′´N16§¿Ù6x
′(s) = x(s) + ∆.
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6¯K
MAX-FLOW
Ñ\ äN = (G , s, t,u), Ù¥kãG = (V ,E ).
ÑÑ lst6fmax.
Ú Ð©zµ-x(e) := 0(e ∈ E )±9fmax := 0.
Ú äµEc16xäNx = (Gx, s, t, u).
Ún 6*¿µXJNx¥vk6*¿´K(åO"§XJ3k6*¿´§KÀÙπ, â(62)ªÚ(63)ª?x6x
′.
-fmax := fmax + ∆, x := x′£Ú"
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graphmaxflow
[MaxFlow, FlowMatrix, Cut] =
graphmaxflow(G, SNode, TNode)
This procedure calculates the maximum flow of directed graph G fromnode SNode to node TNode.
Input G is an N-by-N sparse matrix that represents a directed graph.Nonzero entries in matrix G represent the capacities of the edges.
Output MaxFlow is the maximum flow, and FlowMatrix is a sparse matrixwith all the flow values for every edge. FlowMatrix(X,Y) is the flow fromnode X to node Y.
Output Cut is a logical row vector indicating the nodes connected toSNode after calculating the minimum cut between SNode and TNode. Ifseveral solutions to the minimum cut problem exist, then Cut is a matrix.
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graphmaxflow
Create a directed graph with six nodes and eight edges.
cm = sparse([1 1 2 2 3 3 4 5],[2 3 4 5 4 5 6 6],...
[2 3 3 1 1 1 2 3],6,6);
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graphmaxflow
View the graph with the original capacities.
h = view(biograph(cm,[],’ShowWeights’,’on’))
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graphmaxflow
Calculate the maximum flow in the graph from node 1 to node 6.
[M,F,K] = graphmaxflow(cm,1,6);
Notice that K is a two-row matrix because there are two possible solutionsto the minimum cut problem.
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graphmaxflow
View the graph with the calculated maximum flows.
g = biograph(F,[],’ShowWeights’,’on’); view(g)
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graphmaxflow
Show one solution to the minimum cut problem in the graph.
set(g.Nodes(K(1,:)),’Color’,[1 0 0]); view(g)
Notice that in the three edges that connect the source nodes (red) to thedestination nodes (yellow), the original capacities and the calculatedmaximum flows are the same.
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graphmaxflow
Show another solution to the minimum cut problem in the graph.
set(g.Nodes(K(2,:)),’Color’,[0 1 0]); view(g)
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¤6¯K
ÄäN = (G , s, t, c,u)þ6x, Ù¥6½
x(s) = f ∗.
36x(s) = f ∗ØLä6^e§¦Ñ¦¤6Ò´¤¢¤6¯K"
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¤6¯K
¤6¯KêÆ.µ
min∑e∈E
c(e)x(e)
s.t.∑
e∈Out(v)
x(e)−∑
e∈In(v)
x(e) = 0, v ∈ V − s, t∑e∈Out(s)
x(e)−∑
e∈In(s)
x(e) = f ∗
0 ≤ x(e) ≤ u(e), e ∈ E .
(64)
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¤6¯K
3¤6¯K¥§±kõ :Ú®:"XJ¤6¯Käkk :si , i = 1, · · · , k±9g6Ñþx(si ) = f ∗(si ), kk ′®:tj , j = 1, · · · , k ′±9g6Ñþx(tj) = f ∗(tj), K±Ú\å :sÚå®:t, ¿V\(k + k ′)^>
(s, si ) : c(s, si ) = 0, u(s, si ) = f ∗(si ), i = 1, · · · , k ;
(tj , t) : c(tj , t) = 0, u(tj , t) = f ∗(tj), j = 1, · · · , k ′.
…ks
…
'kt
1t
ts
1s
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¤6¯K
¤6¯KäkéÐ.zUåµ
ÏLÚ\åÚEA>£9ÙþzD¤§òá´»¯KÚ6¯K=z¦)AϤ6¯K"
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¤6¯K
¤6¯KäkéÐ.zUåµ
£¤ÏLÚ\åª:t ′, \\lv ∈ VÑu>(v , t ′)÷v
c(v , t ′) = 0, u(v , t ′) = 1.
@o§á´»¯K¤¦)lst ′äk6f ∗ = n¤6¯K"ùp§n = |V |, ¿b½ä>Nþ∞"
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¤6¯K
¤6¯KäkéÐ.zUåµ
£¤ÏLÚ\å©:s ′, ¿EXeü^>µ
(s ′, s); c(s ′, s) = 0, u(s ′, s) =∞
(s ′, t); c(s ′, t) = 1, u(s ′, t) =∞
@o§6¯K¤¦)ls ′t¤6¯K"ùp§½äpc(e) = 0(e ∈ E ), ¿6·þ.£'X
∑e∈E u(e)¤
ls ′Ñu6f ∗.
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¤Ì6¯K
XØÄA½!: :½ö®:§¿3>e ∈ E þÚ\6e.l = (l(e) | e ∈ E ), l äN = (G , c, l,u).
¦3ùäþ¦¤6§=¤¢¤Ì6¯Kµ
min∑e∈E
c(e)x(e)
s.t.∑
e∈Out(v)
x(e)−∑
e∈In(v)
x(e) = 0, v ∈ V
l(e) ≤ x(e) ≤ u(e), e ∈ E .
(65)
[göSK]
¤Ì6¯K.zUå¤6¯Kd"
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¤Ì6¯K
¤Ì6¯K (65)'u¤6¯K§Ùå^\'§Ï äkߦ)£ã`:"
[?§öS]
Nï¤Ì6¯KïÄ?Ч¿ÀJ¢yÙ¦)"
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Ù£ä`z¤
Exercise 2.1: âéónاÑá´»¯K«¦)"
Exercise 2.2: y²6½nµ?¿6ä6þuTäNþ"
Exercise 2.3: y(63)ªÑx′´äN16§¿Ù6
x′(s) = x(s) + ∆.
Exercise 2.4: ÏLE`²¤6¯KÙAϹ¹µá´¯KÚ6¯K"
Exercise 2.5: Áy²¤Ì6¯K¤6¯Käkd.zUå"
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Ù£ä`z¤
Project 2.6: Nï¿¢y¤Ì6¯Krõªm¦)"¦J§S èÚ§Sw"
§SwJj£)Øu¤µ
¯K£ã
n
§SH
§SÿÁ
(Ø
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1 XØ
2 55y
3 ä`z
4 Ä5y
5 55yÄ:nØ
6 Ãå`z
7 g5y
8 5å`z
9 o(
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Ä5y
Ä5yïÄ´/ûüL§`z0§äk2A^µ§¿®ïáînØÄ:"
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Ä5y
á´»¯K~f
1 3
2
4
5
6
7
7
8
5
12
8
9
7
13
9
6
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Ä5y
©ãÄ
1 3
2
4
5
6
7
7
8
5
12
8
9
7
13
9
6
3
2
4
5
6
0
f_0
7
f_1
12
f_2
17
21
f_3
8
5
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Ä5y
48§(Recursive Equation)
fi (xi ) = min
(xi−1,xi )∈Ed(xi−1, xi ) + fi−1(xi−1), i = 1, 2, 3,
f0(x0) = 0.
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Ä5y
Ä5yA'µ
Stages
Alternatives
States
Recursive Equations
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Ä5y
/ÀÔC1(Knapsack/Cargo-Loading)
A general (n-Item, W -LB) knapsack problem can be represented by thefollowing Integer Linear Programming:
max z =n∑
i=1rimi
s.t.n∑
i=1wimi ≤W
m1, · · · ,mn ∈ Z+ ∪ 0
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Ä5y
1. Stage i is represented by item i , i = 1, 2, · · · , n.
2. The alternatives at stage i are represented by mi , the number of unitsof item i included in the knapsack. It follows thatmi = 0, 1, · · · , bWwi
c.3. The state at stage i is represented by xi , the total weight assigned to
stages (items) i , i + 1, · · · , and n. This definition reflects the fact thatthe weight constraint is the only restriction that links all n stagestogether.
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Ä5y
48(backward recursion)
fi (xi ) = max
mi = 0, 1, · · · , bWwic
xi ≤ W
rimi + fi+1(xi+1), i = n, · · · , 1
fn+1(xn+1) = 0.
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Ä5y
48(backward recursion)
fi (xi ) = max
mi = 0, 1, · · · , bWwic
xi ≤ W
rimi + fi+1(xi − wimi ), i = n, · · · , 1
fn+1(xn+1) = 0.
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Ä5y
~fµ7-ton Vessel
Item i 1 2 3
wi 2 3 1
ri 31 47 15
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Ä5y
#.(Equipment Replacement Model)
b·Ä´ªÝnc#¯K"zcзIû½´Ä3c2¦^c½ö#"-r(t), c(t),s(t)©OL«t-c#cEÂ\§c$E¤9Ùí",§35yÏS?Ûc#¤´I .
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Ä5y
#.
Stage i is represented by year i where i = 1, · · · , n.
The alternatives at stage i are either keeping or replacing the machineat the start of year i .
The state at stage i is the age of the machine at the start of year i .
Recursive equation:
fi (t) = max
r(t)− c(t) + fi+1(t + 1), if Keepingr(0)− c(0) + s(t)− I + fi+1(1), if Replacing
fn+1(t) ≡ 0
where fi (t) is defined as the maximum net income for yearsi , i + 1, · · · , n by given a t-year-old machine at the start of year i .
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Ä5y
#.µ,úiyk3c#§I½54c(n = 5)#`üÑ"Túi5½6c#7L±"#¤´$100,000. eLÑ´#¯K'êâ§Ù¥t´Åìc#§r(t), c(t), s(t) ©OL«tc#ÅìcEÂ\§c$E¤9í"
Table: #¯KêâL
t r(t) c(t) s(t)
0 20,000 200 –1 19,000 600 80,0002 18,500 1,200 60,0003 17,200 1,500 50,0004 15,500 1,700 30,0005 14,000 1,800 10,0006 12,200 2,200 5,000
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 198 / 422
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Ä5y
#.
3
1
2
3
4
5
6
1 1 1
2 2
3
4
end
1 2 3 4 5Decision Year
1
2
6
4
5
Mac
hine
Age K
R
S
S
S
S
R
R
R
K
K
K
K
K
R RR
R R
R
K K
K
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 199 / 422
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Ù£Ä5y¤
Exercise 3.1: yk2c#§,5½5c#7L"35yÏ#¤©O´
(p1, p2, p3, p4, p5) = (100, 105, 110, 115, 120).
ÁïXe#Ä5y.¿¦Ù`#üÑ"
Table: ÊcÏ#
c#t ívt $1¤^ct0 - 301 50 402 25 503 10 754 5 905 2 -
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 200 / 422
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Ù£Ä5y¤
Exercise 3.2: Nï©ÛÄ5y¯K¦)9ÙmE,5"
Project 3.3[ÀÀÀ]: ¢yÄ5y«¯¿J§S èÚ§Sw"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 201 / 422
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1 XØ
2 55y
3 ä`z
4 Ä5y
5 55yÄ:nØ
6 Ãå`z
7 g5y
8 5å`z
9 o(
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 202 / 422
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55y
min f (x)s.t. x ∈ S ⊂ Rn.
(66)
3d§8I¼êf´½Â3Rnþ¢¼ê§S´ûüCþx8ܧ¡¯K1(feasible region).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 203 / 422
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55y
`z¯Klá5þ±©üaµa´äkëYCþ¯K§,a´lÑCþ¯K£=|Ü`z¯K¤"
55yáuëY.`z¯KƧÏ~1Sd|§5£ã§=
S = x ∈ Rn | gi (x) ≥ 0, i = 1, · · · ,m; hj(x) = 0, j = 1, · · · , `.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 204 / 422
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55y
min f (x)s.t. gi (x) ≥ 0, i = 1, · · · ,m
hj(x) = 0, j = 1, · · · , `(67)
ùp§f (x), gi (x), hj(x)Ñ´nCþ!¢!(½¼ê§k´5"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 205 / 422
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55y
¦)55y¯K£=éÑÙ`)¤§d'ïÄ©ü¡µ´ïÄ`)5§´Ok5¼¯K)"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 206 / 422
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£`)½Â
÷vå^x ∈ Sx¡¯K1)(feasible solution), XJ1)x∗ ∈ S?Ú÷v
f (x∗) 6 f (x),∀x ∈ S . (68)
K¡x∗¯K(66)Û`)(global optimal solution). ,§3¹1)x∗ ∈ S·U(x∗)p§¤á
f (x∗) 6 f (x), ∀x ∈ S ∩ U(x∗). (69)
d¡x∗¯K(66)ÛÜ`)(local optimal solution).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 207 / 422
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`5^
`5^µ¯K`)¤÷v7½ö¿©^"
`5^ò«¦)O!©ÛJø7ØnØÄ:"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 208 / 422
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Ãå¯K4^
7^µ8I¼êf (x)3:x?§ex ´ÛÜ4:§K∇f (x) = 0.
7^µ8I¼êf (x)3:x?g§ex´ÛÜ4:§K∇f (x) = 0, ¿HesseÝ∇2f (x) ≥ 0.
¿©^µ8I¼êf (x)3:x?g§e∇f (x) = 0∇2f (x) > 0, Kx´ÛÜ4:"
¿^µf (x)´½Â3Rnþà¼ê§Kx´N4:£Û`)¤¿^´∇f (x) = 0.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 209 / 422
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å¯K`5^
111
x ∈ S , d ∈ Rn´"þ"e3δ > 0¦µ
x + λd ∈ S ,∀λ ∈ (0, δ) (70)
K¡d´S3x?1"
PS3x?¤k18ÜF (x, S).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 210 / 422
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å¯K`5^
eeeüüü
f (x)´Rnþ¢¼ê§x ∈ Rn§d´"þ"e3δ > 0¦µ
f (x + λd) < f (x),∀λ ∈ (0, δ) (71)
K¡d¼êf (x)3x?eü"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 211 / 422
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å¯K`5^
eeeüüü888fff888
XJf (x)´¼ê§∇f (x)Td < 0. w,§d?d f (x) 3x ?eü"Pù8Ü
D(x, f ) = d | ∇f (x)Td < 0.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 212 / 422
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å¯K`5^
(((ØØØ
éu¯Kminf (x) | x ∈ S, x ∈ S , f (x)3x?"XJx´¯KÛÜ`)§K
F (x, S) ∩ D(x, f ) = ∅. (72)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 213 / 422
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å¯K`5^
PI(x) = i ∈ 1, · · · ,m | gi (x) = 0,
Df = D(x, f ) = d | ∇f (x)Td < 0, (73)
Fg = F (x, g) = d | ∇gi (x)Td > 0, i ∈ I(x), (74)
Fh = F (x, h) = d | ∇hj(x)Td = 0, j = 1, · · · , `. (75)
x¯K(67)ÛÜ`)§fÚgi , i ∈ I(x)3:x§gi , i /∈ I(x)3:xëY§hj3:xëY§∇hj(x)`j=1
5Ã'§KDf ∩ Fg ∩ Fh = ∅. (76)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 214 / 422
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å¯K`5^
Fritz-John^µ3¯K(67)¥§x1:§fÚgi , i ∈ I(x)3:x§gi , i /∈ I(x)3:xëY§hj3:xëY"XJx´ÛÜ`)§K3Ø"êλ0, λi , i ∈ I(x) Úµj , j = 1, · · · , ` ¦
λ0∇f (x)−∑
i∈I(x)
λi∇gi (x)−∑j=1
µj∇hj(x) = 0. (77)
Ù¥λ0 ≥ 0, λi ≥ 0, i ∈ I(x).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 215 / 422
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å¯K`5^
y²(1)µXJ∇hj(x)`j=15'§K3Ø"
êµj , j = 1, · · · , ` ¦ ∑j=1
µj∇hj(x) = 0.
ù-λ0 = 0, λi = 0, i ∈ I(x), (ؤá"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 216 / 422
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å¯K`5^
y²(2)µXJ∇hj(x)`j=15Ã'§K7kDf ∩ Fg ∩ Fh = ∅.=ت|
∇f (x)Td < 0∇gi (x)Td > 0, i ∈ I(x)∇hj(x)Td = 0, j = 1, · · · , `
(78)
Ã)"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 217 / 422
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å¯K`5^
y²(2)Yµ-A´±∇f (x),−∇gi (x), i ∈ I(x) |¤Ý§B´±−∇hj(x), j = 1, · · · , `|¤Ý"u´
ATd < 0BTd = 0
(79)
Ã)"
ey Ap1 + Bp2 = 0p1 ≥ 0
(80)
k)"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 218 / 422
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å¯K`5^
y²(2)Yµy½Â
S1 = ( y1
y2) | y1 = ATd, y2 = BTd,d ∈ Rn,
S2 = ( y1
y2) | y1 < 0, y2 = 0.
w,S1ÚS2à8§S1 ∩ S2 = ∅.
dà8©l½n§é∀d ∈ Rn, ∀(y1
y2) ∈ CL(S2), 3"þ(
p1
p2)
¦pT1 ATd + pT
2 BTd ≥ pT1 y1 + pT
2 y2.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 219 / 422
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å¯K`5^
y²(2)Yµ
Äk-y2 = 0, dd?¿5£d = 0¤9y1 < 0, =⇒ p1 ≥ 0.
2-(y1
y2) = (
00
) ∈ CL(S2), =⇒ pT1 ATd + pT
2 BTd ≥ 0.
d = −(Ap1 + Bp2), =⇒ Ap1 + Bp2 = 0.
nþ¤ã§=(80)k)"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 220 / 422
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å¯K`5^
y²(2)Yµ
rp1©þPλ0Úλi , i ∈ I(x), p2©þPµj , j = 1, · · · , `. á=
λ0∇f (x)−∑i∈I (x)
λi∇gi (x)−∑j=1
µj∇hj(x) = 0. (81)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 221 / 422
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å¯K`5^
Kuhn-Tucker^µxå¯K(67)1:§fÚgi , i ∈ I(x) 3:x§gi , i /∈ I(x) 3:xëY§hj3:xëY§þ8∇gi (x), i ∈ I(x);∇hj(x), j = 1, · · · , ` 5Ã'"XJx´ÛÜ`)§K3êλi ≥ 0 Úµj ¦
∇f (x)−∑
i∈I(x)
λi∇gi (x)−∑j=1
µj∇hj(x) = 0. (82)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 222 / 422
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å¯K`5^
½ÂLagrange¼êL(x, λ, µ) = f (x)−m∑i=1
λigi (x)−∑j=1
µjhj(x).
ex¯KÛÜ`)§K3¦fþλ ≥ 0, µ¦
∇xL(x, λ, µ) = 0.
d§7^L
(K− T)
∇xL(x, λ, µ) = 0gi (x) ≥ 0, i = 1, · · · ,mλigi (x) = 0, i = 1, · · · ,mλi ≥ 0, i = 1, · · · ,mhj(x) = 0, j = 1, · · · , `
(83)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 223 / 422
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eü
nØþ±^`5^¦/55y¯K0`)§3¢¥¿Ø¢1"
3¦)`z¯K~^O´/Seü0"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 224 / 422
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N
NµA´½Â3mXþ:::888N§=éz:x(k) ∈ X, ²A^):8A(x(k)) ⊂ X, ?¿ÀJ:x(k+1) ∈ A(x(k))x(k) Y:"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 225 / 422
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4N
Ú\¤¢45§Ù¢´::NëY5í2"
XÚY©O´mEpÚEq¥48§A : X −→ Y:8N"XJx(k) ∈ X, x(k) → x, y(k) ∈ A(x(k)), y(k) → y %ºXy ∈ A(x), K¡NA3x ∈ X?´4"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 226 / 422
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)8Ü
ïÄÂñ5§Äk²()8ÜVg"
3Nõ¹e§¦):ÂñuÛ`)´4(J"Ïd§r÷v,^:8½Â)8Ü"S:áuù8ܧÒÊS"
~X§3Ãå`z¯K¥§±½Â)8Ü
Ω = x | ‖∇f (x)‖ = 0,
3å`z¯K¥§)8Ü
Ω = x | x´K-T:.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 227 / 422
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eü¼ê
/§eüo´,¼ê3SL§¥¼ê~éX3å§ÏdIÑeü¼êVg"
Ω ⊂ X)8ܧAXþN§ψ(x)´½Â3XþëY¢¼ê§e÷v
x /∈ Ωy ∈ A(x)§ψ(y) < ψ(x)
x ∈ Ωy ∈ A(x)§ψ(y) ≤ ψ(x)
K¡ψ´'u)8ÜΩÚAeü¼ê"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 228 / 422
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Âñ5
Ω)8ܧAXþN"e±?¿Ð©:x(0) ∈ Y ⊂ X Ñu§)S?Âñf4áu)8ܧK¡NA3Y þÂñu)8ÜΩ.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 229 / 422
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Âñ5
½½½nnnµµµ AXþ§Ω)8ܧ½Ð©:x(0) ∈ X, ?1XeSµXJx(k) ∈ Ω, KÊS¶ÄKx(k+1) ∈ A(x(k)),k := k + 1, E±þL§"ù)SSx(k). qµ
1 Sx(k)¹uX;f8¥¶
2 3ëY¼êψ, §´'uΩÚAeü¼ê¶3 NA3ΩÖ8þ´4"
KSx(k)?Âñf4áuΩ.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 230 / 422
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Âñ5½ny²
yyy²²²µµµ kySx(k)éAeü¼êêψ(x(k))k4"
x(k)¹uX;f8§ÏdkÂñfx(k)K , Ù4x ∈ X.dψëY5§ψ(x(k)) −→ ψ(x), k ∈ K . =é∀ε > 0,∃N¦k ≥ N§k0 < ψ(x(k))− ψ(x) < ε, k ∈ K . AO/§0 < ψ(x(N))− ψ(x) < ε.
qdψeü5§ψ(x(k))− ψ(x(N)) < 0,∀k > N. u´k
0 < ψ(x(k))− ψ(x) = ψ(x(k))− ψ(x(N)) + ψ(x(N))− ψ(x) < ε,∀k > N.
= limk→∞
ψ(x(k)) = ψ(x).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 231 / 422
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Âñ5½ny²
yyy²²²£££UUU¤¤¤µµµ eyx ∈ Ω£y¤"
bx /∈ Ω, ÄSx(k+1)K . du§¹u;8§¤±3Âñfx(k+1)K , K ⊂ K , Ù4x ∈ X. w,£Óncãy²¤ lim
k→∞ψ(x(k+1)) = ψ(x). dψ(x(k))45§
ψ(x) = ψ(x).
,§ék ∈ K ⊂ Kk
x(k) −→ x, x(k+1) ∈ A(x(k)), x(k+1) −→ x.
A3ΩÖ8þ´4§x /∈ Ω, ÏdA3x?´4§=kx ∈ A(x).
duψ´)8ÜΩÚAeü¼ê§x /∈ Ω, Kkψ(x) < ψ(x). ùw,gñ§¤±x ∈ Ω.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 232 / 422
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¢^ÂñOK
3Seüp§x(k) ∈ ΩâªS"3¢¥Nõ¹e§ù´4L§§IÃgS"Ïd)û¢S¯K§I5½¢^ªSL§OK§¡ÂñOK½ÊÅOK"
~^ÂñOKk±eA«µ
1 ‖x(k+1) − x(k)‖ < ε or‖x(k+1) − x(k)‖‖x(k)‖
< ε
2 f (x(k))− f (x(k+1)) < ε orf (x(k))− f (x(k+1))
|f (x(k))|< ε
3 ‖∇f (x(k))‖ < ε
3ùp§ε¯k½¿©ê"Ød§±âÂñ½n§½ÑÙ§ÂñOK"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 233 / 422
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喂
µd`IO´Âñ¯ú§Ï~¡ÂñÇ"
½ÂXeµSx(k)Âñx∗, ÷v
0 ≤ ¯limk→∞
‖x(k+1) − x∗‖‖x(k) − x∗‖p
= β <∞ (84)
Kêpþ(.¡Sx(k)Âñ"
e3½Âª(84)¥§p = 1β < 1, K¡S´£Âñ'β¤5Âñ"
e3½Âª(84)¥§p > 1, ½öp = 1, β = 0, K¡S´5Âñ"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 234 / 422
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OE,5
`z±J©P4¯K§, §¤ÕáÆ´3Voc"§DantzigJѦ)55y¯KüX/"d«`z¯KnØ9A^ïÄ×uЧAO´55yduÙ.ÊH5Ú¢^5§'?ÐÚå2À"
X¢S¯K55±93OÅEâíÄe§<m©lE,5ÝïÄ55yÚ55y"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 235 / 422
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S
`zÏ~æ^S¦¯K`)§ÙÄg´µ
½Ð©:x(0) ∈ Rn, Uì,S5K):x(k), ¦x(k)´k¡:§Ù:´`z.¯K`)§x(k)´Ã¡:§§k4:Ù4:´`z.¯K`)"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 236 / 422
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S
ÐSAä;.A´µ
S:x(k)U½/CÛÜ4:x∗§,×Âñux∗. /§éu,«·Iy²ÙS:x(k)à:£=f4:¤ÛÜ4:"3¢SO¥§½ÂñOK÷v§S=ª"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 237 / 422
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|¢ÚÏf
x(k)1kgS:§d(k)1kg|¢§αk1kgÚÏf§K1k + 1gSµ
x(k+1) = x(k) + αkd(k). (85)
lþãSª±wѧØÓ|¢ÚØÓÚüѤØÓ"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 238 / 422
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|¢ÚÏf
3`z¥§|¢d(k)À´,d¼ê(meritfunction) ψ 3x(k)?eü§=d(k)÷v
∇ψ(x(k))Td(k) < 0. (86)
ÚÏf(½8()`z¯K
minα≥0
ψ(x(k) + αd(k)) (87)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 239 / 422
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`zSÄ(
(a) ½Ð©:x(0)
(b) O|¢d(k), =E,d¼êψ3x(k):?eü|¢¶
(c) (½ÚÏfαk , ¦Td¼êk,«§Ýeü¶
(d) S#§-x(k+1) = x(k) + αkd(k).
(e) äÊÅOK§ex(k+1)÷v,«ª^§KÊS§Cq`)x = x(k+1). ÄK§£(b)E±þÚ½"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 240 / 422
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1 XØ
2 55y
3 ä`z
4 Ä5y
5 55yÄ:nØ
6 Ãå`z
7 g5y
8 5å`z
9 o(
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 241 / 422
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Ãå`z
Ãå`z¯Kminx∈Rn
f (x) (88)
Ù8I¼êf´½Â3Rnþ¢¼ê§ûüCþx8Ü´mRn.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 242 / 422
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FÝa¦)
FÝþ∇f (x)´¼êf3:x?O\¯§§¤`zóä"¢SþéÃå`z¯K§[¤¦)¥õáue¡FÝa"
GRADIENT £££FFFÝÝÝaaa¤¤¤
(0) ЩzµÀ·Ð©:x(0) ∈ Rn, -k := 0.
(1) O|¢µ|^·½é¡HkO|¢þd(k) := −Hk∇f (x(k)). £XJ∇f (x(k)) = 0, K(åO¤
(2) (½ÚÏfµ)`z¯Kminα≥0
f (x(k) + αd(k)), ¦ÑÚ
α = αk , -x(k+1) = x(k) + αkd(k), k := k + 1, £1(1)Ú"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 243 / 422
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E|¢µKFÝ
Ãå`z¯Kµminx∈Rn
f (x)
f (x) = f (x(k)) +∇f (x(k))T (x− x(k)) + O(‖x− x(k)‖2) (89)
KFÝd(k) = −∇f (x(k)),
Kαkv§oU¦
f (x(k) + αkd(k)) < f (x(k)).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 244 / 422
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E|¢µÚî
f (x) = f (x(k)) +∇f (x(k))T (x− x(k))
+1
2(x− x(k))T∇2f (x(k))(x− x(k)) + O(‖x− x(k)‖3)
(90)
|¢d(k) = −G−1
k ∇f (x(k)),
Ù¥Gk = ∇2f (x(k))¼êf3x(k):?HesseÝ"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 245 / 422
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(½ÚÏfµ|¢
3Sª¥§ÏL)`z¯K
minα≥0
ϕ(α) = f (x(k) + αd(k)) (91)
(½ÚÏf¡|||¢¢¢(Line Search).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 246 / 422
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(½ÚÏfµ|¢
e±¯K(91)`)Ú§d¡°°°(((|||¢¢¢(Exact LineSearch).
²~^°(|¢k7©ÚS"=¦`´°(|¢§ÏLkgO¦Ñ¯K(91)î)´ØU§¢Sþ3kv°ÝCq)§Òæ^§Ú"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 247 / 422
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(½ÚÏfµ|¢
3¢SO¥§ Ø´¦)`z¯K(91), ´éÑ÷v,·^oÑCq)Ú§d¡°°°(((|||¢¢¢(InexactLine Search).
°(|¢'§3éõ¹eæ^°(|¢±JpNOÇ"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 248 / 422
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(½ÚÏfµ|¢
αk´¦f (x(k) + αd(k)) = f (x(k))
êα.
u´§·ò3«m[0, αk ]S¦÷v·^ÉÚÏf§=α ∈ [0, αk ].
ϕ
α α
)0(ϕ
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 249 / 422
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(½ÚÏfµ|¢
Goldstein(1965) conditions:
ϕ(α) ≤ ϕ(0) + ραϕ′(0) (92)
ϕ(α) ≥ ϕ(0) + (1− ρ)αϕ′(0) (93)
Ù¥ρ ∈ (0, 1/2)´½ëê"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 250 / 422
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(½ÚÏfµ|¢
Goldstein(1965) conditions:
ϕ
)0()0( 'ραϕϕ +
α
)0(ϕ
)0()1()0( 'αϕρϕ −+
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 251 / 422
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(½ÚÏfµ|¢
Wolfe(1968)-Powell(1976) conditions:
ϕ(α) ≤ ϕ(0) + ραϕ′(0) (94)
ϕ′(α) ≥ σϕ′(0) (95)
Ù¥σ ∈ (ρ, 1)´,½ëê"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 252 / 422
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(½ÚÏfµ|¢
Wolfe(1968)-Powell(1976) conditions:
ϕ
)0()0( 'ραϕϕ +
α
)0(ϕ
)0(:slope 'σϕ
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 253 / 422
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(½ÚÏfµ|¢
3éõ¢S¥§ª(95)~rzV>^¤
|ϕ′(α)| ≤ −σϕ′(0) (96)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 254 / 422
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(½ÚÏfµ|¢
ÄuWolfe-PowellOK°(|¢µ
(0) ½Ð©|¢«m[0, α], ±9ρ ∈ (0, 1/2), σ ∈ (ρ, 1).Oϕ0 = ϕ(0) = f (x(k)), ϕ
′0 = ϕ
′(0) = ∇f (x(k))Td(k).
¿-a1 = 0, a2 = α, ϕ1 = ϕ0, ϕ′1 = ϕ
′0.
À·α ∈ (a1, a2).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 255 / 422
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(½ÚÏfµ|¢
ÄuWolfe-PowellOK°(|¢µ
(1) Oϕ = ϕ(α) = f (x(k) + αd(k)). eϕ(α) ≤ ϕ(0) + ραϕ′(0), K=
1(2)Ú"ÄK§dϕ1, ϕ′1, ϕEü:gõªp(1)(t), ¿
Ù4:
α = a1 +1
2
(a1 − α)2ϕ′1
(ϕ1 − ϕ)− (a1 − α)ϕ′1
.
u´a2 = α, α = α, E1(1)Ú"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 256 / 422
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(½ÚÏfµ|¢
ÄuWolfe-PowellOK°(|¢µ
(2) Oϕ′
= ϕ′(α) = ∇f (x(k) + αd(k))Td(k). eϕ
′(α) ≥ σϕ′(0), KÑ
Ñαk = α, ¿Ê|¢"ÄK§dϕ,ϕ′, ϕ′1Eü:gõ
ªp(2)(t), ¿Ù4:
α = α− (a1 − α)ϕ′
ϕ′1 − ϕ
′ .
u´a1 = α, α = α, ϕ1 = ϕ,ϕ′1 = ϕ
′, £1(1)Ú"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 257 / 422
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(½ÚÏfµ|¢
[gKµÑþãÄuWolfe-PowellOK°(|¢¥õªp(1)(t), p(2)(t)äNLª"]
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 258 / 422
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ÛÂñ
l?¿Ð©:Ñu§XJ,S):4£à:¤§3·b½eyð¯K`)£½ö½:¤§K¡TSäkÛÂñ5(Global Convergence).
dé§XJ=3)NCÀЩ:§â±y¤)¤:ÂñuT)§K¡ùSkÛÜÂñ5(Local Convergence).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 259 / 422
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ÛÂñ
y²Seü5§·A¦þ;|¢KFÝA/§=¦d(k) lg(k) = ∇f (x(k))"Ä
K§g(k)Td(k)Cu"§d(k)AØ´eü"
d§·bd(k)−g(k)Yθk÷v
θk ≤π
2− µ, ∀k (97)
Ù¥µ > 0£kÃ'¤"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 260 / 422
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ÛÂñ
w,θk ∈ [0, π/2), Ù½Â
cos θk =−g(k)Td(k)
‖g(k)‖‖d(k)‖=−g(k)T s(k)
‖g(k)‖‖s(k)‖(98)
ùps(k) = αkd(k) = x(k+1) − x(k).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 261 / 422
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ÛÂñ
e¡Ñ«ÚOKeeüÛÂñ5(Ø"
ÛÛÛÂÂÂñññ555½½½nnnµµµ
∇f (x)3Y²8L(x(0)) = x | f (x) ≤ f (x(0)) þ3ëY"eü|¢d(k)−∇f (x(k)) mYθk÷vª(97), Ù¥Úαkdn«(½µ
(1) °(|¢
(2) GoldsteinOK(92),(93)
(3) Wolfe-PowellOK(94),(95)
@o§½öé,kk∇f (x(k)) = 0, ½öf (x(k))→ −∞,½ö∇f (x(k))→ 0.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 262 / 422
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ÛÂñ
ÛÛÛÂÂÂñññ555yyy²²²µµµ£y²Wolfe-PowellOK/¤
bé¤kk, g(k) = ∇f (x(k)) 6= 0 Úf (x(k))ke.§
f (x(k))− f (x(k+1))→ 0. dª(94)§−g(k)T s(k) → 0.
£y¤eg(k) → 0ؤá§@o3ε > 0Úfx(k)k∈K¦‖g(k)‖ ≥ ε. l d
−g(k)T s(k) = ‖g(k)‖‖s(k)‖ cos θk ≥ ε‖s(k)‖ sinµ
±9ª(97)k‖s(k)‖ → 0.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 263 / 422
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ÛÂñ
ÛÛÛÂÂÂñññ555yyy²²²£££YYY¤¤¤µµµ
qÏg(x) = ∇f (x)3L(x(0))þëY§¤±
g(k+1)T s(k) = g(k)T s(k) + o(‖s(k)‖)⇓
g(k+1)T s(k)
g(k)T s(k)→ 1.
(99)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 264 / 422
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ÛÂñ
ÛÛÛÂÂÂñññ555yyy²²²£££YYY¤¤¤µµµ
ùWolfe-PowellOKª(95)
g(k+1)T s(k)
g(k)T s(k)≤ σ < 1 (100)
gñ"Ïdkg(k) → 0
[gKµÖ¿y²ÄuGoldsteinOK°(|¢ÛÂñ5"]
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 265 / 422
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eü
eüKFÝS|¢§ÙSª
x(k+1) = x(k) − αk∇f (k(k)).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 266 / 422
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eü
µµµ
(0) ÀЩ:x(0), ªØε > 0, -k := 0.
(1) Og(k) = ∇f (x(k)). e‖g(k)‖ < ε, KÊS¿ÑÑx(k).ÄK?11(2)Ú"
(2) -d(k) = −g(k), ¿d|¢(½ÚÏfαk¦
f (x(k) + αkd(k)) = minα>0
f (x(k) + αd(k)).
(3) S#x(k+1) = x(k) + αkd(k), k := k + 1, £1(1)Ú"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 267 / 422
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eü
eeeüüüÛÛÛÂÂÂñññ555½½½nnnµµµ
f (x) ∈ C 1, 3eü¥æ^£°(½°(¤|¢§K)S:x(k)zà:Ñ´7:"
Project: ?§¢yÄuWolfe-PowellOK°(|¢"¦J§S èÚ§Sw"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 268 / 422
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eü
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 269 / 422
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eü
/§eük5ÂñÝ"
Xe~f´~ͶÿÁ¼ê(Rosenbrock function)
f (x1, x2) = 100(x2 − x21 )2 + (1− x1)2.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 270 / 422
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Úî
f (x)´g¢¼ê§3x(k)NCTaylorÐmCq
f (x(k) + s) ≈ q(k)(s) = f (x(k)) + g(k)T s +1
2sTGks (101)
Ù¥g(k) = ∇f (x(k)),Gk = ∇2f (x(k)).
òq(k)(s)4zBs = −G−1
k g(k). (102)
þªÑ|¢−G−1k g(k)¡Úî(Newton Direction).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 271 / 422
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Úî
38I¼ê´½g¼ê
f (x) =1
2xTGx− cTx
¹e(G½)§é?¿xk∇2f (x) = G .
31gSp-H0 = G−1, Kk
d(0) = −H0∇f (x(0)) = −G−1(Gx(0) − c) = −(x(0) − x∗).
ùp§x∗ = G−1c´¯K`)"ex(0) 6= x∗, Úα0 = 1, u´x(1) = x(0) + α0d(0) = x∗. dd§Ø+Щ:x(0)XÛ§3gS=`)x∗.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 272 / 422
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Úî
â±þ¯¢§±@=¦éu5¼êf (x), 3S¥-|¢
d(k) = −∇2f (x(k))−1∇f (x(k))
´Ü·"
AO/§Úαk ≡ 1Súª
x(k+1) = x(k) + d(k) = x(k) − G−1k g(k). (103)
ùÒ´²;ÚîS
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 273 / 422
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Úî
éu½g¼ê ó§ÚîÚ=`)"éug¼ê§Úî¿ØUy²kgS¦`)"du8I¼ê34:NC^g¼êÐ/Cq§Ð©:C4:§ÚîÂñݬé¯"
±y²ÚîÛÜÂñ5ÚÂñÇ"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 274 / 422
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Úî
ÚÚÚîîîÂÂÂñññ½½½nnnµµµ
f ∈ C 2, x(k)¿©Cx∗, Ù¥∇f (x∗) = 0. XJ∇2f (x∗)½§8I¼êHesseÝG (x)÷vLipschitz^§=3β > 0¦é¤k(i , j)k
|Gij(x)− Gij(y)| ≤ β‖x− y‖. (104)
Kék , ÚîS(103)k½Â§¤Sx(k)Âñx∗, äkÂñÇ"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 275 / 422
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Úî
yyy²²²µµµ
Pg(x) = ∇f (x), Ïf ∈ C 2, ·k
g(x− h) = g(x)− G (x)h + O(‖h‖2).
-x = x(k),h = h(k) = x(k) − x∗ \þª
0 = g(x∗) = g(x(k) − h(k)) = g(x(k))− G (x(k))h(k) + O(‖h(k)‖2). (105)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 276 / 422
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Úî
yyy²²²£££YYY¤¤¤µµµ
duG (x)÷vLipschitz^§´y[G (x(k))
]−1k."§(105)ü>Ó
¦±[G (x(k))
]−1
0 =[G (x(k))
]−1g(x(k))− h(k) + O(‖h(k)‖2)
= x∗ − (x(k) −[G (x(k))
]−1g(x(k))) + O(‖h(k)‖2)
= x∗ − x(k+1) + O(‖h(k)‖2)
= −h(k+1) + O(‖h(k)‖2)
¤±‖h(k+1)‖ = O(‖h(k)‖2), =ÚîSäkÂñÇ"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 277 / 422
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Úî
yyy²²²µµµ
éuÚîS§·P
x(k+1) = x(k) − G−1k g(k) , A(x(k)). (106)
5¿g(x∗) = 0, G (x∗)½£Ûɤ§kA(x∗) = x∗.
u´dx(k+1) − x∗ = A(x(k))−A(x∗)
‖x(k+1) − x∗‖ = ‖A(x(k))−A(x∗)‖≤ ‖A′(x∗)(x(k) − x∗)‖+
1
2‖A′′(x)‖‖x(k) − x∗‖2,
Ù¥x ux(k)Úx∗mãþ"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 278 / 422
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Úî
yyy²²²£££YYY¤¤¤µµµ
w,A′(x) = [x− G (x)−1g(x)]
′= −[G (x)−1]
′g(x)
¤±A′(x∗) = 0. l k
‖h(k+1)‖ = ‖x(k+1) − x∗‖ ≤ γ‖x(k) − x∗‖2 = γ‖h(k)‖2
Ù¥~êγ=6uf (x)3x∗NCnê"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 279 / 422
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Úî
3ª(103)ÚîSp§XJÀЩ:x(0)Ø3)x∗NC§@o)¤:x(k)7Âñu`)"
yÛÂñ5§k7éÚî,U?"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 280 / 422
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Úî
'X§3Úî¥æ^|¢5(½Ú"
ZZZÚÚÚîîîµµµ
(0) ÀЩ:x(0), ªØε > 0, -k := 0.
(1) Og(k) = ∇f (x(k)). e‖g(k)‖ < ε, ÊS¿ÑÑx(k).ÄK?11(2)Ú"
(2) )5§|Gkd = −g(k), ¦ÑÚîd(k).
(3) æ^|¢(½ÚÏfαk , -x(k+1) = x(k) + αkd(k),k := k + 1, £1(1)Ú"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 281 / 422
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Úî
Úî¡Ì(J´HesseÝGk = ∇2f (x(k))ؽ"ùCq.ؽk4:§=g¼êq(k)(s)´Ã."
Ñù(J§<JÑéõ?"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 282 / 422
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Úî
Goldstein & Price (1967)
d(k) =
−G−1
k g(k), if cos θk > η
−g(k), otherwise
(107)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 283 / 422
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Úî
Levenberg(1944), Marquardt(1963), Goldfeld et. al(1966)
(Gk + µk I )d(k) = −g(k) (108)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 284 / 422
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Úî
x´¼êfؽ:§ed÷v
dT∇2f (x)d < 0,
K¡df3x?KÇ"
HesseÝ∇2f (x(k))ؽ§KÇ´?Úî,«å»"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 285 / 422
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[Úî
ÚîâÑ`:´ÛÜÂñ鯣äkÂñǤ§$^ÚîIO§ 8I¼êHesseÝ∇2f (x(k)) U½§$ÛÉ"Ñù":§<JÑ[Úî"ÙÄg´µ^عêÝHkCqÚî¥HesseÝ_G (x(k))−1.
dECqÝØÓ§òÑyØÓ[Úî"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 286 / 422
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[Úî
£ÚîS Gkd = −g(k)
x(k+1) = x(k) + αkd(k)
EHesseÝ_G−1k CqHk , ·k©Û∇2f (x(k))
∇f (x(k))'X"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 287 / 422
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[Úî
1kgSx(k+1), ò8I¼êf (x)3x(k+1)?TaylorÐmµ
f (x) ≈ f (x(k+1)) +∇f (x(k+1))T (x− x(k+1))
+1
2(x− x(k+1))T∇2f (x(k+1))(x− x(k+1)),
?Úk
∇f (x) ≈ ∇f (x(k+1)) +∇2f (x(k+1))(x− x(k+1)),
u´-x = x(k)
∇f (x(k)) ≈ ∇f (x(k+1)) +∇2f (x(k+1))(x(k) − x(k+1)).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 288 / 422
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[Úî
Ps(k) = x(k+1) − x(k), y(k) = ∇f (x(k+1))−∇f (x(k)), Kk
∇2f (x(k+1))s(k) ≈ y(k) or ∇2f (x(k+1))−1y(k) ≈ s(k).
ù§OÑs(k)Úy(k)§þªO3x(k+1)?HesseÝ_"·knd¦3S¥EÑHesseÝ_CqHk+1, ¦Ù÷v
Hk+1y(k) = s(k). (109)
Ï~rª(109)¡^§¡[Úî^"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 289 / 422
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[Úî
[[[ÚÚÚîîîSSSªªªµµµ
(0) ÀЩ:x(0), -H0 = I , k := 0.
(1) O|¢d(k) = −Hk∇f (x(k)).
(2) æ^|¢(½ÚÏfαk , -x(k+1) = x(k) + αkd(k).
(3) Äux(k)x(k+1)FÝCz§#HesseÝ_Cq§=(½÷v^Hk+1. k := k + 1, £1(1)Ú"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 290 / 422
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[Úî
e¡·Ò5?ØNE9(½÷v[Úî^HesseÝ_CqHk+1.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 291 / 422
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[Úî
Hk´1kgSHesseÝ_Cq§·F"±Hk5)Hk+1, =
Hk+1 = Hk + Ek ,
Ù¥Ek´$Ý"
d§æ^é¡(SR1)
Hk+1 = Hk + auuT , (a ∈ R,u ∈ Rn).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 292 / 422
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[Úî
d[Úî^(109)
Hk+1y(k) = Hky(k) + (auTy(k))u = s(k)
u7s(k) − Hky(k)§b½s(k) − Hky(k) 6= 0.
Øu = s(k) − Hky(k), da =1
uTy(k), l
Hk+1 = Hk +(s(k) − Hky(k))(s(k) − Hky(k))T
(s(k) − Hky(k))Ty(k). (110)
þª¡é¡"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 293 / 422
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[Úî
gggªªª555
½½½Âµµµ XJ«SU3(kÚSég¼ê4:§K¡ù«äkgª5"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 294 / 422
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[Úî
é¡âÑ5µ
1 ég¼êäk¢D5§=Hky(`) = s(`), ` = 0, 1, · · · , k − 1.
2 äkgª5§=éug¼êØI?1|¢ äknÚª5§Hn = [∇2f (x∗)]−1.
[gKµy²é¡[Úîþã5"]
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 295 / 422
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[Úî
é¡":´§ØU±SÝHk+1½5"
=(s(k) − Hky(k))Ty(k) > 0§é¡âU±½5" ù^ éJy§=¦(s(k) − Hky(k))Ty(k) > 0 ÷v§§Uél êþ(J"
ùѦé¡[ÚîA^kÛ5"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 296 / 422
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[Úî
æ^é¡(SR2)
Hk+1 = Hk + auuT + bvvT ,
¿¦[Úî^(109)¤á§Kk
Hk+1y(k) = Hky(k) + (auTy(k))u + (bvTy(k))v = s(k).
ùpu, vw,Ø´(½§k«²wÀJ´µu = s(k), auTy(k) = 1;
v = Hky(k), bvTy(k) = −1.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 297 / 422
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[Úî
Ïdk
Hk+1 = Hk +s(k)s(k)T
s(k)Ty(k)− Hky(k)y(k)THk
y(k)THky(k). (111)
þª¡DFP(Davidon-Fletcher-Powell)úª§dDavidon(1959)Jѧ²Fletcher & Powell(1963)?U 5"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 298 / 422
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[Úî
DFP(111)´;.[Úîúª§§kéõ5"
£¤éug¼ê£æ^°(|¢¤
1 ¢D5§=Hky(`) = s(`), ` = 0, 1, · · · , k − 1.
2 gª5§=Hn = [∇2f (x∗)]−1.
3 Ý5§=H0 = I§S)Ý"
£¤éu5¼ê
1 ±½5§Ï d(k)o´eü"
2 zgSI3n2 + O(n)g¦$"
3 äk5ÂñÝ"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 299 / 422
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[Úî
[[[ÚÚÚîî£¤¤¤^
Hk+1y(k) = s(k)
Ù¥Hk+1´HesseÝ_Cq¶
Bk+1s(k) = y(k)
Ù¥Bk+1´HesseÝCq"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 300 / 422
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[Úî
dé¡Ú[Úî^Hk+1y(k) = s(k)HkDFPúª
H(DFP)k+1 = Hk +
s(k)s(k)T
s(k)Ty(k)− Hky(k)y(k)THk
y(k)THky(k).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 301 / 422
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[Úî
BFGS (Broyden-Fletcher-Goldfarb-Shanno)
aq/§·l[Úî^Bk+1s(k) = y(k)'uBké¡úª
B(BFGS)k+1 = Bk +
y(k)y(k)T
y(k)T s(k)− Bks(k)s(k)TBk
s(k)TBks(k). (112)
r(112)ª¡'uBkBFGS"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 302 / 422
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[Úî
XJ·éBkBFGS/¦_0§Ò±'uHkBFGSúª
H(BFGS)k+1 = Hk + (1 +
y(k)THky(k)
s(k)Ty(k))
s(k)s(k)T
s(k)Ty(k)
−Hky(k)s(k)T + s(k)y(k)THk
s(k)Ty(k).
(113)
[ gKµÑH(BFGS)k+1 é¡A)§=a,u, b, v. ]
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 303 / 422
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[Úî
?Ú§eò(113)ª¥H ↔ B, s↔ yp§B'uBkDFPúª
B(DFP)k+1 = Bk + (1 +
s(k)TBks(k)
y(k)T s(k))y(k)y(k)T
y(k)T s(k)
−Bks(k)y(k)T + y(k)s(k)TBk
y(k)T s(k).
(114)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 304 / 422
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[Úî
¦_úª
Sherman-Morrison½½½nnnµµµ A ∈ Rn×n´Ûɧu, v ∈ Rn´?¿þ"e1 + vTA−1u 6= 0, KAA + uvTÛɧÙ_±L«
(A + uvT )−1 = A−1 − A−1uvTA−1
1 + vTA−1u. (115)
[ gKµ|^¦_úª§dH(DFP)k+1 íB
(DFP)k+1 ]
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 305 / 422
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[Úî
?Úë]
R. Fletcher, Practical Methods of Optimization (2nd Edition). JohnWiley & Sons, 1987.
D. C. Liu and J. Nocedal, On the Limited Memory Method for LargeScale Optimization. Mathematical Programming B, 45(3), pp.503-528, 1999.
...
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ÝFÝ
ÝÝÝ
½½½Âµµµ G´n × n½§Rn¥?|"þd(0),d(1), · · · ,d(k), XJd(i)TGd(j) = 0(i 6= j),K¡d(0),d(1), · · · ,d(k)´G−Ý"
w,Ý´Vgí2§G = I§Ý="
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 307 / 422
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ÝFÝ
ÝÝÝ£££aaa¤¤¤µµµ
(0) ½½G , ÀЩ:x(0), Og(0) = ∇f (x(0)) ¿Ed(0)¦
g(0)Td(0) < 0. -k := 0.
(1) ¦°(|¢Úαk , =αk = argminα>0
f (x(k) + αd(k)).
(2) #S:x(k+1) = x(k) + αkd(k), ¿Ed(k+1)¦
d(k+1)TGd(j) = 0, j = 0, 1, · · · , k .
(3) k := k + 1, £1(1)Ú"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 308 / 422
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ÝFÝ
Ý´lïÄg¼ê4z¯K¥)§§±í2?ng¼ê4z¯K"
Ý5´§1°(|¢§SÒäkgª5"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 309 / 422
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ÝFÝ
ÝÝÝÄÄĽ½½nnnµµµ îàg¼êf (x) =1
2xTGx + cTx,
Ý1°(|¢§KzÚS:x(k+1)´f (x)356/
V = x | x = x(0) +k∑
j=0
βjd(j),∀βj ∈ R
¥4:"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 310 / 422
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ÝFÝ
yyy²²²µµµ Ý)G -Ýd(0),d(1), · · · ,d(k).dݽ§d(0),d(1), · · · ,d(k)5Ã'"
e¡yµé¤kk < n¤á
g(k+1)Td(j) = 0, j = 0, 1, · · · , k .
=3:x(k+1)?¼êFÝg(k+1) = ∇f (x(k+1)) fmspand(0),d(1), · · · ,d(k)"
dd´Ñ½n(Ø"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 311 / 422
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ÝFÝ
yyy²²²£££YYY¤¤¤µµµ d°(|¢§é∀j ¤á
g(j+1)Td(j) = 0.
AO/§j = k §g(k+1)Td(k) = 0.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 312 / 422
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ÝFÝ
yyy²²²£££YYY¤¤¤µµµ ¯¢þ§du
y(k) = g(k+1) − g(k) = G (x(k+1) − x(k)) = Gs(k) = αkGd(k).
j < kk
g(k+1)Td(j) = g(j+1)Td(j) +k∑
i=j+1y(i)Td(j)
= g(j+1)Td(j) +k∑
i=j+1αid
(i)TGd(j)
= 0 + 0= 0
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 313 / 422
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ÝFÝ
yyy²²²£££YYY¤¤¤µµµ nÜþã§l y²
g(k+1)Td(j) = 0, j = 0, 1, · · · , k .
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 314 / 422
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ÝFÝ
íííØØصµµ éuîàg¼ê§e÷X|Ý|¢§²kÚS74:"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 315 / 422
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ÝFÝ
duÝäkgª5§<F"UÑäN£áuÝa¤"ÏL?Ueü§¦Ù|¢äkÝ5§ùB´ÝFÝ"
e¡·kég¼ê§ÑÝFÝäN£ã"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 316 / 422
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ÝFÝ
g¼êf (x) =1
2xTGx + cTx, Ù¥G´n × n½§c´nþ"
¼êfFÝþ
g(x) = ∇f (x) = Gx + c.
d(0) = −g(0), Ïx(1) = x(0) + α0d(0)¥Úα0d°(|¢û½§
¤±g(1)Td(0) = 0.
yd(1) = −g(1) + β(1)0 d(0), ÀJβ
(1)0 ¦d(1)TGd(0) = 0, =
β(1)0 =
g(1)Tg(1)
g(0)Tg(0).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 317 / 422
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ÝFÝ
Ón§-d(2) = −g(2) + β(2)0 d(0) + β
(2)1 d(1), ÀJβ
(2)0 , β
(2)1 ¦
d(2)TGd(j) = 0, j = 0, 1. l k
β(2)0 = 0,
β(2)1 =
g(2)Tg(2)
g(1)Tg(1).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 318 / 422
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ÝFÝ
/§31kgS¥§-
d(k) = −g(k) +k−1∑j=0
β(k)j d(j),
ÀJβ(k)j ¦d(k)TGd(j) = 0, j = 0, 1, · · · , k − 1, Kk
β(k)j =
g(k)TGd(j)
d(j)TGd(j)=
g(k)T (g(j+1) − g(j))
d(j)T (g(j+1) − g(j)).
qdug(k)Tg(j) = 0, j = 0, 1, · · · , k − 1,
β(k)j = 0, j = 0, 1, · · · , k − 2
β(k)k−1 =
g(k)Tg(k)
g(k−1)Tg(k−1).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 319 / 422
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ÝFÝ
éééggg¼¼¼êêêÝÝÝFFFÝÝÝ (Fletcher & Reeves, 1964)
(0) ½Ð©:x(0), Og(0) = g(x(0)), -d(0) = −g(0), k := 0.
(1) S#x(k+1) = x(k) + αkd(k), Ù¥αk =g(k)Tg(k)
d(k)TGd(k).
(2) Og(k+1) = g(x(k+1)), EÝFÝ
d(k+1) = −g(k+1) + βkd(k), Ù¥βk =g(k+1)Tg(k+1)
g(k)Tg(k).
(3) k := k + 1, £1(1)Ú"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 320 / 422
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ÝFÝ
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 321 / 422
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ÝFÝ
ÝÝÝFFFÝÝÝ555½½½nnnµµµ 8I¼êf (x) =1
2xTGx + cTx, Kæ^°(
|¢ÝFݲm ≤ nÚSª§é¤k1 ≤ k ≤ m¤áe'Xªµ
d(k)TGd(j) = 0, g(k)Tg(j) = 0, j = 0, 1, · · · , k − 1
d(k)Tg(k) = −g(k)Tg(k)
spang(0), g(1), · · · , g(k) = spang(0),Gg(0), · · · ,G kg(0)spand(0),d(1), · · · ,d(k) = spand(0),Gg(0), · · · ,G kg(0)
[gKµy²þã½n...]
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 322 / 422
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ÝFÝ
òÝFÝí2g¼ê4z¯K§ÙS
x(k+1) = x(k) + αkd(k).
Úαkd°(½ö°(|¢û½§ d(k+1)EXeµ
d(k+1) = −g(k+1) + βkd(k).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 323 / 422
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ÝFÝ
Ù¥
βk :=g(k+1)Tg(k+1)
g(k)Tg(k)(Fletcher− Reeves)
βk :=g(k+1)T (g(k+1) − g(k))
d(k)T (g(k+1) − g(k))(Hestenes− Stiefel)
βk :=g(k+1)T (g(k+1) − g(k))
g(k)Tg(k)(Polak− Ribiere− Polyak)
βk :=g(k+1)Tg(k+1)
−d(k)Tg(k)(Dixon)
βk :=g(k+1)Tg(k+1)
d(k)T (g(k+1) − g(k))(Dai−Yuan)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 324 / 422
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ÝFÝ
éug¼ê§ÝFÝSnÚ±¤)|¢d(k+1) = −g(k+1) + βkd(k) UØ2´eü£d°(|¢E¤¤"
Ïd§nÚ±·AT±Ï5æ^eü|¢§=-d(`n) = −g(`n), ` = 1, 2, . . .
ù«üÑ¡éÄüѧùÝFÝ¡éÄÝFÝ"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 325 / 422
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ÝFÝ
Xþ¤ãÝFÝSéu5¼êz´ì·^§S#ÚÏfÃwªL§I1êCq°(|¢"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 326 / 422
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ÝFÝ
duznÚS1éÄüѧeP#éÄ:z(j),Ky²ùngS:5Âñ"É¢SOØK§3éõ/e=Uaq5ÂñÇ"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 327 / 422
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ÝFÝ
l¢SOÇ9½55w§ÝFÝ7'[ÚîÐ"´§ÝFÝ¥|¢O==^8I¼êFݧ Ø7[Úî@3zgS¥#HesseÝ£½Ù_¤Cq¿PÁ"¤±§¯K5 kDÕ(§ÝFÝkp1OÐ?"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 328 / 422
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ÝFÝ
The preconditioned conjugate gradient method
3õê¹e§(ÝFݯÂñ§ý^?n´7"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 329 / 422
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ÝFÝ
?Úë]
M. Hestenes and E. Stiefel, Methods of conjugate gradients forsolving linear systems. Journal of Research of the National Bureau ofStandards, 49 (6), 1952.
K. Atkinson, An Introduction to Numerical Analysis (2nd Edition).John Wiley & Sons, 1988.
M. Avriel, Nonlinear Programming: Analysis and Methods. DoverPublishing§2003.
G. Golub and C. Van Loan, Matrix Computations (3rd Edition).Johns Hopkins University Press.
...
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 330 / 422
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&6
ySÛÂñ5§c·æ^|¢üÑ"
|¢üÑk(½|¢d(k), ,÷XùÀJ·ÚÏfαk , #S:x(k+1) = x(k) + αkd(k).
y3§·?Ø,«ÛÂñüÑ— &6(Trust-Region Method).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 331 / 422
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&6
ySÛÂñ5§c·æ^|¢üÑ"
|¢üÑk(½|¢d(k), ,÷XùÀJ·ÚÏfαk , #S:x(k+1) = x(k) + αkd(k).
y3§·?Ø,«ÛÂñüÑ— &6(Trust-Region Method).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 331 / 422
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&6
ySÛÂñ5§c·æ^|¢üÑ"
|¢üÑk(½|¢d(k), ,÷XùÀJ·ÚÏfαk , #S:x(k+1) = x(k) + αkd(k).
y3§·?Ø,«ÛÂñüÑ— &6(Trust-Region Method).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 331 / 422
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&6
&6Äk½ÂcS:x(k)
Ωk = x ∈ Rn | ‖x− x(k)‖ ≤ ek,
ùpΩk¡&6§ek´&6»"
b½3ùp§g.q(k)(s)´8I¼êf (x)Ü·Cq§K3&6¥4zg.§Cq4:s(k), ¿x(k+1) = x(k) + s(k).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 332 / 422
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&6
&6|^g.3&6S¦Ús(k), ¦8I¼êeü'|¢k"
&6Ø=äkÛÂñ5§ ئ8I¼êHesseÝ£½ÙCq¤´½"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 333 / 422
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&6
&6f¯K
min q(k)(s) = f (x(k)) + g(k)T s +1
2sTBks
s.t. ‖s‖ ≤ ek .(116)
Ù¥s = x− x(k), g(k) = ∇f (x(k)), é¡Bk´HesseÝ∇2f (x(k))½ÙCq§ek > 0&6»§‖ · ‖,ê"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 334 / 422
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&6
XÛÀJ&6»ek?
·òâg.q(k)(s)é8I¼êf (x)[ܧÝg·A/N&6»"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 335 / 422
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&6
f¯K(116))s(k), -8I¼êeüþ
Actk = f (x(k))− f (x(k) + s(k))
¢Seüþ§-g.¼êeüþ
Prek = q(k)(0)− q(k)(s(k))
ýÿeüþ"½Â'
rk =ActkPrek
=f (x(k))− f (x(k) + s(k))
q(k)(0)− q(k)(s(k)).
§ïþg.8I¼êm§Ý"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 336 / 422
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&6
rkC1, L²g.¼êq(k)(s)8I¼êf5§ÝЧd±O»ek±*&6"
XJrk > 0ØC1, ·±&6»ekØC"
XJrkC"½K§L²q(k)(s)8I¼êf5§ÝØn§Ò~»ek± &6"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 337 / 422
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&6
&&&666
(0) ½Ð©:x(0), &6»þ.e,ε > 0, 0 < γ1 < γ2 < 1, 0 < η1 < 1 < η2. e0 ∈ (0, e), -k := 0.
(1) XJ‖g(k)‖ ≤ ε, ÊS"ÄK§¦)&6f¯K(116)s(k).
(2) O'rk , #S:
x(k+1) =
x(k) + s(k) if rk > 0,
x(k) otherwise.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 338 / 422
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&6
&&&666
(3) N&6Ȥ-
ek+1 =
η1ek if rk < γ1,ek if γ1 ≤ rk < γ2,min(η2ek , e) if rk ≥ γ2.
(4) k := k + 1, £1(1)Ú"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 339 / 422
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&6
&&&666ÛÛÛÂÂÂñññ555½½½nnnµµµ
Y²8L(x(0)) = x | f (x) ≤ f (x(0))k.§f (x)3ÙþC 2ëY§Kd&6)SS3à:x∞, ÷vÚ7^§=
g∞ = ∇f (x∞) = 0, G∞ = ∇2f (x∞) ≥ 0.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 340 / 422
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&6
3&6¥'Ú´§)&6f¯K(116).
ùp·0«¦)&6f¯K§=dPowell (1970) JÑò"
¤¢ò§´ëCauchy :£deü)4:¤ÚÚî:£dÚî)4:¤§Ùë&6>.:x(k+1).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 341 / 422
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&6
òã«
)(kx
Nx
Cx
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 342 / 422
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&6
éug.
q(k)(−αg(k)) = f (x(k))− α‖g(k)‖2 +1
2α2g(k)TBkg(k),
°(|¢ÚÏfL
αk =‖g(k)‖2
g(k)TBkg(k).
u´Cauchy Ú
s(k)C = −αkg(k) = − ‖g(k)‖2
g(k)TBkg(k)g(k).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 343 / 422
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&6
XJ‖s(k)C ‖ = ‖αkg(k)‖ ≥ ek ,
s(k) = − ek‖g(k)‖
g(k),
Bx(k+1) = x(k) − ek
‖g(k)‖g(k).
XJ‖s(k)C ‖ = ‖αkg(k)‖ < ek , 2OÚîÚ
s(k)N = −B−1
k g(k).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 344 / 422
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&6
XJ‖s(k)N ‖ ≤ ek ,
s(k) = s(k)N = −B−1
k g(k),
ÄK§s(k) = s
(k)C + λ(s
(k)N − s
(k)C ),
Ù¥λ¦‖s(k)
C + λ(s(k)N − s
(k)C )‖ = ek .
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 345 / 422
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&6
nþ¤ã§·
x(k+1) =
x(k) − ek
‖g(k)‖g(k) ‖s(k)
C ‖ ≥ ek ,
x(k) + s(k)C + λ(s
(k)N − s
(k)C ) ‖s(k)
C ‖ < ek‖s(k)N ‖ > ek ,
x(k) − B−1k g(k) ‖s(k)
C ‖ < ek‖s(k)N ‖ ≤ ek .
(117)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 346 / 422
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&6
ò÷ve5µ
1) ÷XCauchy :x(k+1)C ÚÚî:x
(k+1)N ë§x(k)ålüNO
\¶
2) ÷XCauchy :x(k+1)C ÚÚî:x
(k+1)N ë§f¯K.¼êü
N~"
[gKµy²þã5...]
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 347 / 422
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Ù£Ãå`z¤
Exercise 5.1: ÑþãÄuWolfe-PowellOK°(|¢¥õªp(1)(t), p(2)(t)äNLª"
Exercise 5.2: y²ÄuGoldsteinOK°(|¢ÛÂñ5"
Exercise 5.3: Áò5§|¦F (x) = 0ÚîS§^u¦)Ãå`z¯Kminx∈Rn f (x). ÑASª§¿`²nd"
Exercise 5.4: y²é¡[Úîäkgª5Ú¢D5"
Exercise 5.5: |^¦_úª£Sherman-Morrison½n¤§dH
(DFP)k+1 íB
(DFP)k+1 .
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 348 / 422
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Ù£Ãå`z¤
Exercise 5.6: y²ÝFÝ5½n"
Exercise 5.7: 3&6¥§Ñ«N&6»g·Aª"
Exercise 5.8: y²ò£&6¤f¯K.¼êüN5"
Project 5.9: ?§¢yÄu°(|¢(line search),«£|¢¤Seü§¿^u¦)Ãå`z¯K"¦J§Sè§^rH9ÿÁw"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 349 / 422
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1 XØ
2 55y
3 ä`z
4 Ä5y
5 55yÄ:nØ
6 Ãå`z
7 g5y
8 5å`z
9 o(
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 350 / 422
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g5y
g5y(Quadratic Programming)´§3Cþ5ªÚ/½Øªe¦g¼ê4:¯K
min Q(x) =1
2xTGx + cTx
s.t. aTi x = bi , i ∈ E = 1, · · · ,me
aTi x ≥ bi , i ∈ I = me + 1, · · · ,m
(118)
·b½G顧ai (i ∈ E)´5Ã'"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 351 / 422
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g5y
g5yåUØN§Uvkk§ù¡g5y¯KÃ)"
XJÝG½§¯K(118)´àg5y¯K§§?¿ÛÜ)´N)"
XJÝG½§¯K(118)´½g5y¯K§3)=´"
XJÝGؽ§¯K(118)´g5y¯K§kUÑyN)ÛÜ)"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 352 / 422
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g5y
ªåg5y¯K
min Q(x) =1
2xTGx + cTx
s.t. Ax = b(119)
ùpA´m × nݧØ5rank(A) = m.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 353 / 422
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g5y
k«Ä©)x =
(xBxN
), Ù¥xB ∈ Rm, xN ∈ Rn−m, ¦ÙåÝ
éA©¬A = (AB ,AN)¥AB_"u´§ªå^¤
xB = A−1B (b− ANxN),
¿òþª\8I¼ê¥Ãå¯K
minxN∈Rn−m
1
2xTN GNxN + cTNxN . (120)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 354 / 422
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g5y
3þª¥
GN = GNN − GNBA−1B AN − AT
NA−TB GBN + AT
NA−TB GBBA
−1B AN ,
cN = cN − ATNA−TB cB + GNBA
−1B b− AT
NA−TB GBBA
−1B b,
±9éA©¬/ª
G =
(GBB GBN
GNB GNN
), c =
(cBcN
).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 355 / 422
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g5y
(1) XJGN½§KÃå¯K)/Ñ
x∗N = −G−1N cN ,
?Ú¯K(119))
x∗ =
(x∗Bx∗N
)=
(A−1B b0
)+
(A−1B AN
−I
)G−1N cN .
x∗éALagrange¦fþλ∗, Kk
Gx∗ + c = ATλ∗ =⇒ λ∗ = A−TB (GBBx∗B + GBNx∗N + cB).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 356 / 422
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g5y
(2) XJGN´½§K3(I − GN G+N )cN = 0§Ãå¯Kk
.§§)L«
x∗N = −G+N cN + (I − G+
N GN)y,
Ù¥y ∈ Rn−m?¿þ§G+NL«GN2Â_Ý"d§¯
K)x∗ÚA`¦fλ∗aq(½"
(I − GN G+N )cN = 0ؤá§KíÑÃå¯KÃe.§l
¯KÃe."
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 357 / 422
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g5y
(3) XJGNؽ£=3KA¤§w,Ãå¯KÃe.§¯KØ3k`)"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 358 / 422
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g5y
þãØv?´§ABCÛɧN´êOؽ"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 359 / 422
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g5y
222ÂÂÂ
Z = zm+1, · · · , zn)mKer(A)|ħY = y1, · · · , ym´ûmRn/Ker(A)|ħK∀x ∈ RnXe©)L
x = Y xY + ZxZ .
l k
Ax = b =⇒ AY xY + AZxZ = b =⇒ xY = (AY )−1b,
¤±x = Y (AY )−1b + ZxZ ,
Ù¥xZ ∈ Rn−m´gdCþ"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 360 / 422
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g5y
222ÂÂÂ
òþª\8I¼ê¥Ãå¯K
minxZ∈Rn−m
1
2xTZ (ZTGZ )xZ + [ZTGY (AY )−1b + ZTc]TxZ . (121)
b½ZTGZ½§Kk
x∗Z = −(ZTGZ )−1ZT [GY (AY )−1b + c].
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 361 / 422
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g5y
222ÂÂÂ
l ¯K`)
x∗ = Y (AY )−1 − Z (ZTGZ )−1ZT [GY (AY )−1b + c],
ALagrange¦f
λ∗ = (AY )−TY T (Gx∗ + c).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 362 / 422
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g5y
Lagrange´Äu¦)1S(K-T):§=Lagrange¼ê½:"
éuªå¯K(119), ÙLagrange¼ê½:Ò´Xe5§|)
Gx + c = ATλ,Ax = b.
¤Ý/ª (G −AT
−A 0
)(xλ
)= −
(cb
).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 363 / 422
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g5y
Ý
(G −AT
−A 0
)_§K3Ý
U ∈ Rn×n,V ∈ Rm×m,W ∈ Rm×n ¦(G −AT
−A 0
)−1
=
(U W T
W V
)l ¦¯K)
x∗ = −Uc−W Tb,λ∗ = −W c− Vb.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 364 / 422
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g5y
þãLagrange¥ÝÛÉ5¿Ø½¦G−13§^ØÓÑ©¬ÝU,V ,WL/ª§l ØÓOúª"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 365 / 422
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g5y
G_§A1÷§K(AG−1AT )−13§ØJyU = G−1 − G−1AT (AG−1AT )−1AG−1,V = −(AG−1AT )−1,W = −(AG−1AT )−1AG−1.
u´·¦)úªx∗ = −G−1c + G−1AT (AG−1AT )−1(AG−1c + b),λ∗ = (AG−1AT )−1(AG−1c + b).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 366 / 422
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g5y
XJY ,Z÷v(Y ,Z ) =
(AB
)−1
, =AY = Im×m,AZ = 0.
e,kZTGZ_§K
(G −AT
−A 0
)_"d
U = Z (ZTGZ )−1ZT ,V = −Y TGPTY ,W = −Y TP.
Ù¥P = I − GZ (ZTGZ )−1ZT .
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 367 / 422
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g5y
ÄuATQR©)§Ñ(Y ,Z )«Aϵ
AT = Q
(R0
)= (Q1,Q2)
(R0
),
=
A = (RT , 0)
(QT
1
QT2
).
Ù¥Qn × n§Rm ×mþn"u´-Y = Q1R
−T ,Z = Q2, Kk
AY = RTQT1 Q1R
−T = Im×m, AZ = RTQT1 Q2 = 0m×(n−m).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 368 / 422
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g5y
g5y
min Q(x) =1
2xTGx + cTx
s.t. aTi x = bi , i ∈ E = 1, · · · ,me
aTi x ≥ bi , i ∈ I = me + 1, · · · ,m
(122)
*þ§ØÈ4تå3)NCØå^§KØĶ È4تå§du§3)?Ҥ᧷±^ªå5OùÈ4تå"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 369 / 422
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g5y
ÈÈÈ444888ÄÄĽ½½nnnµµµ x∗´g5y¯K(122)ÛÜ4:§Kx∗7´ªå¯K
(EQ)
min Q(x) =
1
2xTGx + cTx
s.t. aTi x = bi , i ∈ E ∪ I(x∗)
ÛÜ4:"§XJx∗´¯K(122)1:§Ó´(EQ)K-T:§ALagrange¦fλ∗÷vλ∗i ≥ 0, i ∈ I(x∗),Kx∗7´¯K(122)K-T:"
[SK6.1µy²þã½n...]
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 370 / 422
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g5y
x(k)cS:§´¯K(122)1:"
PEk = E ∪ I(x(k)), Īå¯K
(EQ1)
min
1
2sTGs + (Gx(k) + c)T s
s.t. aTi s = 0, i ∈ Ek
¦(EQ1))s(k), 9ÙALagrange¦fλ(k)i , i ∈ Ek .
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 371 / 422
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g5y
(a) s(k) 6= 0§x(k)ØU´¯KK-T:"
(b) s(k) = 0§x(k)´¯K
(EQ2)
min
1
2xTGx + cTx
s.t. aTi x = bi , i ∈ Ek
K-T:¶XJλ(k)i ≥ 0, i ∈ I(x(k)), Kx(k)´¯KK-T:"
(c) ÄK§dλ(k)iq
= mini∈I(x(k))
λ(k)i < 0 (½iq, @oXe¯K
(EQ3)
min
1
2sTGs + (Gx(k) + c)T s
s.t. aTi s = 0, i ∈ E = Ek \ iq.
)s´¯K3c:x(k)?1§=aTiq
s ≥ 0.
[gKµy²þã(c)(Ø...]YZW (USTC) OPERATIONS RESEARCH 2019-09-02 372 / 422
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g5y
ÈÈÈ444888(Active Set Method)
(0) Ñ1:x(0), -E0 = E ∪ I(x(0)), k := 0.
(1) ¦)ªå¯K(EQ1)s(k), es(k) 6= 0, =1(3)Ú"
(2) XJλ(k)i ≥ 0, i ∈ I(x(k)), KʶÄKdλ
(k)iq
= mini∈I(x(k))
λ(k)i < 0 (
½iq¿-Ek := Ek \ iq, x(k+1) = x(k), =1(4)Ú"
(3) dαk = min1, mini /∈Ek ,aTi s(k)<0
bi−aTi x(k)
aTi s(k) , Ox(k+1) = x(k) + αks(k). X
Jαk = 1, =1(4)Ú¶Ø,½ép /∈ Ek¦aT
p (x(k) + αks(k)) = bp, ¿-Ek := Ek ∪ p.(4) Ek+1 := Ek , k := k + 1, £1(1)Ú"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 373 / 422
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Ù£g5y¤
Exercise 6.1: y²È48Ľn"
Exercise 6.2: y²cã(c)(Ø"
Exercise 6.3 £££ÀÀÀKKK¤¤¤: Áy²3½UþET ,m÷vXe.)
1
nyTET ,my = min
φ ∈ Wm2 (Ω)
φ(Xi ) = yi , i = 1, · · · , n
|φ|2T ,m
Ù¥y = (y1, · · · , yn)T , T = Xini=1,
|f |2T ,m =1
n
n∑i=1
∑|α|=m
m!α! |D
αf (Xi )|2.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 374 / 422
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1 XØ
2 55y
3 ä`z
4 Ä5y
5 55yÄ:nØ
6 Ãå`z
7 g5y
8 5å`z
9 o(
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 375 / 422
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5å`z
kĪå¯Kmin f (x)s.t. c(x) = 0,
(123)
Ù¥c(x) = (c1(x), · · · , cm(x))T .
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 376 / 422
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5å`z
PA(x) = [∇c(x)]T = (∇c1(x), · · · ,∇cm(x))T .
d`5^µx´ªå¯K(123)K-T:=3¦fλ ∈ Rm¦
∇f (x)− A(x)Tλ = 0,
x´1:§=c(x) = 0.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 377 / 422
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5å`z
u´éá§| ∇f (x)− A(x)Tλ = 0,−c(x) = 0.
·^Newton-RaphsonS¦)þãéá§|"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 378 / 422
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5å`z
PxÚλOOþ©Oδx, δλ, Newton-RaphsonS÷vµ(W (x, λ) −A(x)T
−A(x) 0
)(δx
δλ
)= −
(∇f (x)− A(x)Tλ
−c(x)
), (124)
Ù¥W (x, λ) = ∇2f (x)−m∑i=1
λi∇2ci (x).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 379 / 422
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5å`z
þã¡Lagrange-Newton§@dWilson(1963)JÑ"
Ù¢þ´^Newton-RaphsonS¦¯K(123)Lagrange¼êL(x, λ)½:"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 380 / 422
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5å`z
3d§·½Âd¼ê
ψ(x, λ) = ‖∇f (x)− A(x)Tλ‖2 + ‖c(x)‖2. (125)
w,§ψ(x, λ)´'uLagrange-Newtoneü¼ê§=÷v
∇ψ(x, λ)T(δx
δλ
)= −2ψ(x, λ) ≤ 0.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 381 / 422
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5å`z
Lagrange-Newtonµµµ
(0) ½x(0), λ ∈ Rm, β ∈ (0, 1), ε ≥ 0, -k := 0.
(1) Od¼êψ(x(k), λ(k)), XJψ(x(k), λ(k)) ≤ ε, KʶÄK3(x(k), λ(k))?¦)(124)(δx(k) , δλ(k)), ¿-αk = 1.
(2) eψ(x(k) + αkδx(k) , λ(k) + αkδλ(k)) ≤ (1− βαk)ψ(x(k), λ(k)), =1(3)Ú¶ÄK-αk = 1
4αk , £1(2)Ú"
(3) x(k+1) = x(k) + αkδx(k) , λ(k+1) = λ(k) + αkδλ(k) , k := k + 1, £1(1)Ú"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 382 / 422
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5å`z
Lagrange-NewtonÂñ5(J
½½½nnnµµµ Lagrange-Newton)S:(x(k), λ(k))k.§XJf (x)Úci (x)Ñ´gëY§_Ý(
W (x, λ) −A(x)T
−A(x) 0
)−1
k.§K(x(k), λ(k))?Ûà:Ñ´§ψ(x, λ) = 0§l x(k)à:´¯K(123)K-T:"
5µ3½^e§?Úy²Lagrange-NewtonäkÂñÝ"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 383 / 422
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5å`z
ÅÚg5y
Lagrange-Newtonz´§3ÙÄ:þuÐÑÅÚg5y(Sequential Quadratic Programming Methods)" ö®¤¦)5å`z¯Ka©"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 384 / 422
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5å`z
ÅÚg5y
·òª(124)¤Xe/ªµW (x, λ)δx +∇f (x) = A(x)T (λ+ δλ),c(x) + A(x)δx = 0.
d`5^§δx(k)=eg5y¯K
min1
2dTW (x(k), λ(k))d +∇f (x(k))Td
s.t. c(x(k)) + A(x(k))d = 0(126)
K-T:"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 385 / 422
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5å`z
ÅÚg5y
Lagrange-Newton±n)ÅÚ¦)þãªåg5y"
d(k)´g5y¯K(126)`)§@oS#
x(k+1) = x(k) + αkd(k),
Ù¥αk1kgSÚ"
λ(k)´(126)éALagrange¦fþ§@oék ≥ 1k
λ(k+1) = λ(k) + αk(λ(k) − λ(k)).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 386 / 422
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5å`z
ÅÚg5y
yÄ5å`z¯K
min f (x)s.t. ci (x) = 0, i ∈ E = 1, · · · ,me,
ci (x) ≥ 0, i ∈ I = me + 1, · · · ,m.(127)
aq/§31kgSp¦)f¯K
min1
2dTWkd + g(k)Td
s.t. ci (x(k)) + ai (x(k))Td = 0, i ∈ E ,ci (x(k)) + ai (x(k))Td ≥ 0, i ∈ I.
(128)
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 387 / 422
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5å`z
ÅÚg5y
3ùp§Wk´¯KLagrange¼êHesse½ÙCq§g(k) = ∇f (x(k)), A(x(k)) = (a1(x(k)), · · · , am(x(k)))T = [∇c(x(k))]T .
Pf¯K(128))d(k), ALagrange¦fþλ(k), kWkd(k) + g(k) = A(x(k))T λ(k),
λ(k)i ≥ 0, i ∈ I,
c(x(k)) + A(x(k))d(k) = 0.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 388 / 422
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5å`z
ÅÚg5y
ÅÚg5yS±d(k)|¢"T|¢kéÐ5"§´Nõv¼êeü§~XL1v¼ê
P(x , σ) = f (x) + σ
(me∑i=1
|ci (x)|+m∑
i=me+1
|ci (x)−|
).
Ù¥c(x)−½ÂXeµci (x)− = ci (x), i ∈ E ,ci (x)− = min0, ci (x), i ∈ I.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 389 / 422
![Page 404: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/404.jpg)
5å`z
ÅÚg5y
e¡´Han(1977)JÑÅÚg5yµ
(0) ½x(0),W0 ∈ Rn×n, σ > 0, ρ ∈ (0, 1), ε ≥ 0, -k := 0.
(1) ¦)f¯K(128)Ñd(k), XJ‖d(k)‖ ≤ ε, KʶÄK¦αk ∈ [0, ρ]¦
P(x(k) + αkd(k), σ) ≤ min0≤α≤ρ
P(x(k) + αd(k), σ) + εk .
(2) x(k+1) = x(k) + αkδx(k) , OWk+1, -k := k + 1, £1(1)Ú"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 390 / 422
![Page 405: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/405.jpg)
5å`z
ÅÚg5y
y²cãÅÚg5yÂñ5(JXeµ
½½½nnnµµµb½f (x)Úci (x)ëY§3~êM1,M2 > 0¦
M1‖d‖2 ≤ dTWkd ≤ M2‖d‖2, ∀k ∈ N, ∀d ∈ Rn,
XJ‖λ(k)‖∞ ≤ σþ¤á§KHan(1977)):x(k) ?Ûà:Ñ´¯K(127)K-T:"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 391 / 422
![Page 406: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/406.jpg)
5å`z
v¼ê
éu5å`z¯K
min f (x)s.t. ci (x) = 0, i ∈ E = 1, · · · ,me
ci (x) ≥ 0, i ∈ I = me + 1, · · · ,m(129)
v¼ê§´|^8I¼êf (x)Úå§c(x)¤Eäk/v50¼ê
P(x) = P(f (x), c(x)).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 392 / 422
![Page 407: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/407.jpg)
5å`z
v¼ê
¤¢/v50§=¦é¯K1:x ∈ SþkP(x) = f (x), å^»kP(x) > f (x).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 393 / 422
![Page 408: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/408.jpg)
5å`z
v¼ê
£ãå^»§Ý§·½Âc(x)−Xeµci (x)− = ci (x), i ∈ E ,ci (x)− = min0, ci (x), i ∈ I.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 394 / 422
![Page 409: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/409.jpg)
5å`z
v¼ê
v¼ê8I¼ê/v0Ú§=
P(x) = f (x) + φ(c(x)−).
vφ(c(x)−)´½Â3Rmþ¼ê§§÷v
φ(0) = 0, lim‖c‖→∞
φ(c) = +∞.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 395 / 422
![Page 410: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/410.jpg)
5å`z
v¼ê
XCourantv¼êµ
Pσ(x) = f (x) + σ‖c(x)−‖22,
Ù¥σ > 0´vÏf"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 396 / 422
![Page 411: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/411.jpg)
5å`z
v¼ê
Äüv¼êPσ(x) = f (x) + σ‖c(x)−‖2.
Px(σ)´Ãå¯Kminx∈Rn
Pσ(x)`)§·kXeÚn"
ÚÚÚnnnµµµ ex(σ)Ó´5å`z¯K(129)1:§Kx(σ)´¯K`)"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 397 / 422
![Page 412: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/412.jpg)
5å`z
v¼ê
þãÚnL²§À¿©vÏfσ > 0, KÏL¦)Ãå`z¯KAéAå`z¯K`)"
, 3¢SO¥§(½Ü·σ '(J§Ï~´ÀüNOvÏfSσk.
ÏL¦)XÃå¯K5¼å`z¯K)§ù¡S0Ãå4zEâ(SUMT)"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 398 / 422
![Page 413: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/413.jpg)
5å`z
v¼ê
d§·±Ñv¼êSÚ½µ
(0) ?ÀЩ:x(0), ½Ð©vÏfσ0 > 09β > 1, ε > 0. -k := 0.
(1) ±x(k)ЩS:¦)Ãå¯K4:§=
x(σk) = arg minx∈Rn
Pσk (x).
(2) e‖c(x(σk))−‖ < ε, KÊS¿x(σk)å¯KCq`)¶ÄK§x(k+1) = x(σk), σk+1 = βσk , -k := k + 1£1(1)Ú"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 399 / 422
![Page 414: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/414.jpg)
5å`z
v¼ê
´XenÚn
ÚÚÚnnn1µµµ σk+1 > σk > 0, KkPσk (x(σk)) ≤ Pσk+1(x(σk+1)),
‖c(x(σk))−‖ ≥ ‖c(x(σk+1))−‖, f (x(σk)) ≤ f (x(σk+1)).
ÚÚÚnnn2µµµ -x´¯K(129)`)§Ké?¿σk > 0 ¤á
f (x) ≥ Pσk (x(σk)) ≥ f (x(σk)).
ÚÚÚnnn3µµµ -δ = ‖c(x(σ))−‖, Kx(σ)´å¯K
min f (x)s.t. ‖c(x)−‖ ≤ δ
`)"YZW (USTC) OPERATIONS RESEARCH 2019-09-02 400 / 422
![Page 415: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/415.jpg)
5å`z
v¼ê
[gKµy²þãÚn1...]
[gKµy²þãÚn3...]
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 401 / 422
![Page 416: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/416.jpg)
5å`z
v¼ê
ùpÑÚn2y²µ
dK´Pσk (x(σk)) ≥ f (x(σk)).
Ïx´¯K`)§g,1:§u´σk‖c(x)−‖2 = 0.
qÏx(σk) = arg minx∈Rn
Pσk (x), Kk
f (x) = Pσk (x) ≥ Pσk (x(σk)).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 402 / 422
![Page 417: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/417.jpg)
5å`z
v¼ê
'uv¼êÂñ5§·kXe(J
½½½nnn1µµµ v¼ê¥ε÷v
ε > minx∈Rn‖c(x)−‖,
K7kª"
[gKµy²þã½n...]
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 403 / 422
![Page 418: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/418.jpg)
5å`z
v¼ê
T½nL²§XJå¯K31:§Ké?¿½ε > 0, Ñòkªu¯K
min f (x)s.t. ‖c(x)−‖ ≤ δ
)§δ ≤ ε.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 404 / 422
![Page 419: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/419.jpg)
5å`z
v¼ê
½½½nnn2µµµ XJØkª§K7kminx∈Rn‖c(x)−‖ ≥ ε,
limk→∞
‖c(x(σk))−‖ = minx∈Rn‖c(x)−‖. d§x(σk)?Ûà:x∗Ñ´¯
Kmin f (x)s.t. ‖c(x)−‖ = min
y∈Rn‖c(y)−‖
)"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 405 / 422
![Page 420: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/420.jpg)
5å`z
¦fv¼ê
Qãü§=Īå¯K
min f (x)s.t. c(x) = 0
(130)
Ù¥c(x) = (c1(x), · · · , cme (x))T .
x∗´þã¯K`)λ∗´ALagrange¦f§dKuhn-Tucher½n§x∗7´Lagrange¼ê
L(x, λ∗) = f (x)− (λ∗)Tc(x)
½:" ó§x∗¿Ø´Lagrange¼ê4:"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 406 / 422
![Page 421: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/421.jpg)
5å`z
¦fv¼ê
·Ä¦fv¼ê£¡O2Lagrange¼ê¤
P(x, λ, σ) = L(x, λ) +σ
2‖c(x)‖2
2.
duO2Lagrange¼ê5§vvÏfσ Ø7ªÃ¡§ÒÏL4zP(x, λ, σ)¦¯K`)"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 407 / 422
![Page 422: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/422.jpg)
5å`z
¦fv¼ê
PA(x) = [∇c(x)]T , duc(x∗) = 0, ·´
∇xP(x∗, λ∗, σ) = ∇xL(x∗, λ∗) = 0,
∇2xxP(x∗, λ∗, σ) = ∇2
xxL(x∗, λ∗) + σA(x∗)A(x∗)T .
3x∗?÷v¿©^§=é∀d¦A(x∗)Td = 0"þ§þk
dT∇2xxL(x∗, λ∗)d > 0.
@o3¿©^b½e§éu¿©σ, y
∇2xxP(x∗, λ∗, σ)
´½"YZW (USTC) OPERATIONS RESEARCH 2019-09-02 408 / 422
![Page 423: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/423.jpg)
5å`z
¦fv¼ê
½½½nnnµµµ x∗Úλ∗÷vªå¯K(130)ÛÜ`)¿©^§K3σ¦σ > σ§x∗´¼êP(x, λ∗, σ)îÛÜ4:"
[gKµy²þã½n...]
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 409 / 422
![Page 424: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/424.jpg)
5å`z
¦fv¼ê
·¯k¿Ø`¦fþλ∗, Ïd^¦fλO§O2Lagrangev¼êµ
P(x, λ, σ) = f (x)− λTc(x) +σ
2‖c(x)‖2
2.
üÑ´§k½¿©σÚ¦fþЩOλ, ,3SL§¥?¦fλå㦪`¦fλ∗.
XÛ?º
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 410 / 422
![Page 425: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/425.jpg)
5å`z
¦fv¼ê
ÄuO2Lagrange¼êSµ
(0) ½Ð©:x(0)Ú¦fþЩOλ(0), ½vÏfσ0 > 0, ~êα > 1, 0 < β < 19NNØε > 0. -k := 0.
(1) ±x(k)Ð:¦)Ãå¯K4:§=
x(k+1) = arg minx∈Rn
P(x, λ(k), σ).
(2) e‖c(x(k+1))‖ < ε, KÊS¿x(k+1)¯KCq`)¶ÄK§#¦fþ
λ(k+1) = λ(k) − σc(x(k+1)).
(3) XJ‖c(x(k+1))‖‖c(x(k))‖ ≥ β, Kσ := ασ. -k := k + 1£1(1)Ú"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 411 / 422
![Page 426: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/426.jpg)
5å`z
æN¼ê
ÄÄÄggg 3S¥o´lS:Ñu§¿ÏLÚ\æN¼ê¦±31SÜ?1|¢"Ïd§ù«·^uتå5`z¯K"
min f (x)s.t. gi (x) ≥ 0, i = 1, · · · ,m. (131)
yò1SÜPintS , Ù¥S = x | gi (x) ≥ 0, i = 1, · · · ,m. ±S:¹u1SÜ´½ÂXeæN¼êµ
B(x, θ) = f (x) + θψ(x)
Ù¥æNÏfθ´éê§ψ(x)´ëY¼ê§xªu1>.§ψ(x)→ +∞.
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 412 / 422
![Page 427: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/427.jpg)
5å`z
æN¼ê
ü«æN/ª´µ
ψ(x) =m∑i=1
1
gi (x)and ψ(x) = −
m∑i=1
log gi (x)
ù§xª1>.§¼êB(x, θ)→ +∞. ÄK§duθ駼êB(x, θ)Cquf (x).
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 413 / 422
![Page 428: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/428.jpg)
5å`z
æN¼ê
Ïd§·ÏL¦)e¯K¯K(131)Cq)µ
min B(x, θ)s.t. x ∈ intS
(132)
duψ(x)3§31>./¤/p0§ÏdþãæN¯K(132))x(θ)7¹u1SÜ"
I)º´§æN¯K(132)L¡þwå5E´å`z¯K§§å^'5åE,"´§du¼êψ(x)æN^´gÄ¢y§ÏdlO*:w§¦)(132) Ãå¯K5?n"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 414 / 422
![Page 429: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/429.jpg)
5å`z
æN¼ê
Ïd§·ÏL¦)e¯K¯K(131)Cq)µ
min B(x, θ)s.t. x ∈ intS
(132)
duψ(x)3§31>./¤/p0§ÏdþãæN¯K(132))x(θ)7¹u1SÜ"
I)º´§æN¯K(132)L¡þwå5E´å`z¯K§§å^'5åE,"´§du¼êψ(x)æN^´gÄ¢y§ÏdlO*:w§¦)(132) Ãå¯K5?n"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 414 / 422
![Page 430: I Eâ Œ - USTCstaff.ustc.edu.cn/~yangzw/data/course/OR201909.pdf · 2019-10-14 · Outline 1 XØ 2 ‡55y 3 ˜ä†‘z 4 ˜ 5y 5 ı‡55y˜:nØ 6 ˆ 冑z 7 g5y 8 ı‡5 冑z](https://reader033.vdocument.in/reader033/viewer/2022041808/5e55d34dfc52cc78260759d2/html5/thumbnails/430.jpg)
5å`z
æN¼ê
u´§·±ÑæN¼êOÚ½Xeµ
(0) ½Ð©:x(0) ∈ intS , ЩæNÏfθ0 > 0, β ∈ (0, 1), ε > 0.-k := 0.
(1) ±x(k)ЩS:¦)e¯Kµ
min f (x) + θkψ(x)s.t. x ∈ intS
P¦4:x(θk).
(2) eθkψ(x(θk)) < ε, KÊO¿x(θk)¯KCq`)¶ÄK§x(k+1) = x(θk), θk+1 = βθk , -k := k + 1£1(1)Ú"
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 415 / 422
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5å`z
æN¼ê
½½½nnnµµµ θk > θk+1 > 0, Px(θ) = arg minx
B(x, θ), Kk
B(x(θk), θk) ≥ B(x(θk+1), θk+1),
ψ(x(θk)) ≤ ψ(x(θk+1)),
f (x(θk)) ≥ f (x(θk+1)).
[gKµy²þã½n...]
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S:
min f (x)s.t. cE (x) = 0
cI (x) ≥ 0(133)
min f (x)s.t. cE (x) = 0
cI (x)− s = 0s ≥ 0
(134)
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S:
The Karush-Kuhn-Tucker (KKT) conditions for the nonlinearprogram (134) can be written as
∇f (x)− AE (x)Ty − AI (x)Tz = 0Sz− µ1 = 0
cE (x) = 0cI (x)− s = 0
(135)
with µ = 0, together with s ≥ 0, z ≥ 0.
Here AE (x) and AI (x) are the Jacobian matrices of the functions cE (x)and cI (x), respectively, and y and z are their Lagrange multipliers. Wedefine S and Z to be the diagonal matrices whose diagonal entries aregiven by the vectors s and z, respectively, and let 1 = (1, · · · , 1)T .
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S:
Applying Newton’s method to the KKT system (135), in the variablesx, s, y, z, we obtain
∇2xxL −AE (x)T −AI (x)T 00 Z 0 S
AE (x) 0 0 0AI (x) −I 0 0
px
ps
py
pz
= −
∇f (x)− AE (x)Ty − AI (x)Tz
Sz− µ1cE (x)
cI (x)− s
(136)
where L(x, s, y, z) denotes the Lagrange function
L(x, s, y, z) = f (x)− yTcE (x)− zT (cI (x)− s).
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S:
The system (136) is called the primal-dual system. After the stepp = (px ,ps ,py ,pz) has been determined, we compute the new iterate(x+, s+, y+, z+) as
x+ = x + αmaxs px , s+ = s + αmax
s ps ,y+ = y + αmax
z py , z+ = z + αmaxz pz ,
whereαmaxs = maxα ∈ (0, 1] : s + αps ≥ (1− τ)s,
αmaxz = maxα ∈ (0, 1] : z + αpz ≥ (1− τ)z, (137)
with τ ∈ (0, 1) (A typical value of τ is 0.995). The condition (137), calledthe fraction to the boundary rule, prevents the variables s and z fromapproaching their lower bounds of 0 too quickly.
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Ù£5å`z¤
Exercise 7.1: y²(125)¥½Âψ(x, λ)´'uLagrange-Newtoneü¼ê"
Exercise 7.2: y²v¼ê¦)Ø.Cq¯Kkª5"
Exercise 7.3: Ñå`z¯K¿©`5^§¿^u`²O2Lagrange¼ê4:¯K`)d5"
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o(
Ãå`z¯K
- eü- Úî- [Úî- ÝFÝ- &6
å`z¯K
- Lagrange-Newton- ÅÚg5y- v¼ê- ¦fv¼ê- æN¼ê- S:
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 422 / 422
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o(
.é¯KCqx¦)5m²ï
¦)é5ÀJ
Äu¯KµÐ©)E
YZW (USTC) OPERATIONS RESEARCH 2019-09-02 423 / 422