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Index

a^ 169 xc, 25 a M O l CC,6 (A, <, (8»), 280 CO C, 5 (A, <, (8), (g)), 281 coC, 10 Adif),6 co^iC,263 ^Cc(^o), 166 cowC, 260

core C, 18 ^A, 122 Conv(X), 79 y A, 122 ( o j ^

^A' 175 C°°, 20 -6;. 185 C,6 ^1'109 C(X),263

fie, 83 A, 27 )3„,288 A', 28 fi\nO A'A,28 y6M02,103 A^, 29, 265 /Stagr, 69 A^, 30, 265 /6Lgr'83 A3G,31 ,265

Aurr, 81 A4 , 33, 265 )S:u„, 83 A^;, 66 bd C, 6 A ^ , 69 Bx,2 A<",34,266

348 Index

A02 , 35, 266

A " , 34, 265

A'2, 34, 265

A '3 ,34 ,265

A'^, 127

^Ury 133 A^V, 133

Kj^ 135 A„'^, 135

K'T^ 135 9. /(2o) , 26

8(0, xo), 85

9/(zo), 23

9^<^)/(zo), 39

9 f <»'>/(xo), 321

d is t (C, ,C2) , 13

dist (xo, C), 13

dist (x,y),4

dom / , 22

DC(X) ,216

(£>.), 84

(£>,?), 83

(£»A), 122

( S A ) , 122

(/^k), 185

( 5 ^ ) , 185

(Z)J , 323

(£>2), 323

(Z)f), 323

(£>™), 288

( A U T ) , 80

eca C ,10

eco C, 10

ecowC, 261

epi / , 6

£ ' e A ( W ) , 2 6 2

£ 2 ( 1 ^ ) , 262

£CA(W), 260

£:/C(W), 260

Jco? ^ ^

/co(H^), 261

/^^<^\ 35

/ J '37 />^^37

/ ^ ^ 38

/ ; , 3 8

/ ^ ^ , 3 8

/eq, 26

fcqiW). 262

/eqa(Vy), 262 / ^ ^ ^ \ 38 jrMicp)^ 282, 321

/^(xo; X), 24

/ q ,26

/q ,26

/ qca 9 -^ '

/q(A'A),29,266

/q(A^)^ 263

fq(W), 262

/ ^ s ^ 282

/ ^ , 141

/ ^ « , 3 1 1

/ * , 2 1 , 3 6

/ * * , 2 1 , 3 6

^G(-^O) . 85

^ ( / ' < ) , 239

J^(P<), 232

^ ( P ^ , < ) , 241

G ^ 6 3

^o.i/» 7

'WG,.VO. 63

int C, 6

Index 349

Jix,u(x)),296 {Pe),2U

(PI, 169 (^i), 266 (P^), 101 iK2),266 (P,), 80, 130, 244

(^3)'2^6 (/><), 56, 232 '^(^)'260 (F^), 56, 232

iPK.<),24l Pc(xo),40

r, 170, 171

A^ 102, 103

'= ' 22 e(W),262 y^, 175 ^ i ,109 PC, 38 A' , 84 ^, 2 A", 83 p+, 56 surr, 80 /;++, 282

i (A) ,38 R\2 L(A)',38 L, ,84 a(X*,X),7 L",294 cf(X,X*),7 L^,294 SAf),6 L%„, 294 Supp / , 22 Lsurr, 81 supp / , 2

5 G ( / ) , 4 7

M A , 264 M G ( / ) , 101 0 '239

N{C\ Co), 25 u*, 76, 132 Ar,(C; Co), 25 U^j, 17 iv,(C;xo),25 f/|.d,8 N{C; xo), 25 f/|,,, 8

W,90 ^ ' 233 i ^^ 92 S2G,$, 102, 170 ^^(;,Q), 6

^G,w, 103, 171 fi„„109 i;(2), 80, 130,244 OG{XO), 100 V*, , 17

OG(/), 148 Vl„ 8

^Id. 8 (P),47,55 V,67,90 (/>.), 214 ^ ^ 9 2 {(P), (£>)}, 52 {(P), (£>„)}, 288 X*{f,k),\99

350 Index

+,20 +,20 X, 56 x,282 0 ^ 326 0,324 0,281 0,281

abstract convex analysis, 27, 259 addition, 1

lower, 20 upper, 20

adjoint of a mapping, 76, 132 associated pair, 301 axiomatic characterization of:

Fenchel-Moreau conjugations, 280

dual objective functions, 285, 291, 299

the Lagrangian functions associated with -, 285, 292, 299

barrier cone, 63 biconjugate, biconjugate function, 21

Fenchel-Moreau, 36 bipolar theorem, 19

canonical enlargement, 281 cavern, 6 conjugate, conjugate function:

concave, 21 convex, 21 Fenchel, 21 Fenchel-Moreau, 35 ofElQortobi, 282 of Rubinov and §im§ek, 282 of typeLau, 38

conjugation, 35, 281 surrogate, 298

constraint, constraint set, 47 abstract, 55 d.c, 214 essential, 61

general, 71 inessential, 64 primal, 47 structured, 54, 131, 190 surrogate,48, 60, 81, 102, 176

constraint qualification, 50 of Attouch-Brezis, 50 Slater, 57, 78, 236 (/,/)-, 227

constraints multifunction, see perturbation multifunction

coupling function, 35 additive, 307 multiplicative, 304 natural, 35

critical point, 247

d.c. optimization, 213 abstract, 275

deviation, 85 directional derivative, 24 distance

between two subsets, 13 between x and y, 4 from the empty set, 13 fromjco to C, 13

domain, effective domain, 22 dual constraint set, 48, 122, 184 dual objective function, 48, 72, 80,

103, 122, 130, 171, 175, 184, 192, 244

A-, 122, 185 dual of a polarity, 28 dual problem, duality, 46

unperturbational Lagrangian, 48, 50,56, 127, 135, 189, 198

perturbational Lagrangian, 72, 130 unperturbational surrogate, 60, 83,

102, 122, 170, 175, 185 perturbational surrogate, 80, 132 for d.c. infimization, 218, 226,

236, 244 for abstract quasi-convex

supremization, 267, 270

Index 351

for abstract reverse convex infimization 271, 273

for abstract d.c. infimization, 275 associated with a Lagrangian

function, 52 for infimization problems

involving maximum operators, 248, 252

TT-, 84

6>-, 83 A-, 122 PES-, perturbational extended

surrogate dual, 297 decomposed, 297

(Wz)-, 298 anomalous, 323 m-Lagrangian, 288 perturbational conjugate, 297 quasi-convex, 80

duality: for best approximation, 40 for worst approximation, 87 strong, 51, 69, 73, 83, 180,288 weak, 51, 69, 73, 114,178,181 V-, 280, 325 *-, associated with a binary

operation, 258, 296 (*, 5)-, 282, 326

duality equality: Lagrangian, 128 strong, 68 surrogate, 128 weak, 69, 178

duality gap, 51 duality inequality, 49

element: of best approximation, 40 of ^-approximation, 6:-best

approximation, 283 of worst approximation, 85

environment, 301 epigraph, 6 epigraphic methods, 48, 50 8-solution, £-optimal solution, 284

£-subdifferential, 25 excess, see deviation extremality relations, 247

farthest point, see element of worst approximation

Fenchel equality, 23 Fenchel inequality, 23 function:

Of-Holder continuous, 262 with constant N, 262

A'A-quasi-convex, 29 abstract convex, 260 abstract quasi-convex, 260 abstract d.c, 275 additive, 7 affine, 7 best, 207, 313 concave, 21 continuous, 6

at a point, 6 convex, 21 d.c, diff-convex, 213 elementary, 260 evenly quasi-coaffine, 27 evenly quasi-convex, 26

strongly, 139 homogeneous, 6 indicator, 24 Lagrange-type, 309 linear, 6 lower semicontinuous, 6

at a point, 5 A1-quasi-convex, 263 maximal, 45 min-type, 305

additive, 307 optimal, optimal dual solution, 99,

148, 165 optimal value, marginal, 74, 295 positively homogeneous, 21 pseudoconjugate, 38 proper, 21 quasi-concave, 26 quasi-convex, 26

352 Index

evenly, 26 strongly evenly, 139 strongly, 142

/?-evenly convex, 127 regular weak separation, RWS,

303 regular with respcect io(p, 116 regular with respect to if, 268 strictly increasing along segments

from 0, 142 subdifferentiable, 24 sublinear, 21 tangentially convex, 312 upper semicontinuous, 6

at a point, 6 W-convex, 261 VK-evenly quasi-coaffine, 262 V^-evenly quasi-convex, 262 W-quasi-convex, 262

group: lattice ordered, 281

conditionally complete, 281

half-space: closed, 8 open, 8 quasi-support, 16

half-space theorem: for infimization, 65 for quasi-convex supremization,

116, 119 for reverse convex infimization,

183 hull of a function:

A'A-quasi-convex, 29 evenly quasi-coaffine, 27 evenly quasi-convex, 26 lower semicontinuous convex, 22 A^-quasi-convex, 242 quasi-convex, 26

lower semi-continuous quasi-convex, 33

W-convex, 241 W-evenly quasi-coaffine, 242

W-evenly quasi-convex, 241 W-quasi-convex, 241

hull of a set: A A-convex, 28 closed convex, 10 convex, 5 evenly coaffine, 10 evenly convex, 10 A^-convex, 263 V^-convex, 260 W-evenly coaffine, 260 ly-evenly convex, 260

hull operator, 28 hyperplane, 8

best, 207 optimal, 99, 166 quasi-support, 11 support, 11

hyperplane theorem (of surrogate duahty):

for convex infimization, 63 for quasi-convex supremization,

163 for reverse convex infimization,

172

image set, 287, 303 incidence triple, 285 inequality constraint, 56, 198, 225,

232 inf-sup theorem:

of Moreau, 27 of Sion-Kneser-Fan, 27

infimal convolution, 236 inner product, 4 instance of a problem, 300 inversion, 314

Lagrangian, Lagrangian function: associated with a perturbation, 244 augmented, 51 for infimization, 52 for perturbed infimization, 73 for perturbed supremization, 130 for d.c. infimization, 244

Index 353

of Kurcyusz, 294 of type I, 322 of type II, 322 quasi-convex, 81 surrogate, 269

Lagrangian duality theorem, 48, 81, 127,135,189,228,244

general, 287 lattice:

conditionally complete, 281 level set, 6 linearization, 222

mapping: antitone, 28 convex, 54

vector-valued, 320 min-max equality:

combinatorial, 285 of all-cardinality

covering-packing type, 286 for A-cover packings, 286 for ^-colorings, 286 for weighted A-covers, 286 for weighted 5-packings, 286

minimax theorem, 27 minimum principle of Bauer, 311 Moore-Smith closure operator, 29 multiplication:

lower, 282 upper, 282 with a scalar, 1

nearest point, see element of best approximation

norm, 2 normal cone, 24

extended, 25

objective function, 47 open ball, 5 opposite element, 1 optimization problem, xi

convex, xi anticonvex, xi

convex-anticonvex, xi abstract d.c, 259 d.c.,213 extremal, 301 involving maximum operators,

247

penalize, 50 penalty term, 50 perturbation:

horizontal, 80, 293 normal, 292 vertical, 80, 293

perturbation function, parameterization, 72

objective-function separated, 298 perturbation multifunction, 294 PMP, Pontryagin maximum

principle, 295 abstract, 295

PMP„,296 polar set, 19 polarity, 27 primal parameters, 300 primal problem:

of infimization, 47 perturbed, parameterized, 72, 80,

129, 244, 293 structured, 54

constrained, 47 unconstrained, 47 of convex infimization, 47

structured, 55 of quasi-convex infimization, 47 of quasi-convex supremization,

101 of reverse convex infimization,

169 of d.c. infimization, diff-convex,

213 unconstrained, 213 constrained, d.c. constrained,

214 with a d.c. inequality constraint,

225

354 Index

with finitely many d.c. inequality constraints, 232

primal-dual pair, 51, 288 program, programming problem, 55

convex, 55 linear, 55

abstract, 301 algebraic, 301

mathematical, 132 minimization, in an environment,

301 quasi-convex, 82

quasi-conjugate, quasi-conjugate function:

of Greenberg-Pierskalla, 37 second, 37

normalized, 37 ofThach, 141,310

quasi-subdifferential, 300 quasi-support half-space, 16 quasi-support hyperplane, 11

reduction principle, 44, 64, 88, 106, 123, 125, 155, 173, 188

reverse convex best approximation, 153

saddle-value, 293 scheme of formal replacements, 299 semiconjugate, 38 separation:

min-type, 305 nonlinear, 287 by a function, 8

strict, 8 strong, 279

by a hyperplane, 8 strict, 8

by a set, 263 set:

abstract convex, 260 A^A-convex, 29 Chebyshev, 153 comprehensive, 307

conical, 266 conormal, reverse normal, 304 convex, 5 evenly coaffine, 10 evenly convex, 10 linearly open, 18 7V(-convex, 263 normal, 304 of ^-normal directions, 25 of extended £-normal directions,

25 of perturbations, of parameters,

72, 129, 244 polar, 19 proximinal, 149 semi-Chebyshev, 313 support, 22, 280

(X*, /?)-, 22 /^-evenly convex, 126 M -convex, 260 \y-evenly coaffine, 260 ly-evenly convex, 260

Slater condition, see Slater constraint qualification

solution, optimal solution: for quasi-convex minimization, 47 for quasi-convex maximization,

101, 137 for reverse convex minimization,

169, 203 for d.c. minimization, 221 global, 284,318 local, 284, 318 primal, 54, 75 dual, optimal dual, 54, 75 of (D^), 288

space: Banach, 2 conjugate, 7 Euclidean, 2 Hilbert, 4 linear, 2 locally convex, 5 normed linear, 2

complete, 2

Index 355

topological linear, 5 strong CHIP, strong conical hull

intersection property, 28 subdifferential, 22

of Balder, 300 ofThach, 141,283 surrogate, 300 with respect to a conjugation, 39,

283 with respect to a conjugation of

type Lau, 283 with respect to a primal-dual pair

of optimization problems, 300 (*,^K326

substitution method, 64, 128, 189, 198

subtraction, 2 system, 54

canonical, 323 constant time linear control, see

linear system convex, 54 equlibrium, 323, linear, 54

constitutively, 323 fully, 295 geometrically, 295

polar, 295

potential, 323 strictly reflexive, 323

target set, 54, 131 non-one point, 295

theorem of: separation, 8 strict separation, 9 Fenchel-Moreau, 22 Moreau and Pshenichnyi, 24 Moreau-Rockafellar, 32 Fenchel-Rockafellar, 76 Pshenichnyi-Rockafellar, 52

topology: general, 5 Hausdorff, 5 norm, 5 weak, 7 weak*, 7

unit ball, 2

value, optimal value, 51 surrogate dual, 80

vector operations, 1, 7

weak alternative theorem, 304