Structure of Valid Inequalities for Mixed Integer Conic Programs

Transcription

1 Structure of Valid Inequalities for Mixed Integer Conic Programs Fatma Kılınç-Karzan Tepper School of Business Carnegie Mellon University 18 th Combinatorial Optimization Workshop Aussois, France January 6-10, 2014 F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 1 / 34

2 Outline Mixed integer conic optimization Motivation Problem setting Structure of linear valid inequalities K-minimal valid inequalities K-sublinear valid inequalities Necessary conditions Sufficient conditions Disjunctive cuts for Lorentz cone (joint work with Sercan Yıldız) Connection to the new framework Deriving the nonlinear valid inequality F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 2 / 34

3 Mixed Integer Conic Programming Mixed Integer Linear Program min s.t. c T x Ax b x Z d R n d F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 3 / 34

4 Mixed Integer Conic Programming Mixed Integer Linear Program min c T x s.t. Ax b R m + x Z d R n d min c T F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 3 / 34

5 Mixed Integer Conic Programming Mixed Integer Linear Program Mixed Integer Convex Program min c T x s.t. Ax b R m + x Z d R n d min s.t. c T x x Q x Z d R n d where Q is a closed convex set min c T min c T Q F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 3 / 34

6 Mixed Integer Conic Programming Mixed Integer Linear Program Mixed Integer Conic Program min c T x s.t. Ax b R m + x Z d R n d min s.t. c T x Ax b K x Z d R n d where K is a convex cone min c T x 3 x 1 x 2 0 F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 4 / 34

7 Mixed Integer Conic Programming Mixed Integer Linear Program Mixed Integer Conic Program min c T x s.t. Ax b R m + x Z d R n d min s.t. c T x Ax b K x Z d R n d where K is a convex cone min c T Nonnegative orthant R m + = {y Rm : y i 0 i} Lorentz cone L m = {y R m : y m Positive semidefinite cone y y 2 m 1 } S m + = {X Rm m : a T Xa 0 a R m } F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 4 / 34

8 Mixed Integer Conic Programming Mixed Integer Linear Program Mixed Integer Conic Program min c T x s.t. Ax b R m + x Z d R n d min s.t. c T x Ax b K x Z d R n d where K is a convex cone min c T min c T F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 4 / 34

9 Motivation A good understanding of mixed integer linear programs (MIPs) Well known classes of valid inequalities, closures,... [C-G cuts, MIR inequalities, Split cuts, Disjunctive cuts,...] Characterization of minimal and extremal inequalities for linear MIPs F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 5 / 34

10 Motivation A good understanding of mixed integer linear programs (MIPs) Well known classes of valid inequalities, closures,... [C-G cuts, MIR inequalities, Split cuts, Disjunctive cuts,...] Characterization of minimal and extremal inequalities for linear MIPs Recent and growing interest in mixed integer conic programs (MICPs) Development of general classes of valid inequalities for conic sets [Extensions of C-G cuts, MIR inequalities, Split cuts, Intersection cuts, Disjunctive cuts,...] Recent results on C-G closures for convex sets, conic MIP duality F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 5 / 34

11 Motivation A good understanding of mixed integer linear programs (MIPs) Well known classes of valid inequalities, closures,... [C-G cuts, MIR inequalities, Split cuts, Disjunctive cuts,...] Characterization of minimal and extremal inequalities for linear MIPs Recent and growing interest in mixed integer conic programs (MICPs) Development of general classes of valid inequalities for conic sets [Extensions of C-G cuts, MIR inequalities, Split cuts, Intersection cuts, Disjunctive cuts,...] Recent results on C-G closures for convex sets, conic MIP duality Success of solvers depend on identifying and efficiently separating strong valid inequalities F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 5 / 34

12 Is it possible to develop a framework to establish strength of valid linear inequalities for MICPs? F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 6 / 34

13 Problem Setting We are interested in the convex hull of the following set: where S(A, K, B) = {x E : Ax B, x K} E is a finite dimensional Euclidean space with inner product, A is a linear map from E to R m B R m is a given set of points (can be finite or infinite) K E is a full-dimensional, closed, convex and pointed cone Nonnegative orthant, R n + := {x R n : x i 0 i} Lorentz cone, L n := {x R n : x n x x n 1 2 } Positive semidefinite cone, S n + := {X R n n : a T Xa 0 a R n } F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 7 / 34

14 Representation flexibility S(A, K, B) = {x E : Ax B, x K} This set captures the essential structure of MICPs (can also be used as a natural relaxation): F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 8 / 34

15 Representation flexibility S(A, K, B) = {x E : Ax B, x K} This set captures the essential structure of MICPs (can also be used as a natural relaxation): {(y, v) R k + Z q : Wy + Hv b K} where K E is a full-dimensional, closed, convex, pointed cone. F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 8 / 34

16 Representation flexibility S(A, K, B) = {x E : Ax B, x K} This set captures the essential structure of MICPs (can also be used as a natural relaxation): {(y, v) R k + Z q : Wy + Hv b K} where K E is a full-dimensional, closed, convex, pointed cone. Define ( ) y x =, A = [ W, Id ], and B = b HZ q, z where Id is the identity map in E. Then we arrive at S(A, K, B) = {x (R k E) : Ax B, x (R k + K) } }{{} :=K F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 8 / 34

17 Representation flexibility S(A, K, B) = {x E : Ax B, x K} This set captures the essential structure of MICPs (can also be used as a natural relaxation): {(y, v) R k + Z q : Wy + Hv b K} where K E is a full-dimensional, closed, convex, pointed cone. Define ( ) y x =, A = [ W, Id ], and B = b HZ q, z where Id is the identity map in E. Then we arrive at S(A, K, B) = {x (R k E) : Ax B, x (R k + K) } }{{} :=K Many more examples involving modeling disjunctions, complementarity relations, Gomory s Corner Polyhedron (1969), etc... F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 8 / 34

18 Some Notation S(A, K, B) = {x E : Ax B, x K} E is a finite dimensional Euclidean space with inner product, A is a linear map from E to R m A denotes the conjugate linear map from R m to E Ker(A) := {x E : Ax = 0} Im(A) := {Ax : x E} F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 9 / 34

19 Some Notation S(A, K, B) = {x E : Ax B, x K} E is a finite dimensional Euclidean space with inner product, A is a linear map from E to R m A denotes the conjugate linear map from R m to E Ker(A) := {x E : Ax = 0} Im(A) := {Ax : x E} B R m is a given set of points (can be finite or infinite) K E is a full-dimensional, closed, convex and pointed cone and its dual cone is given by K := {y E : y, x 0 x K}. F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 9 / 34

20 Some Notation S(A, K, B) = {x E : Ax B, x K} E is a finite dimensional Euclidean space with inner product, A is a linear map from E to R m A denotes the conjugate linear map from R m to E Ker(A) := {x E : Ax = 0} Im(A) := {Ax : x E} B R m is a given set of points (can be finite or infinite) K E is a full-dimensional, closed, convex and pointed cone K := the cone dual to K Ext(K) := the set of extreme rays of K F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 9 / 34

21 Goal S(A, K, B) = {x E : Ax B, x K} We are interested in conv(s(a, K, B)) F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 10 / 34

22 Goal S(A, K, B) = {x E : Ax B, x K} We are interested in conv(s(a, K, B)) conv(s(a, K, B)) conv(s(a, K, B)) = intersection of all linear valid inequalities (v.i.) µ, x η 0 for S(A, K, B): F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 10 / 34

23 Goal S(A, K, B) = {x E : Ax B, x K} We are interested in conv(s(a, K, B)) conv(s(a, K, B)) conv(s(a, K, B)) = intersection of all linear valid inequalities (v.i.) µ, x η 0 for S(A, K, B): C(A, K, B) = convex cone of all linear valid inequalities for S(A, K, B) = {(µ; η 0 ) : µ E, µ 0, < η 0 inf µ, x } x S(A,K,B) }{{} :=µ 0 Goal: Study C(A, K, B) in order to characterize the properties of the linear v.i., and identify the necessary and/or sufficient ones defining conv(s(a, K, B)) F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 10 / 34

24 Which Inequalities Should We Really Care About? S(A, K, B) = {x E : Ax B, x K} C(A, K, B) = convex cone of all linear valid inequalities (v.i.) C(A, K, B) F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 11 / 34

25 Which Inequalities Should We Really Care About? S(A, K, B) = {x E : Ax B, x K} C(A, K, B) = convex cone of all linear valid inequalities (v.i.) Cone implied inequality, (δ; 0) for any δ K \ {0}, is always valid. Cone implied inequalities C(A, K, B) F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 11 / 34

26 Which Inequalities Should We Really Care About? S(A, K, B) = {x E : Ax B, x K} C(A, K, B) = convex cone of all linear valid inequalities (v.i.) Definition An inequality (µ; η 0 ) C(A, K, B) is a K-minimal valid inequality if for all ρ such that ρ K µ and ρ µ, we have ρ 0 < η 0. F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 11 / 34

27 Which Inequalities Should We Really Care About? S(A, K, B) = {x E : Ax B, x K} C(A, K, B) = convex cone of all linear valid inequalities (v.i.) Definition An inequality (µ; η 0 ) C(A, K, B) is a K-minimal valid inequality if for all ρ such that ρ K µ and ρ µ, we have ρ 0 < η 0. C m (A, K, B) = cone of K-minimal valid inequalities F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 11 / 34

28 Which Inequalities Should We Really Care About? S(A, K, B) = {x E : Ax B, x K} C(A, K, B) = convex cone of all linear valid inequalities (v.i.) C m (A, K, B) = cone of K-minimal valid inequalities C(A, K, B) C m (A, K, B) Cone implied inequalities F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 11 / 34

29 K-minimal Inequalities Definition An inequality (µ; η 0 ) C(A, K, B) is a K-minimal valid inequality if for all ρ such that ρ K µ and ρ µ, we have ρ 0 < η 0. A K-minimal v.i. (µ; η 0 ) cannot be written as a sum of a cone implied inequality and another valid inequality. F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 12 / 34

30 K-minimal Inequalities Definition An inequality (µ; η 0 ) C(A, K, B) is a K-minimal valid inequality if for all ρ such that ρ K µ and ρ µ, we have ρ 0 < η 0. A K-minimal v.i. (µ; η 0 ) cannot be written as a sum of a cone implied inequality and another valid inequality. Cone implied inequality (δ; 0) for any δ K \ {0} is never minimal. F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 12 / 34

31 K-minimal Inequalities Definition An inequality (µ; η 0 ) C(A, K, B) is a K-minimal valid inequality if for all ρ such that ρ K µ and ρ µ, we have ρ 0 < η 0. A K-minimal v.i. (µ; η 0 ) cannot be written as a sum of a cone implied inequality and another valid inequality. Cone implied inequality (δ; 0) for any δ K \ {0} is never minimal. K-minimal v.i. exists if and only if valid equations of form δ, x = 0 with δ K \ {0}. F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 12 / 34

32 K-minimal Inequalities Cone implied inequalities (δ; 0) with δ K \ {0} are always valid but are never K-minimal. yet they can still be extremal in the cone C(A, K, B). are not particularly interesting as they are already included in the description (x K). F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 13 / 34

33 K-minimal Inequalities Cone implied inequalities (δ; 0) with δ K \ {0} are always valid but are never K-minimal. yet they can still be extremal in the cone C(A, K, B). are not particularly interesting as they are already included in the description (x K). C(A, K, B) C m (A, K, B) Cone implied inequalities F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 13 / 34

34 K-minimal Inequalities Cone implied inequalities (δ; 0) with δ K \ {0} are always valid but are never K-minimal. yet they can still be extremal in the cone C(A, K, B). are not particularly interesting as they are already included in the description (x K). Theorem: [Sufficiency of K-minimal Inequalities] Whenever there is at least one K-minimal v.i., C m (A, K, B), i.e., δ K \ {0} such that δ, x = 0 for all x S(A, K, B), then C(A, K, B) is generated by K-minimal v.i. and cone implied v.i. F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 13 / 34

35 Necessary Conditions for K-minimality What can we say about µ for a v.i. (µ; η 0 ) C m (A, K, B)? Proposition If (µ; η 0 ) C m (A, K, B), then for all linear maps Z : K K s.t. AZ A, we have µ Zµ K \ {0}. F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 14 / 34

36 Necessary Conditions for K-minimality Proposition If (µ; η 0 ) C m (A, K, B), then for all linear maps Z : K K s.t. AZ A, we have µ Zµ K \ {0}. More on F K := {(Z : E E) : Z is linear, and Z v K v K} F K is the cone of K K positive maps. They also appear in applications of robust optimization, quantum physics,... F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 14 / 34

37 Necessary Conditions for K-minimality Proposition If (µ; η 0 ) C m (A, K, B), then for all linear maps Z : K K s.t. AZ A, we have µ Zµ K \ {0}. More on F K := {(Z : E E) : Z is linear, and Z v K v K} When K = R n +, F K = {Z R n n : Z ij 0 i, j} F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 14 / 34

38 Necessary Conditions for K-minimality Proposition If (µ; η 0 ) C m (A, K, B), then for all linear maps Z : K K s.t. AZ A, we have µ Zµ K \ {0}. More on F K := {(Z : E E) : Z is linear, and Z v K v K} When K = R n +, F K = {Z R n n : Z ij 0 i, j} When K = L n, F K has a semidefinite programming representation (Hildebrand 2006, 2008). [ In particular when Z = ab T with a, b L n, then Z F L n. ] F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 14 / 34

39 Necessary Conditions for K-minimality Proposition If (µ; η 0 ) C m (A, K, B), then for all linear maps Z : K K s.t. AZ A, we have µ Zµ K \ {0}. More on F K := {(Z : E E) : Z is linear, and Z v K v K} But in general they are hard to describe: [ Deciding whether a given linear map takes S n + to itself is an NP-hard problem. (Ben Tal & Nemirovski, 1998) ] F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 14 / 34

40 Valid Inequalities C(A, K, B) C m (A, K, B) Cone implied inequalities F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 15 / 34

41 Valid Inequalities C(A, K, B) C m (A, K, B) Cone implied inequalities F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 15 / 34

42 K-sublinear Inequalities Definition (µ; η 0 ) is a K-sublinear v.i. if it satisfies (A.1) (A.2) 0 µ, u for all u s.t. Au = 0 and α, v u + v K v Ext(K) holds for some α Ext(K ), η 0 µ, x for all x S(A, K, B). F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 16 / 34

43 K-sublinear Inequalities Definition (µ; η 0 ) is a K-sublinear v.i. if it satisfies (A.1) (A.2) 0 µ, u for all u s.t. Au = 0 and α, v u + v K v Ext(K) holds for some α Ext(K ), η 0 µ, x for all x S(A, K, B). Condition (A.1) implies (A.0) 0 µ, u for all u K such that Au = 0. F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 16 / 34

44 K-sublinear Inequalities Definition (µ; η 0 ) is a K-sublinear v.i. if it satisfies (A.1) (A.2) 0 µ, u for all u s.t. Au = 0 and α, v u + v K v Ext(K) holds for some α Ext(K ), η 0 µ, x for all x S(A, K, B). Condition (A.1) implies (A.0) µ Im(A ) + K F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 16 / 34

45 K-sublinear Inequalities Definition (µ; η 0 ) is a K-sublinear v.i. if it satisfies (A.1) (A.2) 0 µ, u for all u s.t. Au = 0 and α, v u + v K v Ext(K) holds for some α Ext(K ), η 0 µ, x for all x S(A, K, B). Condition (A.1) implies (A.0) µ Im(A ) + K Proposition Any valid inequality (µ; η 0 ) C(A, K, B) satisfies condition (A.0). F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 16 / 34

46 K-sublinear Inequalities Theorem C m (A, K, B) C a (A, K, B). C a (A, K, B) C(A, K, B) C m (A, K, B) Cone implied inequalities F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 17 / 34

47 K-sublinear Inequalities Theorem C m (A, K, B) C a (A, K, B). Theorem [K.-K. & Steffy]: Sufficiency of K-sublinear Inequalities C(A, K, B) is generated by K-sublinear v.i. and cone implied v.i. without any restrictions on set S(A, K, B). F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 17 / 34

48 K-sublinear Inequalities C a (A, K, B) C(A, K, B) C m (A, K, B) Cone implied inequalities This is all fine but the definition of K-sublinearity is much less apparent, how can we hope to verify it? F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 18 / 34

49 Relations to Support Functions Consider any v.i. (µ; η 0 ) and let D µ = {λ R m : µ A λ K }. Remark Let σ D ( ) be the support function of D µ, i.e., σ D (h) = sup{ h, λ : λ D µ }. λ For all v.i. (µ; η 0 ), we have D µ, σ D (0) = 0, and σ D (Ad) µ, d for all d K. F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 19 / 34

50 Necessary Conditions for K-sublinearity Remark For any µ Im(A ) + K, D µ σ D (Az) µ, z holds z K; and for any η 0 inf b B σ D (b), we have (µ; η 0 ) C(A, K, B). F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 20 / 34

51 Necessary Conditions for K-sublinearity Remark For any µ Im(A ) + K, D µ σ D (Az) µ, z holds z K; and for any η 0 inf b B σ D (b), we have (µ; η 0 ) C(A, K, B). Proposition [Necessary Condition for K-Sublinearity] Suppose µ satisfies condition (A.0), i.e., µ Im(A ) + K. Then µ Im(A ), we have σ D (Az) = µ, z for all z K. For any (µ; η 0 ) C(A, K, B), if S(A, K, B) is closed, then there exists z Ext(K) s.t. σ D (Az) = µ, z. F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 20 / 34

52 Sufficient Condition for K-sublinearity Proposition [Sufficient Condition for K-Sublinearity] Let (µ; η 0 ) C(A, K, B) and suppose that x i Ext(K) s.t. σ D (Ax i ) = µ, x i for all i I and i I x i int(k), then (µ; η 0 ) C a (A, K, B). F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 21 / 34

53 Refinement of K-sublinearity for K = R n + Refinement of the condition (A.1) for K-sublinearity leads to (A.0) 0 µ, u for all u R n + such that Au = 0, and (A.1i) for all i = 1,..., n, µ i µ, u for all u R n + such that Au = A i F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 22 / 34

54 Refinement of K-sublinearity for K = R n + Refinement of the condition (A.1) for K-sublinearity leads to (A.0) 0 µ, u for all u R n + such that Au = 0, and (A.1i) for all i = 1,..., n, µ i µ, u for all u R n + such that Au = A i Hence for K = R n +, we obtain identically the class of subadditive v.i. defined by Johnson 1981 via a much simplified analysis. In fact Johnson 1981 also shows that one needs to verify only a finitely many of these requirements (A.1i) for u satisfying a minimal dependence condition. F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 22 / 34

55 Refinement of K-sublinearity for K = R n + Refinement of the condition (A.1) for K-sublinearity leads to (A.0) 0 µ, u for all u R n + such that Au = 0, and (A.1i) for all i = 1,..., n, µ i µ, u for all u R n + such that Au = A i Hence for K = R n +, we obtain identically the class of subadditive v.i. defined by Johnson 1981 via a much simplified analysis. In fact Johnson 1981 also shows that one needs to verify only a finitely many of these requirements (A.1i) for u satisfying a minimal dependence condition. Proposition [Necessary Condition for R n +-Sublinearity] Let K = R n +. For any (µ; η 0 ) C a (A, K, B), we have σ D (Az) = µ, z for all z Ext(K), i.e., σ D (A i ) = µ i for all i = 1,..., n where A i is the i th column of the matrix A. F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 22 / 34

56 Sufficient Condition for K-sublinearity Proposition [Sufficient Condition for K-Sublinearity] Let (µ; η 0 ) C(A, K, B) and suppose that x i Ext(K) s.t. σ D (Ax i ) = µ, x i for all i I and i I x i int(k), then (µ; η 0 ) C a (A, K, B). Complete characterization of R n +-sublinear inequalities: F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 23 / 34

57 Sufficient Condition for K-sublinearity Proposition [Sufficient Condition for K-Sublinearity] Let (µ; η 0 ) C(A, K, B) and suppose that x i Ext(K) s.t. σ D (Ax i ) = µ, x i for all i I and i I x i int(k), then (µ; η 0 ) C a (A, K, B). Complete characterization of R n +-sublinear inequalities: All R n +-sublinear inequalities are generated by sublinear (subadditive and positively homogeneous, in fact also piecewise linear and convex) functions, i.e., support functions σ Dµ ( ) of D µ. [ This recovers a number of results from Johnson 81, and Conforti et al. 13. ] F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 23 / 34

58 Sufficient Condition for K-sublinearity Proposition [Sufficient Condition for K-Sublinearity] Let (µ; η 0 ) C(A, K, B) and suppose that x i Ext(K) s.t. σ D (Ax i ) = µ, x i for all i I and i I x i int(k), then (µ; η 0 ) C a (A, K, B). Complete characterization of R n +-sublinear inequalities: All R n +-sublinear inequalities are generated by sublinear (subadditive and positively homogeneous, in fact also piecewise linear and convex) functions, i.e., support functions σ Dµ ( ) of D µ. [ This recovers a number of results from Johnson 81, and Conforti et al. 13. ] This is the underlying source of a cut generating function view for linear MIPs. F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 23 / 34

59 Sufficient Condition for K-minimality Theorem [Sufficient Condition for K-Minimality] Suppose that C m (A, K, B). Consider any (µ; η 0 ) C a (A, K, B) s.t. η 0 = inf b B σ D (b) and x i K s.t. i x i int(k), Ax i = b i with b i B satisfying σ D (b i ) = η 0 and µ, x i = η 0, then (µ; η 0 ) C m (A, K, B). F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 24 / 34

60 Sufficient Condition for K-minimality Theorem [Sufficient Condition for K-Minimality] Suppose that C m (A, K, B). Consider any (µ; η 0 ) C a (A, K, B) s.t. η 0 = inf b B σ D (b) and x i K s.t. i x i int(k), Ax i = b i with b i B satisfying σ D (b i ) = η 0 and µ, x i = η 0, then (µ; η 0 ) C m (A, K, B). For general cones K other than R n +, unfortunately there is a gap between the current necessary condition and the sufficient condition for K-minimality. F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 24 / 34

61 A Simple Example K = L 3, A = [1, 0, 0] and B = { 1, 1}., i.e., S(A, K, B) = {x R 3 : x 1 { 1, 1}, x 3 x x 2 2 } x 3 (0, 0) x 1 x 2 conv(s(a, K, B)) = {x R 3 : 1 x 1 1, x x 2 2 } F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 25 / 34

62 A Simple Example K = L 3, A = [1, 0, 0] and B = { 1, 1}., i.e., S(A, K, B) = {x R 3 : x 1 { 1, 1}, x 3 x x 2 2 } conv(s(a, K, B)) = {x R 3 : 1 x 1 1, x x 2 2 } K-minimal inequalities are: (a) µ (+) = (1; 0; 0) with η (+) 0 = 1 and µ ( ) = ( 1; 0; 0) with η ( ) 0 = 1; (b) µ (t) = (0; t; t 2 + 1) with η (t) 0 = 1 for all t R. F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 25 / 34

63 A Simple Example K = L 3, A = [1, 0, 0] and B = { 1, 1}., i.e., S(A, K, B) = {x R 3 : x 1 { 1, 1}, x 3 x x 2 2 } conv(s(a, K, B)) = {x R 3 : 1 x 1 1, x x 2 2 } K-minimal inequalities are: (a) µ (+) = (1; 0; 0) with η (+) 0 = 1 and µ ( ) = ( 1; 0; 0) with η ( ) 0 = 1; (b) µ (t) = (0; t; t 2 + 1) with η (t) 0 = 1 for all t R. (these can be expressed as a single conic inequality x x 2 2.) F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 25 / 34

64 A Simple Example K = L 3, A = [1, 0, 0] and B = { 1, 1}., i.e., S(A, K, B) = {x R 3 : x 1 { 1, 1}, x 3 x x 2 2 } conv(s(a, K, B)) = {x R 3 : 1 x 1 1, x x 2 2 } K-minimal inequalities are: (a) µ (+) = (1; 0; 0) with η (+) 0 = 1 and µ ( ) = ( 1; 0; 0) with η ( ) 0 = 1; (b) µ (t) = (0; t; t 2 + 1) with η (t) 0 = 1 for all t R. (these can be expressed as a single conic inequality x x 2 2.) Linear inequalities in (b) cannot be generated by any cut generating function ρ( ), i.e., ρ(a i ) = µ (t) i is not possible for any function ρ( ). F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 25 / 34

65 A Simple Example K = L 3, A = [1, 0, 0] and B = { 1, 1}., i.e., S(A, K, B) = {x R 3 : x 1 { 1, 1}, x 3 x x 2 2 } conv(s(a, K, B)) = {x R 3 : 1 x 1 1, x x 2 2 } K-minimal inequalities are: (a) µ (+) = (1; 0; 0) with η (+) 0 = 1 and µ ( ) = ( 1; 0; 0) with η ( ) 0 = 1; (b) µ (t) = (0; t; t 2 + 1) with η (t) 0 = 1 for all t R. (these can be expressed as a single conic inequality x x 2 2.) Linear inequalities in (b) cannot be generated by any cut generating function ρ( ), i.e., ρ(a i ) = µ (t) i is not possible for any function ρ( ). One cannot hope to develop a strong conic dual for problems of form min x { c, x : Ax = b, x K, x i is integer for all i I }. [Sharp contrast to the results of Dey, Moran & Vielma 12. ] F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 25 / 34

66 Can we derive conic valid inequalities for S(A, K, B) using this framework? F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 26 / 34

67 Disjunctive Cuts for Lorentz Cone, L n Start with a simple set for x, i.e., x K = L n Consider a two-term disjunction of form either π T 1 x π 1,0 or π T 2 x π 2,0 must hold. F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 27 / 34

68 Disjunctive Cuts for Lorentz Cone, L n Start with a simple set for x, i.e., x K = L n Consider a two-term disjunction of form either π T 1 x π 1,0 or π T 2 x π 2,0 must hold. Let S i := {x : π T i x π i,0, x K}. F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 27 / 34

69 Disjunctive Cuts for Lorentz Cone, L n Start with a simple set for x, i.e., x K = L n Consider a two-term disjunction of form either π T 1 x π 1,0 or π T 2 x π 2,0 must hold. Let S i := {x : π T i x π i,0, x K}. S 1 S 2 Without loss of generality assume that π 1,0, π 2,0 {0, ±1} and S 1 and S 2. F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 27 / 34

70 Disjunctive Cuts for Lorentz Cone, L n Start with a simple set for x, i.e., x K = L n Consider a two-term disjunction of form either π T 1 x π 1,0 or π T 2 x π 2,0 must hold. Let S i := {x : π T i x π i,0, x K}. By setting [ π T A = 1 π T 2 we arrive back at and S(A, K, B) = S 1 S 2. ] {[ π1,0 + R, and B = + R ] [ S(A, K, B) = {x R n : Ax B, x K} R π 2,0 + R + ]} F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 27 / 34

71 Disjunctive Cuts for Lorentz Cone, L n The cases of S 1 S 2 or S 2 S 1 are not interesting, so we assume Assumption The disjunction π1 T x π 1,0 and π2 T x π 2,0 satisfy {β R n + : βπ 1,0 π 2,0, π 2 βπ 1 L n } =, and {β R n + : βπ 2,0 π 1,0, π 1 βπ 2 L n } =. F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 28 / 34

72 Disjunctive Cuts for Lorentz Cone, L n Convex nonlinear valid inequalities for L n via a disjunctive argument [Sketch of derivation]: Given a disjunction π T 1 x π 1,0 and π T 2 x π 2,0 with π 1,0, π 2,0 {0, ±1} Characterize the structure of linear L n -minimal valid inequalities F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 29 / 34

73 Disjunctive Cuts for Lorentz Cone, L n Convex nonlinear valid inequalities for L n via a disjunctive argument [Sketch of derivation]: Given a disjunction π T 1 x π 1,0 and π T 2 x π 2,0 with π 1,0, π 2,0 {0, ±1} Characterize the structure of linear L n -minimal valid inequalities Proposition [K.-K., Yıldız] For all L n -minimal valid linear inequalities µ x µ 0 for conv(s 1 S 2 ) there exists α 1, α 2 bd(l n ), and β 1, β 2 (R + \ {0}) s.t. µ = α 1 + β 1 π 1, µ = α 2 + β 2 π 2, min{π 1,0 β 1, π 2,0 β 2 } = µ 0 = min{π 1,0, π 2,0 }, and at least one of β 1 and β 2 is equal to 1. F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 29 / 34

74 Disjunctive Cuts for Lorentz Cone, L n Convex nonlinear valid inequalities for L n via a disjunctive argument [Sketch of derivation]: Given a disjunction π T 1 x π 1,0 and π T 2 x π 2,0 with π 1,0, π 2,0 {0, ±1} Characterize the structure of linear L n -minimal valid inequalities Based on their characterization, e.g., (β 1, β 2 ) values, group all of the linear L n -minimal valid linear inequalities via an optimization problem over µ F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 29 / 34

75 Disjunctive Cuts for Lorentz Cone, L n Convex nonlinear valid inequalities for L n via a disjunctive argument [Sketch of derivation]: Given a disjunction π T 1 x π 1,0 and π T 2 x π 2,0 with π 1,0, π 2,0 {0, ±1} Characterize the structure of linear L n -minimal valid inequalities Based on their characterization, e.g., (β 1, β 2 ) values, group all of the linear L n -minimal valid linear inequalities via an optimization problem over µ Turns out to be a nonconvex optimization problem, but it has a tight relaxation F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 29 / 34

76 Disjunctive Cuts for Lorentz Cone, L n Convex nonlinear valid inequalities for L n via a disjunctive argument [Sketch of derivation]: Given a disjunction π T 1 x π 1,0 and π T 2 x π 2,0 with π 1,0, π 2,0 {0, ±1} Characterize the structure of linear L n -minimal valid inequalities Based on their characterization, e.g., (β 1, β 2 ) values, group all of the linear L n -minimal valid linear inequalities via an optimization problem over µ Turns out to be a nonconvex optimization problem, but it has a tight relaxation Process the relaxation of this problem by taking its dual, etc., to arrive at the following main result F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 29 / 34

77 Disjunctive Cuts for Lorentz Cone, L n Disjunction: either π T 1 x π 1,0 or π T 2 x π 2,0 Theorem [K.-K., Yıldız] Let σ = min{π 1,0, π 2,0 }. For any β > 0 such that βπ 1 π 2 / ±int(l n ), the following convex inequality is valid for conv(s 1 S 2 ): 2σ (βπ 1 + π 2 ) x ((βπ 1 π 2 ) x) 2 + N(β) (xn 2 x 2 2 ) where N(β) := β π 1 π (βπ 1,n π 2,n ) 2, and exactly captures all linear v.i. corresponding to β 1 = β and β 2 = 1. F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 30 / 34

78 Disjunctive Cuts for Lorentz Cone, L n 2σ (βπ 1 + π 2 ) x ((βπ 1 π 2 ) x) 2 + N(β) (x 2 n x 2 2 ) Theorem [K.-K., Yıldız] In certain cases such as conv(s 1 S 2 ) is closed, Splits, i.e., π 1 = απ 2 for some α > 0, and π 1,0 = π 2,0 = σ with σ = 1 it is sufficient (for conv(s 1 S 2 )) to consider only one inequality with β = 1. F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 31 / 34

79 Disjunctive Cuts for Lorentz Cone, L n 2σ (βπ 1 + π 2 ) x ((βπ 1 π 2 ) x) 2 + N(β) (x 2 n x 2 2 ) Theorem [K.-K., Yıldız] In certain cases such as conv(s 1 S 2 ) is closed, Splits, i.e., π 1 = απ 2 for some α > 0, and π 1,0 = π 2,0 = σ with σ = 1 it is sufficient (for conv(s 1 S 2 )) to consider only one inequality with β = 1. For splits with rhs σ = 1, it is exactly the following conic quadratic inequality x 2(π 1 x σ) ( ( π 1 π 2 ) N(1) x n + 2(π 1 x σ) ) (π 1,n π 2,n ) 2 N(1) F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 31 / 34

80 Disjunctive Cuts for Lorentz Cone, L n Disjunction: x 3 1 or 2x 1 + 2x 3 1 conv(s 1 S 2 ) = { } x L 3 : 2 (2x 1 + 3x 2 ) ( 2x 1 x 3 ) (x 23 x 21 x 22 ) F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 32 / 34

81 Disjunctive Cuts for Lorentz Cone, L n Disjunction: x 3 1 or 2x 1 + 2x 3 1 conv(s 1 S 2 ) = { } x L 3 : 2 (2x 1 + 3x 2 ) ( 2x 1 x 3 ) (x 23 x 21 x 22 ) F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 32 / 34

82 Disjunctive Cuts for Lorentz Cone, L n Disjunction: x 3 1 or x 2 0 conv(s 1 S 2 ) = {x L 3 : x 2 1, 1 + x 1 x 3 } 1 max{0, x 2 } 2 F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 32 / 34

83 Final remarks Introduce a unifying framework for defining K-minimal and K-sublinear inequalities for conic MIPs F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 33 / 34

84 Final remarks Introduce a unifying framework for defining K-minimal and K-sublinear inequalities for conic MIPs Understand when K-minimal inequalities exist and are sufficient Characterize structure of K-minimal and K-sublinear inequalities Necessary, and also sufficient conditions Relation with support functions of certain structured sets F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 33 / 34

85 Final remarks Introduce a unifying framework for defining K-minimal and K-sublinear inequalities for conic MIPs Understand when K-minimal inequalities exist and are sufficient Characterize structure of K-minimal and K-sublinear inequalities Necessary, and also sufficient conditions Relation with support functions of certain structured sets Captures previous results from the MIP literature, i.e., K = R n + (i.e., Johnson 81 and Conforti et al. 13) F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 33 / 34

86 Final remarks Introduce a unifying framework for defining K-minimal and K-sublinear inequalities for conic MIPs Understand when K-minimal inequalities exist and are sufficient Characterize structure of K-minimal and K-sublinear inequalities Necessary, and also sufficient conditions Relation with support functions of certain structured sets Captures previous results from the MIP literature, i.e., K = R n + (i.e., Johnson 81 and Conforti et al. 13) For K = L n, by studying structure of linear K-minimal inequalities from this framework, we derive explicit expressions for conic cuts Covers most of the recent results on conic MIR, split, and two-term disjunctive inequalities (i.e., Belotti et al. 11, Andersen & Jensen 13, Modaresi et al. 13) Much more intuitive and elegant derivations covering new cases F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 33 / 34

87 Thank you! F. Kılınç-Karzan (CMU) Structure of Valid Inequalities for MICPs 34 / 34