positive polynomials and convergence of lp and sdp relaxationsschweigh/presentations/control.pdf ·...
TRANSCRIPT
![Page 1: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/1.jpg)
Positive polynomials andconvergence of LP and SDP
relaxations
Markus Schweighofer
Universitat Konstanz
ETHZ¨urich, May 31, 2005
1
![Page 2: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/2.jpg)
Notation for the whole talk
![Page 3: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/3.jpg)
Notation for the whole talk
• X1, . . . , Xn variables
![Page 4: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/4.jpg)
Notation for the whole talk
• X1, . . . , Xn variables
• X := X1 when n = 1
![Page 5: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/5.jpg)
Notation for the whole talk
• X1, . . . , Xn variables
• X := X1 when n = 1, (X, Y ) := (X1, X2) when n = 2, . . .
![Page 6: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/6.jpg)
Notation for the whole talk
• X1, . . . , Xn variables
• X := X1 when n = 1, (X, Y ) := (X1, X2) when n = 2, . . .
• R[X1, . . . , Xn] polynomial ring
![Page 7: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/7.jpg)
Notation for the whole talk
• X1, . . . , Xn variables
• X := X1 when n = 1, (X, Y ) := (X1, X2) when n = 2, . . .
• R[X1, . . . , Xn] polynomial ring
• f ∈ R[X1, . . . , Xn] an arbitrary polynomial
![Page 8: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/8.jpg)
Notation for the whole talk
• X1, . . . , Xn variables
• X := X1 when n = 1, (X, Y ) := (X1, X2) when n = 2, . . .
• R[X1, . . . , Xn] polynomial ring
• f ∈ R[X1, . . . , Xn] an arbitrary polynomial
• g1, . . . , gm ∈ R[X1, . . . , Xn] polynomials defining. . .
![Page 9: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/9.jpg)
Notation for the whole talk
• X1, . . . , Xn variables
• X := X1 when n = 1, (X, Y ) := (X1, X2) when n = 2, . . .
• R[X1, . . . , Xn] polynomial ring
• f ∈ R[X1, . . . , Xn] an arbitrary polynomial
• g1, . . . , gm ∈ R[X1, . . . , Xn] polynomials defining. . .
• . . . the set S := {x ∈ Rn | g1(x) ≥ 0, . . . , gm(x) ≥ 0}
2
![Page 10: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/10.jpg)
Notation for the whole talk
• X1, . . . , Xn variables
• X := X1 when n = 1, (X, Y ) := (X1, X2) when n = 2, . . .
• R[X] := R[X1, . . . , Xn] polynomial ring
• f ∈ R[X] an arbitrary polynomial
• g1, . . . , gm ∈ R[X] polynomials defining. . .
• . . . the set S := {x ∈ Rn | g1(x) ≥ 0, . . . , gm(x) ≥ 0}
3
![Page 11: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/11.jpg)
Optimization
We consider the problem of minimizing f on S.
![Page 12: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/12.jpg)
Optimization
We consider the problem of minimizing f on S. So we want tocompute numerically the infimum
f∗ := inf{f(x) | x ∈ S} ∈ R ∪ {±∞}
![Page 13: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/13.jpg)
Optimization
We consider the problem of minimizing f on S. So we want tocompute numerically the infimum
f∗ := inf{f(x) | x ∈ S} ∈ R ∪ {±∞}
and, if possible, a minimizer, i.e., an element of the set
S∗ := {x∗ ∈ S | f(x∗) ≤ f(x) for all x ∈ S}.
4
![Page 14: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/14.jpg)
L P
![Page 15: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/15.jpg)
Linear Programming
minimize f(x)
subject to x ∈ Rn
g1(x) ≥ 0...
gm(x) ≥ 0
where all polynomials f and gi are linear, i.e.,their degree is ≤ 1. In particular, S ⊆ Rn is a polyhedron.
5
![Page 16: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/16.jpg)
Linear Programming
minimize f(x)
subject to x ∈ Rng1(x)
. . .
gm(x)
is psd
where all polynomials f and gi are linear, i.e.,their degree is ≤ 1. In particular, S ⊆ Rn is a polyhedron.
5
![Page 17: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/17.jpg)
S D P
minimize f(x)
subject to x ∈ Rng11(x) . . . g1m(x)
.... . .
...
. . . gmm(x)
is psd
where all polynomials f and gij are linear, i.e.,their degree is ≤ 1.
![Page 18: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/18.jpg)
Semidefinite Programming
minimize f(x)
subject to x ∈ Rng11(x) . . . g1m(x)
.... . .
...
. . . gmm(x)
is psd
where all polynomials f and gij are linear, i.e.,their degree is ≤ 1.
![Page 19: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/19.jpg)
Positive semidefinite matrices and families of vectors
Proposition. A real symmetric k × k matrix is psd if and only ifthere are vectors v1, . . . , vk ∈ Rk such that
M =
〈v1, v1〉 . . . 〈v1, vk〉
......
〈vk, v1〉 . . . 〈vk, vk〉
.
6
![Page 20: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/20.jpg)
Duality
• Every linear program (P ) has an optimal value P ∗.
![Page 21: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/21.jpg)
Duality
• Every linear program (P ) has an optimal value P ∗.
• To every linear program (P ), one can define a dualprogram (D) which is again a linear program.
![Page 22: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/22.jpg)
Duality
• Every linear program (P ) has an optimal value P ∗.
• To every linear program (P ), one can define a dualprogram (D) which is again a linear program.
• If (P ) is a minimization problem, then (D) is a maximizationproblem and weak duality holds:
D∗ ≤ P ∗
![Page 23: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/23.jpg)
Duality
• Every linear program (P ) has an optimal value P ∗.
• To every linear program (P ), one can define a dualprogram (D) which is again a linear program.
• If (P ) is a minimization problem, then (D) is a maximizationproblem and weak duality holds:
D∗ ≤ P ∗
• Strong duality is desired and often holds:
D∗ = P ∗
7
![Page 24: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/24.jpg)
Duality
• Every semidefinite program (P ) has an optimal value P ∗.
• To every semidefinite program (P ), one can define a dualprogram (D) which is again a semidefinite program.
• If (P ) is a minimization problem, then (D) is a maximizationproblem and weak duality holds:
D∗ ≤ P ∗
• Strong duality is desired and often holds:
D∗ = P ∗
7
![Page 25: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/25.jpg)
minimize2d∑
i=0
aixi
subject to x ∈ R
where a0, . . . , a2d ∈ R.
8
![Page 26: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/26.jpg)
minimize2d∑
i=0
aixi
subject to x ∈ R
Note that
1 x x2 . . . xd
x x2 . . . . . .
x2 . . . . . ....
. . .
xd x2d
is psd
where a0, . . . , a2d ∈ R.
8
![Page 27: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/27.jpg)
minimize2d∑
i=0
aixi
subject to x ∈ R
Note that
1 X X2 . . . Xd
1
X
X2
...
Xd
1 x x2 . . . xd
x x2 . . . . . .
x2 . . . . . ....
. . .
xd x2d
is psd
where a0, . . . , a2d ∈ R.
8
![Page 28: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/28.jpg)
(P ) minimize2d∑
i=1
aiyi + a0
subject to y ∈ R2d
1 X X2 . . . Xd
1
X
X2
...
Xd
1 y1 y2 yd
y1 y2. . . . . .
y2. . . . . .
.... . .
yd y2d
is psd
where a0, . . . , a2d ∈ R.
8
![Page 29: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/29.jpg)
Set f :=∑2d
i=0 aiXi and denote by (D) the semidefinite program
dual to (P ).
![Page 30: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/30.jpg)
Set f :=∑2d
i=0 aiXi and denote by (D) the semidefinite program
dual to (P ). Then it is clear that
D∗ ≤ P ∗ ≤ f∗.
![Page 31: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/31.jpg)
Set f :=∑2d
i=0 aiXi and denote by (D) the semidefinite program
dual to (P ). Then it is clear that
D∗ ≤ P ∗ ≤ f∗.
It turns out that (D) can be interpreted as:
(D) maximize µ
subject to f − µ is sos
![Page 32: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/32.jpg)
Set f :=∑2d
i=0 aiXi and denote by (D) the semidefinite program
dual to (P ). Then it is clear that
D∗ ≤ P ∗ ≤ f∗.
It turns out that (D) can be interpreted as:
(D) maximize µ
subject to f − µ is sos
Proposition. For every p ∈ R[X],
p ≥ 0 on R =⇒ p is a sum of two squares in R[X].
![Page 33: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/33.jpg)
Set f :=∑2d
i=0 aiXi and denote by (D) the semidefinite program
dual to (P ). Then it is clear that
D∗ ≤ P ∗ ≤ f∗.
It turns out that (D) can be interpreted as:
(D) maximize µ
subject to f − µ is sos
Proposition. For every p ∈ R[X],
p ≥ 0 on R =⇒ p is a sum of two squares in R[X].
Corollary.D∗ = P ∗ = f∗
9
![Page 34: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/34.jpg)
minimize∑i+j≤4
aijxiyj
subject to x, y ∈ R
where aij ∈ R (i + j ≤ 4).
10
![Page 35: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/35.jpg)
minimize∑i+j≤4
aijxiyj
subject to x, y ∈ R
Note that
1 x y x2 xy y2
x x2 xy x3 x2y xy2
y xy y2 x2y xy2 y3
x2 x3 x2y x4 x3y x2y2
xy x2y xy2 x3y x2y2 xy3
y2 xy2 y3 x2y2 xy3 y4
is psd
where aij ∈ R (i + j ≤ 4).
10
![Page 36: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/36.jpg)
minimize∑i+j≤4
aijxiyj
subject to x, y ∈ R
Note that
1 X Y X2 XY Y 2
1
X
Y
X2
XY
Y 2
1 x y x2 xy y2
x x2 xy x3 x2y xy2
y xy y2 x2y xy2 y3
x2 x3 x2y x4 x3y x2y2
xy x2y xy2 x3y x2y2 xy3
y2 xy2 y3 x2y2 xy3 y4
is psd
where aij ∈ R (i + j ≤ 4).
10
![Page 37: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/37.jpg)
(P ) minimize∑
1≤i+j≤4
aijyij + a00
subject to yij ∈ R (1 ≤ i + j ≤ 4)
1 X Y X2 XY Y 2
1
X
Y
X2
XY
Y 2
1 y10 y01 y20 y11 y02
y10 y20 y11 y30 y21 y12
y01 y11 y02 y21 y12 y03
y20 y30 y21 y40 y31 y22
y11 y21 y12 y31 y22 y13
y02 y12 y03 y22 y13 y04
is psd
where aij ∈ R (i + j ≤ 4).
10
![Page 38: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/38.jpg)
Set f :=∑
i+j≤4 aijXij and denote by (D) the semidefinite
program dual to (P ).
![Page 39: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/39.jpg)
Set f :=∑
i+j≤4 aijXij and denote by (D) the semidefinite
program dual to (P ). Then it is clear that
D∗ ≤ P ∗ ≤ f∗.
![Page 40: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/40.jpg)
Set f :=∑
i+j≤4 aijXij and denote by (D) the semidefinite
program dual to (P ). Then it is clear that
D∗ ≤ P ∗ ≤ f∗.
It turns out that (D) can be interpreted as:
(D) maximize µ
subject to f − µ is sos
![Page 41: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/41.jpg)
Set f :=∑
i+j≤4 aijXij and denote by (D) the semidefinite
program dual to (P ). Then it is clear that
D∗ ≤ P ∗ ≤ f∗.
It turns out that (D) can be interpreted as:
(D) maximize µ
subject to f − µ is sos
Theorem (Hilbert). For every p ∈ R[X, Y ] of degree ≤ 4,
p ≥ 0 on R2 =⇒ p is a sum of three squares in R[X, Y ].
David Hilbert: Ueber die Darstellung definiter Formen als Summevon FormenquadratenMath. Ann. XXXII 342-350 (1888)http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN235181684_0032
11
![Page 42: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/42.jpg)
Set f :=∑
i+j≤4 aijXij and denote by (D) the semidefinite
program dual to (P ). Then it is clear that
D∗ ≤ P ∗ ≤ f∗.
It turns out that (D) can be interpreted as:
(D) maximize µ
subject to f − µ is sos
Theorem (Hilbert). For every p ∈ R[X, Y ] of degree ≤ 4,
p ≥ 0 on R2 =⇒ p is a sum of three squares in R[X, Y ].
Corollary. D∗ = P ∗ = f∗
David Hilbert: Ueber die Darstellung definiter Formen als Summevon FormenquadratenMath. Ann. XXXII 342-350 (1888)http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN235181684_0032
11
![Page 43: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/43.jpg)
The Motzkin polynomial
• Unfortunately, not every polynomial p ∈ R[X1, . . . , Xn] withp ≥ 0 on Rn is a sum of squares of polynomials.
![Page 44: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/44.jpg)
The Motzkin polynomial
• Unfortunately, not every polynomial p ∈ R[X1, . . . , Xn] withp ≥ 0 on Rn is a sum of squares of polynomials.
• The first explicit example was found in 1967 by Motzkin:
p := X4Y 2 + X2Y 4 − 3X2Y 2 + 1
![Page 45: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/45.jpg)
The Motzkin polynomial
• Unfortunately, not every polynomial p ∈ R[X1, . . . , Xn] withp ≥ 0 on Rn is a sum of squares of polynomials.
• The first explicit example was found in 1967 by Motzkin:
p := X4Y 2 + X2Y 4 − 3X2Y 2 + 1
• In fact, there is even no N ∈ N such that p + N is a sum ofsquares in R[X, Y, Z].
![Page 46: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/46.jpg)
The Motzkin polynomial
• Unfortunately, not every polynomial p ∈ R[X1, . . . , Xn] withp ≥ 0 on Rn is a sum of squares of polynomials.
• The first explicit example was found in 1967 by Motzkin:
p := X4Y 2 + X2Y 4 − 3X2Y 2 + 1
• In fact, there is even no N ∈ N such that p + N is a sum ofsquares in R[X, Y, Z].
• Described method always yields certified lower bounds, butthey might by −∞:
−∞ ≤ D∗ = P ∗ ≤ f∗
![Page 47: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/47.jpg)
The Motzkin polynomial
• Unfortunately, not every polynomial p ∈ R[X1, . . . , Xn] withp ≥ 0 on Rn is a sum of squares of polynomials.
• The first explicit example was found in 1967 by Motzkin:
p := X4Y 2 + X2Y 4 − 3X2Y 2 + 1
• In fact, there is even no N ∈ N such that p + N is a sum ofsquares in R[X, Y, Z].
• Described method always yields certified lower bounds, butthey might by −∞:
−∞ ≤ D∗ = P ∗ ≤ f∗
• But there are a lot of remedies...
12
![Page 48: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/48.jpg)
Case where S is compact.
For simplicity, we suppose m = 1 and write g := g1 (technicaldifficulties which are however not very serious otherwise), i.e.
S = {x ∈ Rn | g(x) ≥ 0}.
![Page 49: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/49.jpg)
Case where S is compact.
For simplicity, we suppose m = 1 and write g := g1 (technicaldifficulties which are however not very serious otherwise), i.e.
S = {x ∈ Rn | g(x) ≥ 0}.
Now we get a sequence (Pk)2k≥d of relaxations such that
D∗k ≤ P ∗
k ≤ f∗ and limk→∞
D∗k = lim
k→∞P ∗
k = f∗.
Jean Lasserre: Global optimization with polynomials and theproblem of momentsSIAM J. Optim. 11, No. 3, 796–817 (2001)
13
![Page 50: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/50.jpg)
minimize∑|α|≤d
aαxα11 · · ·xαn
n
subject to x ∈ S
where k ∈ N, 2k ≥ d, aα ∈ R (|α| ≤ k).
14
![Page 51: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/51.jpg)
minimize∑|α|≤d
aαxα11 · · ·xαn
n
subject to x ∈ S
Note that
1 x1 . . . xk
n
x1
......
xkn . . . . . . . . . x2k
n
“localization
matrix”
is psd
where k ∈ N, 2k ≥ d, aα ∈ R (|α| ≤ k).
14
![Page 52: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/52.jpg)
minimize∑|α|≤d
aαxα11 · · ·xαn
n
subject to x ∈ S
Note that
1 X1 . . . Xkn
1
X1
...
Xkn
1 x1 . . . xk
n
x1
......
xkn . . . . . . . . . x2k
n
“localization
matrix”
is psd
where k ∈ N, 2k ≥ d, aα ∈ R (|α| ≤ k).
14
![Page 53: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/53.jpg)
(Pk) minimize∑
1≤|α|≤d
aαyα + a0
subject to yα ∈ R (|α| ≤ k)
1 X1 . . . Xkn
1
X1
...
Xkn
1 y10...0 . . .
y10...0
...
“localization
matrix”
is psd
where k ∈ N, 2k ≥ d, aα ∈ R (|α| ≤ k).
14
![Page 54: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/54.jpg)
Case where S is compact.
Theorem (Schmudgen, Putinar, ...) If f > 0 on S, then f = s + gt
for sums of squares s, t in R[X1, . . . , Xn].
![Page 55: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/55.jpg)
Case where S is compact.
Theorem (Schmudgen, Putinar, ...) If f > 0 on S, then f = s + gt
for sums of squares s, t in R[X1, . . . , Xn].
Corollary (Lasserre). (D∗k)k∈N and (P ∗
k )k∈N are increasingsequences that converge to f∗ and satisfy D∗
k ≤ P ∗k ≤ f∗. How fast?
![Page 56: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/56.jpg)
Case where S is compact.
Theorem (Schmudgen, Putinar, ...) If f > 0 on S, then f = s + gt
for sums of squares s, t in R[X1, . . . , Xn].
Corollary (Lasserre). (D∗k)k∈N and (P ∗
k )k∈N are increasingsequences that converge to f∗ and satisfy D∗
k ≤ P ∗k ≤ f∗. How fast?
Theorem. There exists C ∈ N depending on f and g and c ∈ Ndepending on g such that
f∗ −D∗k ≤
Cc√
kfor big k.
On the complexity of Schmudgen’s PositivstellensatzJournal of Complexity 20, No. 4, 529—543 (2004)
Optimization of polynomials on compact semialgebraic setsSIAM Journal on Optimization 15, No. 3, 805-825 (2005)
15
![Page 57: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/57.jpg)
Further properties of the method for compact S
• Feasible solutions of (Dk) are certified lower bounds of f∗.
![Page 58: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/58.jpg)
Further properties of the method for compact S
• Feasible solutions of (Dk) are certified lower bounds of f∗.
• Method converges from below to f∗.
![Page 59: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/59.jpg)
Further properties of the method for compact S
• Feasible solutions of (Dk) are certified lower bounds of f∗.
• Method converges from below to f∗.
• Method converges to unique minimizers.
Optimization of polynomials on compact semialgebraic setsSIAM Journal on Optimization 15, No. 3, 805-825 (2005)
16
![Page 60: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/60.jpg)
Further properties of the method for compact S
• Feasible solutions of (Dk) are certified lower bounds of f∗.
• Method converges from below to f∗.
• Method converges to unique minimizers. Disadvantage:Possibly from outside the set S.
Optimization of polynomials on compact semialgebraic setsSIAM Journal on Optimization 15, No. 3, 805-825 (2005)
16
![Page 61: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/61.jpg)
Further properties of the method for compact S
• Feasible solutions of (Dk) are certified lower bounds of f∗.
• Method converges from below to f∗.
• Method converges to unique minimizers. Disadvantage:Possibly from outside the set S.
• If there is a unique minimizer and it lies in the interior of S,
Optimization of polynomials on compact semialgebraic setsSIAM Journal on Optimization 15, No. 3, 805-825 (2005)
16
![Page 62: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/62.jpg)
Further properties of the method for compact S
• Feasible solutions of (Dk) are certified lower bounds of f∗.
• Method converges from below to f∗.
• Method converges to unique minimizers. Disadvantage:Possibly from outside the set S.
• If there is a unique minimizer and it lies in the interior of S,then the method produces a sequence of intervals containing f∗
whose endpoints converge to f∗.
Optimization of polynomials on compact semialgebraic setsSIAM Journal on Optimization 15, No. 3, 805-825 (2005)
16
![Page 63: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/63.jpg)
Implementations
• Henrion and Lasserre: GloptiPolyhttp://www.laas.fr/~henrion/software/gloptipoly/
• Prajna, Papachristodoulou, Parrilo: SOSTOOLShttp://control.ee.ethz.ch/~parrilo/sostools/
• Both use the free SeDuMi solver by Jos Sturm
• But they need MATLAB and the MATLAB Symbolic Toolbox
17
![Page 64: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/64.jpg)
Example: The maximum cut problem
Given a graph, i.e., an n ∈ N (number of nodes) and a set
E ⊆ {(i, j) ∈ {1, . . . , n}2 | i < j}
(of edges),
![Page 65: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/65.jpg)
Example: The maximum cut problem
Given a graph, i.e., an n ∈ N (number of nodes) and a set
E ⊆ {(i, j) ∈ {1, . . . , n}2 | i < j}
(of edges), find the maximum cut value, i.e., the maximal possiblenumber of edges that connect nodes with different signs when eachnode is assigned a sign + or −.
![Page 66: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/66.jpg)
Example: The maximum cut problem
Given a graph, i.e., an n ∈ N (number of nodes) and a set
E ⊆ {(i, j) ∈ {1, . . . , n}2 | i < j}
(of edges), find the maximum cut value, i.e., the maximal possiblenumber of edges that connect nodes with different signs when eachnode is assigned a sign + or −.
maximize∑
(i,j)∈E
12(1− xixj)
subject to x2i = 1 for all i ∈ {1, . . . , n}
18
![Page 67: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/67.jpg)
MAXCUT
maximize∑
(i,j)∈E
12(1− xixj)
subject to x ∈ {−1, 1}n
19
![Page 68: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/68.jpg)
MAXCUT
maximize∑
(i,j)∈E
12(1− xixj)
subject to x ∈ {−1, 1}n
Note that
1 x1x2 . . . x1xn
x2x1 1 x2xn
.... . .
...
xnx1 . . . . . . . . . . 1
is psd
19
![Page 69: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/69.jpg)
MAXCUT
maximize∑
(i,j)∈E
12(1− xixj)
subject to x ∈ {−1, 1}n
Note that
X1 . . . . . . . . . Xn
X1
...
...
Xn
1 x1x2 . . . x1xn
x2x1 1 x2xn
.... . .
...
xnx1 . . . . . . . . . . 1
is psd
19
![Page 70: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/70.jpg)
First MAXCUT relaxation
(P1) maximize∑
(i,j)∈E
12(1− yij)
subject to yij ∈ R (1 ≤ i < j ≤ n)
X1 . . . . . . . . . Xn
X1
...
...
Xn
1 y12 . . . y1n
y12 1 y2n
.... . .
...
y1n . . . . . . . . . . 1
is psd
19
![Page 71: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/71.jpg)
MAXCUT
maximize∑
(i,j)∈E
12(1− xixj)
subject to x ∈ {−1, 1}n
20
![Page 72: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/72.jpg)
MAXCUT
maximize∑
(i,j)∈E
12(1− xixj)
subject to x ∈ {−1, 1}n
Note that
1 x1x2 . . . . . . . . . . . .
x2x1 1...
. . ....
. . .
1
is psd
20
![Page 73: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/73.jpg)
MAXCUT
maximize∑
(i,j)∈E
12(1− xixj)
subject to x ∈ {−1, 1}n
Note that
1 X1X2 X1X3 . . . Xn−1Xn
1
X1X2
X1X3
...
Xn−1Xn
1 x1x2 . . . . . . . . . . . .
x2x1 1...
. . ....
. . .
1
is psd
20
![Page 74: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/74.jpg)
Second MAXCUT relaxation
(P2) maximize∑
(i,j)∈E
12(1− yij)
subject to yij ∈ R (1 ≤ i < j ≤ n)
1 X1X2 X1X3 . . . Xn−1Xn
1
X1X2
X1X3
...
Xn−1Xn
1 y12 . . . . . . . . . . . .
y12 1...
. . ....
. . .
1
is psd
20
![Page 75: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/75.jpg)
• The maximum cut problem is NP–complete
![Page 76: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/76.jpg)
• The maximum cut problem is NP–complete
• The first relaxation gives a polynomial time algorithm whichoverestimates the maximum cut value at most by a factor of≈ 1.1382.
![Page 77: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/77.jpg)
• The maximum cut problem is NP–complete
• The first relaxation gives a polynomial time algorithm whichoverestimates the maximum cut value at most by a factor of≈ 1.1382.
• The first relaxation is the famous algorithm of Goemans andWilliamson.
![Page 78: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/78.jpg)
• The maximum cut problem is NP–complete
• The first relaxation gives a polynomial time algorithm whichoverestimates the maximum cut value at most by a factor of≈ 1.1382.
• The first relaxation is the famous algorithm of Goemans andWilliamson. From no polynomial algorithm it is known that ithas a better approximation ratio.
![Page 79: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/79.jpg)
• The maximum cut problem is NP–complete
• The first relaxation gives a polynomial time algorithm whichoverestimates the maximum cut value at most by a factor of≈ 1.1382.
• The first relaxation is the famous algorithm of Goemans andWilliamson. From no polynomial algorithm it is known that ithas a better approximation ratio. Existence of such analgorithm with ratio < 1.0625 implies P = NP (Hastad).
![Page 80: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/80.jpg)
• The maximum cut problem is NP–complete
• The first relaxation gives a polynomial time algorithm whichoverestimates the maximum cut value at most by a factor of≈ 1.1382.
• The first relaxation is the famous algorithm of Goemans andWilliamson. From no polynomial algorithm it is known that ithas a better approximation ratio. Existence of such analgorithm with ratio < 1.0625 implies P = NP (Hastad).
• Solving the second relaxation is a polynomial time algorithmwhich yields the exact value for all planar graphs (consequenceof results of Seymour, Barahona, Mahjoub),
![Page 81: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/81.jpg)
• The maximum cut problem is NP–complete
• The first relaxation gives a polynomial time algorithm whichoverestimates the maximum cut value at most by a factor of≈ 1.1382.
• The first relaxation is the famous algorithm of Goemans andWilliamson. From no polynomial algorithm it is known that ithas a better approximation ratio. Existence of such analgorithm with ratio < 1.0625 implies P = NP (Hastad).
• Solving the second relaxation is a polynomial time algorithmwhich yields the exact value for all planar graphs (consequenceof results of Seymour, Barahona, Mahjoub), and is conjecturedto improve over the GW–algorithm.
![Page 82: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/82.jpg)
• The maximum cut problem is NP–complete
• The first relaxation gives a polynomial time algorithm whichoverestimates the maximum cut value at most by a factor of≈ 1.1382.
• The first relaxation is the famous algorithm of Goemans andWilliamson. From no polynomial algorithm it is known that ithas a better approximation ratio. Existence of such analgorithm with ratio < 1.0625 implies P = NP (Hastad).
• Solving the second relaxation is a polynomial time algorithmwhich yields the exact value for all planar graphs (consequenceof results of Seymour, Barahona, Mahjoub), and is conjecturedto improve over the GW–algorithm.
• The n–th relaxation yields the exact maximum cut value.
21
![Page 83: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/83.jpg)
Exactness of the n-th MAXCUT relaxation
Proposition. Suppose p ∈ R[X1, . . . , Xn] such that
p ≥ 0 on {−1, 1}n.
Then f is a square modulo the ideal
I := (X21 − 1, . . . , X2
n − 1) ⊆ R[X1, . . . , Xn].
![Page 84: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/84.jpg)
Exactness of the n-th MAXCUT relaxation
Proposition. Suppose p ∈ R[X1, . . . , Xn] such that
p ≥ 0 on {−1, 1}n.
Then f is a square modulo the ideal
I := (X21 − 1, . . . , X2
n − 1) ⊆ R[X1, . . . , Xn].
Proof by algebra. By chinese remainder theorem
R[X1, . . . , Xn]/I ∼= R{−1,1}n ∼= R2n
.
![Page 85: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/85.jpg)
Exactness of the n-th MAXCUT relaxation
Proposition. Suppose p ∈ R[X1, . . . , Xn] such that
p ≥ 0 on {−1, 1}n.
Then f is a square modulo the ideal
I := (X21 − 1, . . . , X2
n − 1) ⊆ R[X1, . . . , Xn].
Proof by algebra. By chinese remainder theorem
R[X1, . . . , Xn]/I ∼= R{−1,1}n ∼= R2n
.
Proof by algebraic geometry. I is a zero-dimensional radical ideal.
![Page 86: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/86.jpg)
Exactness of the n-th MAXCUT relaxation
Proposition. Suppose p ∈ R[X1, . . . , Xn] such that
p ≥ 0 on {−1, 1}n.
Then f is a square modulo the ideal
I := (X21 − 1, . . . , X2
n − 1) ⊆ R[X1, . . . , Xn].
Proof by algebra. By chinese remainder theorem
R[X1, . . . , Xn]/I ∼= R{−1,1}n ∼= R2n
.
Proof by algebraic geometry. I is a zero-dimensional radical ideal.
Corollary. D∗n = P ∗
n = f∗
22
![Page 87: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/87.jpg)
The story goes on...
Theorem (Lasserre). For every p ∈ R[X1, . . . , Xn], the following areequivalent:
(i) p ≥ 0 on Rn
![Page 88: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/88.jpg)
The story goes on...
Theorem (Lasserre). For every p ∈ R[X1, . . . , Xn], the following areequivalent:
(i) p ≥ 0 on Rn
(ii) For every ε > 0, there exists N ∈ N such that
p + ε
n∑i=1
N∑k=0
X2ki
k!is sos.
Jean Lasserre: A sum of squares approximation of nonnegativepolynomialshttp://front.math.ucdavis.edu/math.AG/0412398
23
![Page 89: Positive polynomials and convergence of LP and SDP relaxationsschweigh/presentations/control.pdf · Positive polynomials and convergence of LP and SDP relaxations Markus Schweighofer](https://reader030.vdocument.in/reader030/viewer/2022041013/5ec3e323492f2054425f7ace/html5/thumbnails/89.jpg)
The story goes on...
Theorem (Nie, Demmel, Sturmfels). If p > 0 on Rn, then p is sosmodulo its own gradient ideal
I :=(
∂f
∂X1, . . . ,
∂f
∂Xn
).
Nie, Demmel, Sturmfels: Minimizing Polynomials via Sum ofSquares over the Gradient Idealhttp://front.math.ucdavis.edu/math.OC/0411342
24