Scaling relationship between the copositive cone and Parrilo s first level approximation

Transcription

1 Scaling relationship between the copositive cone and Parrilo s first level approximation Peter J.C. Dickinson University of Groningen University of Vienna University of Twente Mirjam Dür University of Groningen University of Trier Luuk Gijben University of Groningen Real World Roland Hildebrand University of Grenoble Weierstrass Institute Friday 17th July 2015 Peter J.C. Dickinson (UTwente) Scaling Parrilo 1/16

2 1 Introduction Copositive Cone Approximations 2 Sum-of-squares Parrilo Cones Scaling Out Scaling In 3 Conclusion Summary Bibliography Peter J.C. Dickinson (UTwente) Scaling Parrilo 2/16

3 Copositive Cone Copositive Cone, COP n := {X S n v T X v 0 for all v R n +}. Proper cone, i.e. closed convex pointed solid cone. Theorem ([Burer, 2009]) Any bounded mixed binary optimisation problem with linear constraints and a quadratic objective function can be reformulated as a copositive optimisation problem. Theorem ([de Klerk and Pasechnik, 2002]) Clique number of a graph G, α(g) := Maximum cardinality of the cliques of G = min λ R {λ λ(e A G ) E COP n }. Peter J.C. Dickinson (UTwente) Scaling Parrilo 3/16

4 Approximations Lemma Any clique of a graph gives a lower bound on its clique number. Lemma If K COP n then α(g) min λ R {λ λ(e A G ) E K}. Theorem ([Diananda, 1962]) S + n + N n COP n with equality iff n 4. Example (Horn matrix, Prof. Alfred Horn, [Diananda, 1962]) COP 5 \ (S N 5) Horn.html Prof. R.W. Cottle confirmed the matrix originated from Prof. Alfred Horn. Peter J.C. Dickinson (UTwente) Scaling Parrilo 4/16

5 1 Introduction Copositive Cone Approximations 2 Sum-of-squares Parrilo Cones Scaling Out Scaling In 3 Conclusion Summary Bibliography Peter J.C. Dickinson (UTwente) Scaling Parrilo 5/16

6 Parrilo Cones Theorem COP n = { } A S n n i,j=1 A ijx i x j 0 for all x R n +. We say that f R[x] is sum-of-squares (SOS) if there exists g 1,..., g k R[x] such that f (x) = k i=1 g i(x) 2. Parrilo cones, 0 th level { Kn 0 := A S n n i,j=1 A ijzi 2z2 j } is SOS. Theorem ([Parrilo, 2000]) We have 1 K 0 n COP n, 2 K r n K r+1 n for all r, 3 For all A int COP n, there exists r s.t. A K r n. [Parrilo, 2000] Peter J.C. Dickinson (UTwente) Scaling Parrilo 6/16

7 Parrilo Cones Theorem COP n = {A S n n i,j=1 A ijz 2 i z2 j 0 for all z Rn }. We say that f R[x] is sum-of-squares (SOS) if there exists g 1,..., g k R[x] such that f (x) = k i=1 g i(x) 2. Parrilo cones, 0 th level { Kn 0 := A S n n i,j=1 A ijzi 2z2 j } is SOS. Theorem ([Parrilo, 2000]) We have 1 K 0 n COP n, 2 K r n K r+1 n for all r, 3 For all A int COP n, there exists r s.t. A K r n. [Parrilo, 2000] Peter J.C. Dickinson (UTwente) Scaling Parrilo 6/16

8 Parrilo Cones Theorem COP n = {A S n n i,j=1 A ijz 2 i z2 j 0 for all z Rn }. We say that f R[x] is sum-of-squares (SOS) if there exists g 1,..., g k R[x] such that f (x) = k i=1 g i(x) 2. Parrilo cones, 0 th level { Kn 0 := A S n n i,j=1 A ijzi 2z2 j } is SOS. Theorem ([Parrilo, 2000]) We have 1 K 0 n COP n, 2 K r n K r+1 n for all r, 3 For all A int COP n, there exists r s.t. A K r n. [Parrilo, 2000] Peter J.C. Dickinson (UTwente) Scaling Parrilo 6/16

9 Parrilo Cones Theorem COP n = {A S n n i,j=1 A ijz 2 i z2 j 0 for all z Rn }. We say that f R[x] is sum-of-squares (SOS) if there exists g 1,..., g k R[x] such that f (x) = k i=1 g i(x) 2. Parrilo cones, r th level { Kn r := A S n (z T z) r n i,j=1 A ijzi 2z2 j } is SOS. Theorem ([Parrilo, 2000]) We have 1 K r n COP n for all r, 2 K r n K r+1 n for all r, 3 For all A int COP n, there exists r s.t. A K r n. [Parrilo, 2000] Peter J.C. Dickinson (UTwente) Scaling Parrilo 6/16

10 Parrilo Cones Theorem COP n = {A S n n i,j=1 A ijz 2 i z2 j 0 for all z Rn }. We say that f R[x] is sum-of-squares (SOS) if there exists g 1,..., g k R[x] such that f (x) = k i=1 g i(x) 2. Parrilo cones, r th level { Kn r := A S n (z T z) r n i,j=1 A ijzi 2z2 j } is SOS. Theorem ([Parrilo, 2000]) We have 1 K r n COP n for all r, 2 K r n K r+1 n for all r, 3 For all A int COP n, there exists r s.t. A K r n. [Parrilo, 2000] Peter J.C. Dickinson (UTwente) Scaling Parrilo 6/16

11 r = 0, 1 Theorem ([Parrilo, 2000]) K 0 n = S + n + N n. Corollary K 0 n = COP n iff n 4. Theorem ([Parrilo, 2000] [Bomze and de Klerk, 2002]) We have X Kn 1 if and only if M 1,..., M n S n s.t. X M i S n + for all i, (M i ) ii = 0 for all i, (M i ) jj + 2(M i ) ij = 0 for all i j, (M i ) jk + (M j ) ik + (M k ) ij 0 for all i < j < k. Theorem ([Dickinson et al., Scaling..., 2013]) Let X S n { 1, +1} n n. Then X COP n X Kn. 1 Proof: M i = X X,i X T,i Q: For a given r, what is the minimum n such that K r n COP n? Peter J.C. Dickinson (UTwente) Scaling Parrilo 7/16

12 r = 0, 1 Theorem ([Parrilo, 2000]) K 0 n = S + n + N n. Corollary K 0 n = COP n iff n 4. Theorem ([Parrilo, 2000] [Bomze and de Klerk, 2002]) We have X Kn 1 if and only if M 1,..., M n S n s.t. X M i S n + for all i, (M i ) ii = 0 for all i, (M i ) jj + 2(M i ) ij = 0 for all i j, (M i ) jk + (M j ) ik + (M k ) ij 0 for all i < j < k. Theorem ([Dickinson et al., Scaling..., 2013]) Let X S n { 1, +1} n n. Then X COP n X Kn. 1 Proof: M i = X X,i X T,i Q: For a given r, what is the minimum n such that K r n COP n? Peter J.C. Dickinson (UTwente) Scaling Parrilo 7/16

13 r = 0, 1 Theorem ([Parrilo, 2000]) K 0 n = S + n + N n. Corollary K 0 n = COP n iff n 4. Theorem ([Parrilo, 2000] [Bomze and de Klerk, 2002]) We have X Kn 1 if and only if M 1,..., M n S n s.t. X M i S n + for all i, (M i ) ii = 0 for all i, (M i ) jj + 2(M i ) ij = 0 for all i j, (M i ) jk + (M j ) ik + (M k ) ij 0 for all i < j < k. Theorem ([Dickinson et al., Scaling..., 2013]) Let X S n { 1, +1} n n. Then X COP n X Kn. 1 Proof: M i = X X,i X T,i Q: For a given r, what is the minimum n such that K r n COP n? Peter J.C. Dickinson (UTwente) Scaling Parrilo 7/16

14 r = 0, 1 Theorem ([Parrilo, 2000]) K 0 n = S + n + N n. Corollary K 0 n = COP n iff n 4. Theorem ([Parrilo, 2000] [Bomze and de Klerk, 2002]) We have X Kn 1 if and only if M 1,..., M n S n s.t. X M i S n + for all i, (M i ) ii = 0 for all i, (M i ) jj + 2(M i ) ij = 0 for all i j, (M i ) jk + (M j ) ik + (M k ) ij 0 for all i < j < k. Theorem ([Dickinson et al., Scaling..., 2013]) Let X S n { 1, +1} n n. Then X COP n X Kn. 1 Proof: M i = X X,i X T,i Q: For a given r, what is the minimum n such that K r n COP n? Peter J.C. Dickinson (UTwente) Scaling Parrilo 7/16

15 Bordering Theorem ([Dickinson and Vera, 2015?]) ( ) A 0 If A COP n \ Kn 0 then B = 0 T COP 0 n+1 \ Kn+1 r for all r. Proof. Equivalently B Kn+1 r A K0 n. Consider B Kn+1 r for some r. Have (z z2 n+1 )r n i,j=1 a ijzi 2z2 j = p k=1 g k(z) 2. Let g k (z 1,..., z n+1 ) = q l=0 zl n+1 g k,l(z 1,..., z n ). Suppose q > r: Considering term containing z 2q n+1 we get 0 = z 2q p n+1 k=1 g k,q(z 1,..., z n ) 2. Therefore w.l.o.g. q = r. Considering terms containing z 2r z 2r n+1 Thus n i,j=1 a ijz 2 i z2 j Corollary n i,j=1 a ijz 2 i z 2 j n+1 get = zn+1 2r p k=1 g k,r (z 1,..., z n ) 2. = p k=1 g k,r (z 1,..., z n ) 2 is SOS and A Kn. 0 For n 6 there exists B COP n such that B / K r n for all r. Peter J.C. Dickinson (UTwente) Scaling Parrilo 8/16

16 Scaling Let D = {Diag(d) d R n ++}. Lemma For A COP n and D D we have DAD COP n. For A S + n and D D we have DAD S + n. For A N n and D D we have DAD N n. Proof. For x R n + have x T (DAD)x = (Dx) T A(Dx) 0. Similarly for S + n. For N n note that this set is closed under multiplication. Corollary For A K 0 n and D D we have DAD K 0 n. Peter J.C. Dickinson (UTwente) Scaling Parrilo 9/16

17 Scaling out Theorem ([Dickinson et al., Scaling..., 2013]) For A S n \ K 0 n and r N, D D s.t. DAD / K r n. Proof. Equivalently: DAD Kn r for all D D A Kn. 0 ( ) I 0 Let D ε = 0 T and q(z ε 1,..., z n ) = n i,j=1 a ijzi 2z2 j. Then if DAD K r n for all D D Corollary D ε AD ε K r n for all ε > 0 (z z 2 n ) r q(z 1,..., z n 1, εz n ) is SOS for all ε > 0 (z z 2 n ε z2 n ) r q(z 1,..., z n ) is SOS for all ε > 0 (εz εz 2 n 1 + z 2 n ) r q(z 1,..., z n ) is SOS for all ε > 0 z 2r n q(z 1,..., z n ) is SOS q(z 1,..., z n ) is SOS A K 0 n. For n 5 and r N, A K r n and D D s.t. DAD COP n \ K r n. Peter J.C. Dickinson (UTwente) Scaling Parrilo 10/16

18 Scaling in: All ones diagonal Lemma Let A COP n. Then 0 e T i Ae i = a ii. If A S n with a ii = 0, then A COP n iff a ij 0 for all j and principal submatrix of A, deleting ith row/column, is copositive. Lemma Let A S n s.t. (A) ii > 0 for all i. Then unique D D s.t. (DAD) ii = 1 for all i. We then have A COP n iff DAD COP n. Proof. For D D, A S n have (DAD) ii = (D) 2 ii (A) ii. Theorem For all A COP n and D D we have DAD COP n Therefore from now on can limit ourselves to testing copositivity of matrices with all diagonal entries equal to one. Peter J.C. Dickinson (UTwente) Scaling Parrilo 11/16

19 Theorem ([Hildebrand, 2012]) Let Θ := {0} {θ R 5 ++ e T θ < π}, and for θ Θ 1 cos θ 1 cos(θ 1 + θ 2) cos(θ 4 + θ 5) cos θ 5 cos θ 1 1 cos θ 2 cos(θ 2 + θ 3) cos(θ 5 + θ 1) S(θ) := cos(θ 1 + θ 2) cos θ 2 1 cos θ 3 cos(θ 3 + θ 4). cos(θ 4 + θ 5) cos(θ 2 + θ 3) cos θ 3 1 cos θ 4 cos θ 5 cos(θ 5 + θ 1) cos(θ 3 + θ 4) cos θ 4 1 Matrices DPS(θ)P T D generate all extreme rays of COP 5 not in K5 0. Theorem ([Dickinson et al., Scaling..., 2013]) PS(θ)P T K5 1 for all θ Θ and permutation matrices P. Peter J.C. Dickinson (UTwente) Scaling Parrilo 12/16

20 α 1 (ϑ) 0 M 1 (ϑ)= β 1 (ϑ) γ 1 (ϑ) 0 α 1 (ϑ) β 1 (ϑ) 0 0, 0 0 γ 1 (ϑ) α 3 (ϑ) β 3 (ϑ) γ 3 (ϑ) M 3 (ϑ)= α 3 (ϑ) , β 3 (ϑ) γ 3 (ϑ) α 5 (ϑ) β 5 (ϑ) γ 5 (ϑ) 0 M 5 (ϑ)= α 5 (ϑ) β 5 (ϑ) γ 5 (ϑ) 0 0 0, M2 (ϑ)= γ 2 (ϑ) α 2 (ϑ) γ 2 (ϑ) β 2 (ϑ) 0 0 α 2 (ϑ) β 2 (ϑ) 0, 0 β 4 (ϑ) γ 4 (ϑ) 0 0 β 4 (ϑ) α 4 (ϑ) M4 (ϑ)= γ 4 (ϑ) , 0 α 4 (ϑ) where for all i = 1,..., 5 (the indices being modulo 5) we define α i (ϑ) = cos(ϑ i 2 + ϑ i 1 + ϑ i ) + cos(ϑ i+1 + ϑ i+2 ), β i (ϑ) = cos(ϑ i+2 ) cos(ϑ i 2 + ϑ i 1 + ϑ i + ϑ i+1 ), γ i (ϑ) = cos(ϑ i 1 + ϑ i + ϑ i+1 ) + cos(ϑ i+2 + ϑ i 2 ). Peter J.C. Dickinson (UTwente) Scaling Parrilo 13/16

21 Theorem ([Hildebrand, 2012]) Let Θ := {0} {θ R 5 ++ e T θ < π}, and for θ Θ 1 cos θ 1 cos(θ 1 + θ 2) cos(θ 4 + θ 5) cos θ 5 cos θ 1 1 cos θ 2 cos(θ 2 + θ 3) cos(θ 5 + θ 1) S(θ) := cos(θ 1 + θ 2) cos θ 2 1 cos θ 3 cos(θ 3 + θ 4). cos(θ 4 + θ 5) cos(θ 2 + θ 3) cos θ 3 1 cos θ 4 cos θ 5 cos(θ 5 + θ 1) cos(θ 3 + θ 4) cos θ 4 1 Matrices DPS(θ)P T D generate all extreme rays of COP 5 not in K5 0. Theorem ([Dickinson et al., Scaling..., 2013]) PS(θ)P T K5 1 for all θ Θ and permutation matrices P. Theorem ([Dickinson et al., Irreducible..., 2013]) For A COP 5 \ K 0 5 have A = DPS(θ)PT D + N for some D D, θ Θ, permutation matrix P and N N n, with diag(n) = 0. N.B. have D = Diag( a 11,..., a 55 ). Corollary ([Dickinson et al., Scaling..., 2013]) Let A S 5 s.t. a ii = 1 for all i. Then A COP 5 A K 1 5. Peter J.C. Dickinson (UTwente) Scaling Parrilo 14/16

22 Theorem ([Hildebrand, 2012]) Let Θ := {0} {θ R 5 ++ e T θ < π}, and for θ Θ 1 cos θ 1 cos(θ 1 + θ 2) cos(θ 4 + θ 5) cos θ 5 cos θ 1 1 cos θ 2 cos(θ 2 + θ 3) cos(θ 5 + θ 1) S(θ) := cos(θ 1 + θ 2) cos θ 2 1 cos θ 3 cos(θ 3 + θ 4). cos(θ 4 + θ 5) cos(θ 2 + θ 3) cos θ 3 1 cos θ 4 cos θ 5 cos(θ 5 + θ 1) cos(θ 3 + θ 4) cos θ 4 1 Matrices DPS(θ)P T D generate all extreme rays of COP 5 not in K5 0. Theorem ([Dickinson et al., Scaling..., 2013]) PS(θ)P T K5 1 for all θ Θ and permutation matrices P. Theorem ([Dickinson et al., Irreducible..., 2013]) For A COP 5 \ K 0 5 have A = DPS(θ)PT D + N for some D D, θ Θ, permutation matrix P and N N n, with diag(n) = 0. N.B. have D = Diag( a 11,..., a 55 ). Corollary ([Dickinson et al., Scaling..., 2013]) Let A S 5 s.t. a ii = 1 for all i. Then A COP 5 A K 1 5. Peter J.C. Dickinson (UTwente) Scaling Parrilo 14/16

23 All-ones on diagonal Theorem ([Dickinson et al., Scaling..., 2013]) Let A S 5 s.t. a ii = 1 for all i. Then A COP 5 A K 1 5. Theorem ([Dickinson and Vera, 2015?]) Let X S n { 1, +1} n n. Then X COP n X K 1 n. Conjecture ([de Klerk and Pasechnik, 2002]) α(g) = min λ R {λ λ(e A G ) E K r n} for r α(g) 1. Proven for α(g) 8 [Laurent and Gvozdenović, 2007]. [ ] Note that λ(e A G ) E = λ 1 for all i = 1,..., n. ii Peter J.C. Dickinson (UTwente) Scaling Parrilo 15/16

24 Theorem For A int COP n, there exists r N s.t. A K r n. For all r N and n 5, there exists A COP n \ K r n. For A S 5 can find D D s.t. A COP 5 iff DAD K 1 5. Peter J.C. Dickinson (UTwente) Scaling Parrilo 16/16

25 Theorem For A int COP n, there exists r N s.t. A K r n. For all r N and n 5, there exists A COP n \ K r n. For A S 5 can find D D s.t. A COP 5 iff DAD K 1 5. P.J.C. Dickinson, M. Dür, L. Gijben and R. Hildebrand. Scaling relationship between the copositive cone and Parrilo s first level approximation. Optimization Letters, 7(8): , P.J.C. Dickinson, M. Dür, L. Gijben and R. Hildebrand. Irreducible elements of the copositive cone. Linear Algebra and its Applications 439(6): , P.J.C. Dickinson and J. Vera. Approximations for Copositive Optimisation closed under scaling. In construction, 2015? Slides of this talk (& bibliography) will be available at Peter J.C. Dickinson (UTwente) Scaling Parrilo 16/16

26 Theorem For A int COP n, there exists r N s.t. A K r n. For all r N and n 5, there exists A COP n \ K r n. For A S 5 can find D D s.t. A COP 5 iff DAD K 1 5. Thank You! P.J.C. Dickinson, M. Dür, L. Gijben and R. Hildebrand. Scaling relationship between the copositive cone and Parrilo s first level approximation. Optimization Letters, 7(8): , P.J.C. Dickinson, M. Dür, L. Gijben and R. Hildebrand. Irreducible elements of the copositive cone. Linear Algebra and its Applications 439(6): , P.J.C. Dickinson and J. Vera. Approximations for Copositive Optimisation closed under scaling. In construction, 2015? Slides of this talk (& bibliography) will be available at Peter J.C. Dickinson (UTwente) Scaling Parrilo 16/16

27 Bibliography: Surveys M. Dür. Copositive Programming - a Survey. In: Recent Advances in Optimization and its Applications in Engineering, Springer Berlin Heidelberg, 3 20, 2010 I.M. Bomze, W. Schachinger and G. Uchida. Think co(mpletely )positive! Matrix properties, examples and a clustered bibliography on copositive optimization. Journal of Global Optimization 52(3): , 2012 P.J.C. Dickinson. The copositive cone, the completely positive cone and their generalisations. PhD Thesis, University of Groningen, Peter J.C. Dickinson (UTwente) Scaling Parrilo 17/16

28 Bibliography: Applications I.M. Bomze, M. Dür, E. de Klerk, C. Roos, A.J. Quist and T. Terlaky. On Copositive Programming and Standard Quadratic Optimization Problems. Journal of Global Optimization 18: , E. de Klerk and D.V. Pasechnik. Approximation of the stability number of a graph via copositive programming. SIAM Journal on Optimization 12(4): , S. Burer. On the Copositive Representation of Binary and Continuous Nonconvex Quadratic Programs. Mathematical Programming 120(2): , Peter J.C. Dickinson (UTwente) Scaling Parrilo 18/16

29 Bibliography: Approximations P.H. Diananda. On nonnegative forms in real variables some or all of which are nonnegative. Mathematical Proceedings of the Cambridge Philosophical Society 58:17 25, P.A. Parrilo. Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. PhD thesis, California Institute of Technology, I.M. Bomze and E. de Klerk. Solving standard quadratic optimization problems via linear, semidefinite and copositive programming. Journal of Global Optimization 24(2): , Peter J.C. Dickinson (UTwente) Scaling Parrilo 19/16

30 Bibliography: Misc R. Hildebrand. The extreme rays of the 5 5 copositive cone. Linear Algebra and its Applications 437(7): , P.J.C. Dickinson, M. Dür, L. Gijben and R. Hildebrand. Scaling relationship between the copositive cone and Parrilo s first level approximation. Optimization Letters 7(8): , P.J.C. Dickinson, M. Dür, L. Gijben and R. Hildebrand. Irreducible elements of the copositive cone. Linear Algebra and its Applications 439(6): , Peter J.C. Dickinson (UTwente) Scaling Parrilo 20/16

31 Bibliography: Misc P.J.C. Dickinson and J. Vera. Approximations for Copositive Optimisation closed under scaling. In construction. M. Laurent and N. Gvozdenović. Semidefinite bounds for the stability number of a graph via sums of squares of polynomials. Mathematical Programming 110(B): , Peter J.C. Dickinson (UTwente) Scaling Parrilo 21/16