Considering Copositivity Locally

Transcription

1 Considering Copositivity Locally Peter J.C. Dickinson Uni. of Groningen Uni. of Vienna Uni. of Twente Roland Hildebrand Uni. of Grenoble Weierstrass Institute Uni. of Grenoble IFORS 2017 Thursday 20th July Local Copositivity Peter J.C. Dickinson 1/17

2 Contents 1 Introduction Feasible directions Faces 2 Feasible directions for copositive cone Copositive Cone Set of Zeros Main Result Simplified Main Result 3 Irreducibility General Nonnegative matrices Positive Semidefinite Matrices 4 End Thank you Bibliography Local Copositivity Peter J.C. Dickinson 2/17

3 Feasible directions Will consider K S n to be a proper cone and A K. Define cone of feasible directions at A in K as: dir (A, K) := {B S n A + δb K for some δ > 0}. A, K Example ( ) 1 0 A = and K = S := {X S 2 v T X v 0 for all v R 2 }. { (b11 ) } b dir (A, K) = 12 b 22 0 b 12 b 22 b 12 = 0 if b 22 = 0 Local Copositivity Peter J.C. Dickinson 3/17

4 Feasible directions Will consider K S n to be a proper cone and A K. Define cone of feasible directions at A in K as: dir (A, K) := {B S n A + δb K for some δ > 0}. dir (A, K) Example ( ) 1 0 A = and K = S := {X S 2 v T X v 0 for all v R 2 }. { (b11 ) } b dir (A, K) = 12 b 22 0 b 12 b 22 b 12 = 0 if b 22 = 0 Local Copositivity Peter J.C. Dickinson 3/17

5 Faces F S n is a face of K if A F, B, C K, A = B + C B, C F. This is a closed convex pointed cone. Denote the minimal face of K containing A by face (A, K) (this is intersection of all faces which contain A). A K \ {0} gives an extreme ray of K iff 1 = dim(face (A, K)) := dim (span[face (A, K)]). Local Copositivity Peter J.C. Dickinson 4/17

6 Faces F S n is a face of K if A F, B, C K, A = B + C B, C F. This is a closed convex pointed cone. Denote the minimal face of K containing A by face (A, K) (this is intersection of all faces which contain A). A K \ {0} gives an extreme ray of K iff 1 = dim(face (A, K)) := dim (span[face (A, K)]). Local Copositivity Peter J.C. Dickinson 4/17

7 Faces F S n is a face of K if A F, B, C K, A = B + C B, C F. This is a closed convex pointed cone. Denote the minimal face of K containing A by face (A, K) (this is intersection of all faces which contain A). A K \ {0} gives an extreme ray of K iff 1 = dim(face (A, K)) := dim (span[face (A, K)]). Local Copositivity Peter J.C. Dickinson 4/17

8 Faces F S n is a face of K if A F, B, C K, A = B + C B, C F. This is a closed convex pointed cone. Denote the minimal face of K containing A by face (A, K) (this is intersection of all faces which contain A). A K \ {0} gives an extreme ray of K iff 1 = dim(face (A, K)) := dim (span[face (A, K)]). Local Copositivity Peter J.C. Dickinson 4/17

9 Faces F S n is a face of K if A F, B, C K, A = B + C B, C F. This is a closed convex pointed cone. Denote the minimal face of K containing A by face (A, K) (this is intersection of all faces which contain A). A K \ {0} gives an extreme ray of K iff 1 = dim(face (A, K)) := dim (span[face (A, K)]). Local Copositivity Peter J.C. Dickinson 4/17

10 Faces F S n is a face of K if A F, B, C K, A = B + C B, C F. This is a closed convex pointed cone. Denote the minimal face of K containing A by face (A, K) (this is intersection of all faces which contain A). A K \ {0} gives an extreme ray of K iff 1 = dim(face (A, K)) := dim (span[face (A, K)]). Local Copositivity Peter J.C. Dickinson 4/17

11 Faces F S n is a face of K if A F, B, C K, A = B + C B, C F. This is a closed convex pointed cone. Denote the minimal face of K containing A by face (A, K) (this is intersection of all faces which contain A). A K \ {0} gives an extreme ray of K iff 1 = dim(face (A, K)) := dim (span[face (A, K)]). Local Copositivity Peter J.C. Dickinson 4/17

12 Faces F S n is a face of K if A F, B, C K, A = B + C B, C F. This is a closed convex pointed cone. Denote the minimal face of K containing A by face (A, K) (this is intersection of all faces which contain A). A K \ {0} gives an extreme ray of K iff 1 = dim(face (A, K)) := dim (span[face (A, K)]). Local Copositivity Peter J.C. Dickinson 4/17

13 Faces F S n is a face of K if A F, B, C K, A = B + C B, C F. This is a closed convex pointed cone. Denote the minimal face of K containing A by face (A, K) (this is intersection of all faces which contain A). A K \ {0} gives an extreme ray of K iff 1 = dim(face (A, K)) := dim (span[face (A, K)]). Local Copositivity Peter J.C. Dickinson 4/17

14 Faces F S n is a face of K if A F, B, C K, A = B + C B, C F. This is a closed convex pointed cone. Denote the minimal face of K containing A by face (A, K) (this is intersection of all faces which contain A). A K \ {0} gives an extreme ray of K iff 1 = dim(face (A, K)) := dim (span[face (A, K)]). Local Copositivity Peter J.C. Dickinson 4/17

15 Faces F S n is a face of K if A F, B, C K, A = B + C B, C F. This is a closed convex pointed cone. Denote the minimal face of K containing A by face (A, K) (this is intersection of all faces which contain A). A K \ {0} gives an extreme ray of K iff 1 = dim(face (A, K)) := dim (span[face (A, K)]). Theorem ([Pataki, 2000]) span [face (A, K)] = dir (A, K) ( dir (A, K)), dir (A, K) = K + span [face (A, K)]. Local Copositivity Peter J.C. Dickinson 4/17

16 Example ( ) 1 0 K = S+ 2 and A =. 0 0 { (b11 ) b dir (A, K) = 12 b 12 b 22 b 22 0 b 12 = 0 if b 22 = 0 span [face (A, K)] = dir (A, K) ( dir (A, K)) {( ) b11 0 = b R}, dir (A, K) = K + span [face (A, K)] {( ) = S+ 2 b b R}. }, Local Copositivity Peter J.C. Dickinson 5/17

17 Contents 1 Introduction Feasible directions Faces 2 Feasible directions for copositive cone Copositive Cone Set of Zeros Main Result Simplified Main Result 3 Irreducibility General Nonnegative matrices Positive Semidefinite Matrices 4 End Thank you Bibliography Local Copositivity Peter J.C. Dickinson 6/17

18 Copositive Cone Copositive Cone, COP n := {X S n v T X v 0 for all v R n +}. Theorem ([de Klerk and Pasechnik, 2002]) Clique number of a graph, G := Maximum cardinality of the cliques of G = min λ R {λ λ(e A G ) E COP n }. S n + + N n COP n Theorem ([Murty and Kabadi, 1987, D. and Gijben, 2014]) Checking copositivity is a co-np-complete problem. N.B. Checking complete positivity is an NP-hard problem. It is an open question whether this problem is in NP, and thus is an NP-complete problem. Local Copositivity Peter J.C. Dickinson 7/17

19 Set of Zeros Set of Zeros V A := {u R n + u T Au = 0, 1 T u = 1}. Lemma For A COP n, v V A have Av 0 and supp(v) supp(av) =. Set of Minimal Zeros, [Hildebrand, 2014] V A min := { u V A v V A s.t. supp(v) supp(u) }. For A COP n, set V A is possibly infinite, but set Vmin A is finite. Lemma For A COP n, u V A there exists v 1,..., v m V A min and θ > 0 s.t. u = i θ iv i. Have supp(u) = i supp(v i ), supp(au) = i supp(av i ), = supp(v i ) supp(av j ) for all i, j. Local Copositivity Peter J.C. Dickinson 8/17

20 Example Consider A = COPn \ (S+ n + N n ), w i = 1 2 (e i + e i+1 ) for i = 1, 2, 3, 4. We have Aw 1 = 1 2 e 4, Aw 2 = 1 2 e 5, Aw 3 = 1 2 e 1, Aw 4 = 1 2 e 2, V A = conv{w 1, w 2 } conv{w 2, w 3 } conv{w 3 w 4 }, Vmin A = {w 1, w 2, w 3, w 4 }. Local Copositivity Peter J.C. Dickinson 9/17

21 Main Result Theorem dir (A, COP n ) = Proof. { } B S n v T Bv 0 for all v V A, (Bv) i 0 for all v V A V B, i / supp(av) : Suppose A + δb COP n for some δ > 0. For v V A have 0 v T (A + δb)v = δv T Bv. For v V A V B V A+δB have 0 ((A + δb)v) i = (Av) i + δ(bv) i. : Comes from considering the KKT optimality conditions for min{v T (A + δb)v 1 T v = 1, v R n +} as δ 0 +. Example For A = n i=2 e ie T i, we have V A = {e 1 }, supp(ae 1 ) = and thus dir (A, COP n ) = {B S n b 11 > 0} {B S n b 11 = 0, b 1i 0 for all i}. Local Copositivity Peter J.C. Dickinson 10/17

22 Simplified Main Result Theorem dir (A, COP n ) = Lemma { } B S n v T Bv 0 for all v V A, (Bv) i 0 for all v V A V B, i / supp(av) span [face (A, COP n )] = dir (A, COP n ) ( dir (A, COP n )) { } = B S n v T Bv = 0 for all v V A, (Bv) i = 0 for all i / supp(av), v V A V B { } = B S n V A V B, (Bv) i = 0 for all i / supp(av), v V A = {B S n (Bv)i = 0 for all i / supp(av), v V A} { } = B S n (Bv)i = 0 for all i / supp(av), v Vmin A, dir (A, COP n ) = COP n + span [face (A, COP n )]. Local Copositivity Peter J.C. Dickinson 11/17

23 Example A = span [face (A, COP n )] = B S 5 : = span A, w i = 1 2 (e i + e i+1 ) for i = 1, 2, 3, 4 V A min = {w 1, w 2, w 3, w 4 } Aw 1 = 1 2 e 4, Aw 2 = 1 2 e 5, Aw 3 = 1 2 e 1, Aw 4 = 1 2 e 2, (Bw 1 ) i = 0 for i = 1, 2, 3, 5, (Bw 2 ) i = 0 for i = 1, 2, 3, 4, (Bw 3 ) i = 0 for i = 2, 3, 4, 5, (Bw 4 ) i = 0 for i = 1, 3, 4, 5, Local Copositivity Peter J.C. Dickinson 12/17

24 Contents 1 Introduction Feasible directions Faces 2 Feasible directions for copositive cone Copositive Cone Set of Zeros Main Result Simplified Main Result 3 Irreducibility General Nonnegative matrices Positive Semidefinite Matrices 4 End Thank you Bibliography Local Copositivity Peter J.C. Dickinson 13/17

25 Irreducibility For A, C COP n we say that A is irreducible w.r.t. C if δ > 0 s.t. A δc COP n.. Lemma A COP n \ {0} gives an extreme ray of COP n iff A is irreducible w.r.t. all C COP n \ {αa α 0}. Local Copositivity Peter J.C. Dickinson 14/17

26 Irreducibility For A, C COP n we say that A is irreducible w.r.t. C if δ > 0 s.t. A δc COP n. Equivalently, C / face (A, COP n ). Lemma A COP n \ {0} gives an extreme ray of COP n iff A is irreducible w.r.t. all C COP n \ {αa α 0}. Local Copositivity Peter J.C. Dickinson 14/17

27 Irreducibility For A, C COP n we say that A is irreducible w.r.t. C if δ > 0 s.t. A δc COP n. Equivalently, C / span[face (A, COP n )]. Lemma A COP n \ {0} gives an extreme ray of COP n iff A is irreducible w.r.t. all C COP n \ {αa α 0}. Local Copositivity Peter J.C. Dickinson 14/17

28 Irreducibility For A, C COP n we say that A is irreducible w.r.t. C if δ > 0 s.t. A δc COP n. Equivalently, C / span[face (A, COP n )]. Lemma A COP n \ {0} gives an extreme ray of COP n iff A is irreducible w.r.t. all C COP n \ {αa α 0}. Lemma For A, C COP n have that A is irreducible w.r.t. C iff v V A min, i {1,..., n} s.t. (Av) i = 0 (Cv) i. Proof. A is not irreducible w.r.t. C C span[face (A, COP n )] (Cv) i = 0 for all i / supp(av), v V A min. Local Copositivity Peter J.C. Dickinson 14/17

29 Irreducible w.r.t. N n Lemma Following are equivalent: 1 A is irreducible w.r.t. C = e 1 e T n + e n e T 1, 2 v V A min, i s.t. (Av) i = 0 (Cv) i = δ i1 v n + δ in v 1, 3 v V A min s.t. (Av) 1 = (Av) n = 0 < v 1 + v n. Proof. 1 2: Previous page 2 3: w.l.o.g. i = 1. Then (Av) 1 = 0 and v n > 0 and (Av) n = : w.l.o.g. v 1 > 0 and let i = n. Theorem A is irreducible w.r.t. all C N n \ {0} iff for all i, j there exists v V A min s.t. (Av) i = (Av) j = 0 < v i + v j. [D. et al., 2013] Local Copositivity Peter J.C. Dickinson 15/17

30 Irreducible w.r.t. S n + Lemma Following are equivalent: 1 A is irreducible w.r.t. C = cc T 0, 2 v V A min, i s.t. (Av) i = 0 (Cv) i = (v T c)c i, 3 v V A min s.t. vt c 0. Proof. 1 2: Previous page 2 3: Trivial 3 2: i s.t. v i, c i 0, and have (Av) i = 0 (v T c)c i. Theorem A is irreducible w.r.t. all C S n + \ {0} iff span(v A min ) = Rn. [Hildebrand, 2014] Local Copositivity Peter J.C. Dickinson 16/17

31 Thank You! P.J.C. Dickinson and R. Hildebrand. Considering Copositivity Locally. Journal of Mathematical Analysis and Applications, 437(2): , P.J.C. Dickinson, M. Dür, L. Gijben and R. Hildebrand. Irreducible elements of the copositive cone. Linear Algebra and its Applications, 439(6): , Slides of this talk (& bibliography) will be available at Local Copositivity Peter J.C. Dickinson 17/17

32 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, Local Copositivity Peter J.C. Dickinson 18/17

33 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): , Local Copositivity Peter J.C. Dickinson 19/17

34 Bibliography: Set of Zeros 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.J.C. Dickinson. An improved characterisation of the interior of the completely positive cone. Electronic Journal of Linear Algebra 20: , R. Hildebrand. Minimal zeros of copositive matrices. Linear Algebra and its Applications 459: , Local Copositivity Peter J.C. Dickinson 20/17

35 Bibliography: Geometry G. Pataki. The Geometry of Semidefinite Programming. In: Handbook of Semidefinite Programming, 2000 Kluwer Academic Publishers, Eds: H. Wolkowicz, R. Saigal and L. Vandenberghe. P.J.C. Dickinson. Geometry of the copositive and completely positive cones. Journal of Mathematical Analysis and Applications 380(1): , P.J.C. Dickinson and R. Hildebrand. Considering Copositivity Locally. Journal of Mathematical Analysis and Applications, 437(2): , Local Copositivity Peter J.C. Dickinson 21/17

36 Bibliography: Misc K.G. Murty and S.N. Kabadi. Some NP-complete problems in quadratic and nonlinear programming. Mathematical Programming 39(2): , 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 L. Gijben. On the computational complexity of membership problems for the completely positive cone and its dual. Computational Optimization and Applications 57(2): , Local Copositivity Peter J.C. Dickinson 22/17