Structure in Mixed Integer Conic Optimization: From Minimal Inequalities to Conic Disjunctive Cuts

Transcription

1 Structure in Mixed Integer Conic Optimization: From Minimal Inequalities to Conic Disjunctive Cuts Fatma Kılınç-Karzan Tepper School of Business Carnegie Mellon University Joint work with Sercan Yıldız (CMU) MINLP Workshop Pittsburgh (PA) June 03, 2014 F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 1 / 27

2 Outline Mixed integer conic optimization (MICP) Problem setting Motivation Structure of linear valid inequalities K-minimal valid inequalities K-sublinear valid inequalities Disjunctive cuts for Lorentz cone (joint work with Sercan Yıldız (CMU)) Structure of valid linear inequalities A class of valid convex inequalities Nice case and nasty case (with examples) F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 2 / 27

3 A Disjunctive Perspective Problem: Study the closed convex hull of 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 [regular cone] F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 3 / 27

4 A Disjunctive Perspective Problem: Study the closed convex hull of 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 [regular cone] Most (if not all!) cutting planes (convexification techniques) in MILPs rely on disjunctive formulations. F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 3 / 27

5 Representation Flexibility S(A, K, B) = {x E : Ax B, x K} This set captures the essential structure of MICPs F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 4 / 27

6 Representation Flexibility S(A, K, B) = {x E : Ax B, x K} This set captures the essential structure of MICPs {(y, v) R k + Z q : W y + H v b K} where K E is a full-dimensional, closed, convex, pointed cone. F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 4 / 27

7 Representation Flexibility S(A, K, B) = {x E : Ax B, x K} This set captures the essential structure of MICPs {(y, v) R k + Z q : W y + H v b K} where K E is a full-dimensional, closed, convex, pointed cone. Define ( ) y x =, A = [ W, Id ], and B = b H Z 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 in Mixed Integer Conic Sets 4 / 27

8 Representation Flexibility S(A, K, B) = {x E : Ax B, x K} Models traditional disjunctive formulations Captures the essential non-convex structure of MICPs Precisely contains famous Gomory s Corner Polyhedron (1969) Arises in modeling complementarity relations Can be used as a natural relaxation for sequential convexification etc., etc... F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 5 / 27

9 Questions of Interest S(A, K, B) = {x E : Ax B, x K} General Questions: When can we study/characterize conv(s(a, K, B)) explicitly? Will (or when will) conv(s(a, K, B)) preserve the nice structural properties we originally had? F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 6 / 27

10 Questions of Interest S(A, K, B) = {x E : Ax B, x K} General Questions: When can we study/characterize conv(s(a, K, B)) explicitly? Will (or when will) conv(s(a, K, B)) preserve the nice structural properties we originally had? Let s first see what we can find out about the structure of linear valid inequalities... F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 6 / 27

11 Structure of Valid Linear Inequalities S(A, K, B) = {x E : Ax B, x K} 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 in Mixed Integer Conic Sets 7 / 27

12 Structure of Valid Linear Inequalities S(A, K, B) = {x E : Ax B, x K} 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) 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 in Mixed Integer Conic Sets 7 / 27

13 Which Inequalities Should We Really Care About? S(A, K, B) = {x E : Ax B, x K} C(A, K, B) = convex cone of (µ; η 0 ) of all linear valid inequalities µ, x η 0 C(A, K, B) F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 8 / 27

14 Which Inequalities Should We Really Care About? S(A, K, B) = {x E : Ax B, x K} C(A, K, B) = convex cone of (µ; η 0 ) of all linear valid inequalities µ, x η 0 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 in Mixed Integer Conic Sets 8 / 27

15 Which Inequalities Should We Really Care About? S(A, K, B) = {x E : Ax B, x K} C(A, K, B) = convex cone of (µ; η 0 ) of all linear valid inequalities µ, x η 0 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 in Mixed Integer Conic Sets 8 / 27

16 Which Inequalities Should We Really Care About? S(A, K, B) = {x E : Ax B, x K} C(A, K, B) = convex cone of (µ; η 0 ) of all linear valid inequalities µ, x η 0 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 in Mixed Integer Conic Sets 8 / 27

17 Which Inequalities Should We Really Care About? S(A, K, B) = {x E : Ax B, x K} C(A, K, B) = convex cone of (µ; η 0 ) of all linear valid inequalities µ, x η 0 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 in Mixed Integer Conic Sets 8 / 27

18 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 in Mixed Integer Conic Sets 9 / 27

19 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 in Mixed Integer Conic Sets 9 / 27

20 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 in Mixed Integer Conic Sets 9 / 27

21 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}. Theorem: [Sufficiency of K-minimal Inequalities] Whenever C m (A, K, B), K-minimal v.i. together with x K constraint are sufficient to describe conv(s(a, K, B)). F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 9 / 27

22 Some Properties We can show that Every valid inequality (µ; η 0 ) C(A, K, B) satisfies condition (A.0) µ Im(A ) + K. F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 10 / 27

23 Some Properties We can show that Every valid inequality (µ; η 0 ) C(A, K, B) satisfies condition (A.0) µ Im(A ) + K. Remark For any µ Im(A ) + K, D µ := {λ R m : µ A λ K } ; and for any η 0 inf b B σ Dµ (b), where σ Dµ := support function of D µ, we have (µ; η 0 ) C(A, K, B). F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 10 / 27

24 Some Properties We can show that Every valid inequality (µ; η 0 ) C(A, K, B) satisfies condition (A.0) µ Im(A ) + K. K-minimal inequalities have more structure, i.e., they are K-sublinear: Definition An inequality (µ; η 0 ) is a K-sublinear v.i. (C a (A, K, B)) if it satisfies (A.1) 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 ), (A.2) η 0 µ, x for all x S(A, K, B). Condition (A.1) implies (A.0). F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 10 / 27

25 On Conditions for K-minimality and K-sublinearity We can establish necessary, and also sufficient conditions for an inequality (µ; η 0 ) to be K-minimal or K-sublinear via its relation with support function σ Dµ of the structured set D µ. F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 11 / 27

26 On Conditions for K-minimality and K-sublinearity We can establish necessary, and also sufficient conditions for an inequality (µ; η 0 ) to be K-minimal or K-sublinear via its relation with support function σ Dµ of the structured set D µ. When K = R n +: K-sublinear v.i. are identical to the class of subadditive v.i. defined by Johnson 81, i.e., condition (A.1) is precisely condition (A.0) and (A.1i) for all i = 1,..., n, µ i µ, u for all u R n + s.t. Au = A i. F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 11 / 27

27 On Conditions for K-minimality and K-sublinearity We can establish necessary, and also sufficient conditions for an inequality (µ; η 0 ) to be K-minimal or K-sublinear via its relation with support function σ Dµ of the structured set D µ. When K = R n +: K-sublinear v.i. are identical to the class of subadditive v.i. defined by Johnson 81, i.e., condition (A.1) is precisely condition (A.0) and (A.1i) for all i = 1,..., n, µ i µ, u for all u R n + s.t. Au = A i. Our sufficient condition for K-sublinearity matches precisely our necessary condition, resulting in (µ; η 0) C a(a, K, B) µ i = σ Dµ (A i ) for all i. All R n +-sublinear (and thus R n +-minimal) inequalities are generated by sublinear functions (subadditive and positively homogeneous, in fact also piecewise linear and convex), 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 in Mixed Integer Conic Sets 11 / 27

28 On Conditions for K-minimality and K-sublinearity We can establish necessary, and also sufficient conditions for an inequality (µ; η 0 ) to be K-minimal or K-sublinear via its relation with support function σ Dµ of the structured set D µ. When K = R n +: K-sublinear v.i. are identical to the class of subadditive v.i. defined by Johnson 81, i.e., condition (A.1) is precisely condition (A.0) and (A.1i) for all i = 1,..., n, µ i µ, u for all u R n + s.t. Au = A i. Our sufficient condition for K-sublinearity matches precisely our necessary condition, resulting in (µ; η 0) C a(a, K, B) µ i = σ Dµ (A i ) for all i. All R n +-sublinear (and thus R n +-minimal) inequalities are generated by sublinear functions (subadditive and positively homogeneous, in fact also piecewise linear and convex), 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 underlies a cut generating function view for MILPs. F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 11 / 27

29 On Conditions for K-minimality and K-sublinearity For general regular cones K other than R n +, unfortunately there is a gap between our current necessary condition and our sufficient condition for K-minimality. Moreover, there is a simple example S(A, K, B) with K = L 3, where a necessary (in terms of convex hull description) family of K-minimal inequalities cannot be generated by any class of cut generating functions. This is in sharp contrast to MILP case, i.e., K = R n +, and, to the strong dual for MICP result of Moran et al. 12, which studied a more restrictive conic setup. F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 12 / 27

30 Questions on Structure General Questions: S(A, K, B) = {x E : Ax B, x K} When can we characterize conv(s(a, K, B)) explicitly? Will (or when will) conv(s(a, K, B)) preserve the nice structural properties we originally had? F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 13 / 27

31 Questions on Structure General Questions: S(A, K, B) = {x E : Ax B, x K} When can we characterize conv(s(a, K, B)) explicitly? Will (or when will) conv(s(a, K, B)) preserve the nice structural properties we originally had? General case is too general for us to answer these questions... In the rest of this talk, we will study a simple (?) yet interesting case Joint work with S. Yıldız F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 13 / 27

32 Disjunctive Cuts for Lorentz Cone, L n Start with a simple set for x, i.e., a regular cone K R n Consider a two-term disjunction: either c1 T x c 1,0 or c2 T x c 2,0 must hold. Let C i := {x : ci T x c i,0, x K}. C 1 C 2 A special case is split disjunctions, i.e., c 1 = τc 2 for some τ > 0, and c 1,0c 2,0 > 0. F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 14 / 27

33 Disjunctive Cuts for Lorentz Cone, L n Start with a simple set for x, i.e., a regular cone K R n Consider a two-term disjunction: either c T 1 x c 1,0 or c T 2 x c 2,0 must hold. Let C i := {x : c T i By setting [ c T A = 1 we arrive at c T 2 x c i,0, x K}. ] {[ {c1,0 } + R, and B = + R ] [ R {c 2,0 } + R + S(A, K, B) = {x R n : Ax B, x K} = C 1 C 2. ]} F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 14 / 27

34 Disjunctive Cuts for Lorentz Cone, L n Start with a simple set for x, i.e., a regular cone K R n Consider a two-term disjunction: either c T 1 x c 1,0 or c T 2 x c 2,0 must hold. Let C i := {x : c T i By setting [ c T A = 1 we arrive at c T 2 x c i,0, x K}. ] {[ {c1,0 } + R, and B = + R ] [ R {c 2,0 } + R + S(A, K, B) = {x R n : Ax B, x K} = C 1 C 2. We are interested in describing conv(s(a, K, B)) in the original space of variables: Is there any structure in conv(s(a, K, B))? Can we preserve the simple conic structure we started out with? ]} F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 14 / 27

35 Disjunctive Cuts for Lorentz Cone, L n Start with a simple set for x, i.e., a regular cone K R n Consider a two-term disjunction: either c T 1 x c 1,0 or c T 2 x c 2,0 must hold. Let C i := {x : c T i By setting [ c T A = 1 we arrive at c T 2 x c i,0, x K}. ] {[ {c1,0 } + R, and B = + R ] [ R {c 2,0 } + R + S(A, K, B) = {x R n : Ax B, x K} = C 1 C 2. ]} Simple set we start out can be U = {x R n : Qx d K} with Q R m n having full row rank. F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 14 / 27

36 Approach Characterize the structure of K-minimal and tight valid linear inequalities F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 15 / 27

37 Approach Characterize the structure of K-minimal and tight valid linear inequalities Using conic duality, we group these linear inequalities appropriately, and thus derive a family of convex valid inequalities sufficient to describe the closed convex hull F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 15 / 27

38 Approach Characterize the structure of K-minimal and tight valid linear inequalities Using conic duality, we group these linear inequalities appropriately, and thus derive a family of convex valid inequalities sufficient to describe the closed convex hull Any structure beyond convexity? Understand when these convex inequalities are conic representable F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 15 / 27

39 Approach Characterize the structure of K-minimal and tight valid linear inequalities Using conic duality, we group these linear inequalities appropriately, and thus derive a family of convex valid inequalities sufficient to describe the closed convex hull Any structure beyond convexity? Understand when these convex inequalities are conic representable When does a single inequality from this family suffice? Characterize when only single inequality from this family is sufficient to describe the closed convex hull F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 15 / 27

40 Setup for Conic Disjunctive Cuts Start with a simple set for x, i.e., K Consider a two-term disjunction: either c T 1 x c 1,0 or c T 2 x c 2,0 must hold. Let C i := {x K : c T i x c i,0 }. F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 16 / 27

41 Setup for Conic Disjunctive Cuts Start with a simple set for x, i.e., K Consider a two-term disjunction: either c T 1 x c 1,0 or c T 2 x c 2,0 must hold. Let C i := {x K : c T i x c i,0 }. WLOG we assume that c 1,0, c 2,0 {0, ±1} and C 1 and C 2 (in fact we assume C 1, C 2 are strictly feasible) C 1 C 2 and C 2 C 1. F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 16 / 27

42 Setup for Conic Disjunctive Cuts Start with a simple set for x, i.e., K Consider a two-term disjunction: either c T 1 x c 1,0 or c T 2 x c 2,0 must hold. Let C i := {x K : c T i x c i,0 }. WLOG we assume that Assumption c 1,0, c 2,0 {0, ±1} and C 1 and C 2 (in fact we assume C 1, C 2 are strictly feasible) C 1 C 2 and C 2 C 1. This assumption is equivalent to The disjunction c T 1 x c 1,0 or c T 2 x c 2,0 satisfies {β R + : βc 1,0 c 2,0, c 2 βc 1 K} =, and {β R + : βc 2,0 c 1,0, c 1 βc 2 K} =. F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 16 / 27

43 Structure of Valid Linear Inequalities Disjunction on K: either c T 1 x c 1,0 or c T 2 x c 2,0 with c 1,0, c 2,0 {0, ±1} Standard Approach For any valid linear inequality, µ x µ 0 for conv(c 1 C 2 ) there exists α 1, α 2 K, and β 1, β 2 R + s.t. µ = α 1 + β 1 c 1, µ = α 2 + β 2 c 2, µ 0 min{β 1 c 1,0, β 2 c 2,0 }. F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 17 / 27

44 Structure of Valid Linear Inequalities Disjunction on K: either c T 1 x c 1,0 or c T 2 x c 2,0 with c 1,0, c 2,0 {0, ±1} Proposition For any K-minimal and tight valid linear inequality, µ x µ 0 for conv(c 1 C 2 ) there exists α 1, α 2 bd(k), and β 1, β 2 (R + \ {0}) s.t. µ = α 1 + β 1 c 1, µ = α 2 + β 2 c 2, min{c 1,0 β 1, c 2,0 β 2 } = µ 0 = min{c 1,0, c 2,0 }, and at least one of β 1 and β 2 is equal to 1. F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 17 / 27

45 Deriving a Nonlinear (but Convex) Valid Inequality when K = L n Assume c 1,0 c 2,0 β 2 = 1 F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 18 / 27

46 Deriving a Nonlinear (but Convex) Valid Inequality when K = L n Assume c 1,0 c 2,0 β 2 = 1 Consider the set of undominated v.i. µ T x µ 0 for given β 1 = β > 0 and β 2 = 1, i.e., µ 0 = min{c 1,0, c 2,0 } = c 2,0 and µ M(β, 1):={µ R n : α 1, α 2 bd L n s.t. µ = α 1 + βc 1 = α 2 + c 2 } F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 18 / 27

47 Deriving a Nonlinear (but Convex) Valid Inequality when K = L n Assume c 1,0 c 2,0 β 2 = 1 Consider the set of undominated v.i. µ T x µ 0 for given β 1 = β > 0 and β 2 = 1, i.e., µ 0 = min{c 1,0, c 2,0 } = c 2,0 and µ M(β, 1):={µ R n : α 1, α 2 bd L n s.t. µ = α 1 + βc 1 = α 2 + c 2 } = {µ R n : µ (β c 1 c 2 ) µ n (βc 1,n c 2,n )= M2 }, µ β c 1 2 =µ n βc 1,n where M := (β 2 c c 2 2 2) (β 2 c 2 1,n c 2 2,n). F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 18 / 27

48 Deriving a Nonlinear (but Convex) Valid Inequality when K = L n Assume c 1,0 c 2,0 β 2 = 1 Consider the set of undominated v.i. µ T x µ 0 for given β 1 = β > 0 and β 2 = 1, i.e., µ 0 = min{c 1,0, c 2,0 } = c 2,0 and µ M(β, 1):={µ R n : α 1, α 2 bd L n s.t. µ = α 1 + βc 1 = α 2 + c 2 } = {µ R n : µ (β c 1 c 2 ) µ n (βc 1,n c 2,n )= M2 }, µ β c 1 2 =µ n βc 1,n where M := (β 2 c c 2 2 2) (β 2 c 2 1,n c 2 2,n). x conv(c 1 C 2 ) x L n and µ x c 2,0 µ M(β, 1) F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 18 / 27

49 Deriving a Nonlinear (but Convex) Valid Inequality when K = L n Assume c 1,0 c 2,0 β 2 = 1 Consider the set of undominated v.i. µ T x µ 0 for given β 1 = β > 0 and β 2 = 1, i.e., µ 0 = min{c 1,0, c 2,0 } = c 2,0 and µ M(β, 1):={µ R n : α 1, α 2 bd L n s.t. µ = α 1 + βc 1 = α 2 + c 2 } = {µ R n : µ (β c 1 c 2 ) µ n (βc 1,n c 2,n )= M2 }, µ β c 1 2 =µ n βc 1,n where M := (β 2 c c 2 2 2) (β 2 c 2 1,n c 2 2,n). x conv(c 1 C 2 ) x L n and µ x c 2,0 µ M(β, 1) x L n { and inf µ x : µ M(β, 1) } c 2,0 µ F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 18 / 27

50 Deriving a Nonlinear (but Convex) Valid Inequality when K = L n Assume c 1,0 c 2,0 β 2 = 1 Consider the set of undominated v.i. µ T x µ 0 for given β 1 = β > 0 and β 2 = 1, i.e., µ 0 = min{c 1,0, c 2,0 } = c 2,0 and µ M(β, 1):={µ R n : α 1, α 2 bd L n s.t. µ = α 1 + βc 1 = α 2 + c 2 } = {µ R n : µ (β c 1 c 2 ) µ n (βc 1,n c 2,n )= M2 }, µ β c 1 2 =µ n βc 1,n where M := (β 2 c c 2 2 2) (β 2 c 2 1,n c 2 2,n). x conv(c 1 C 2 ) x L n and µ x c 2,0 µ M(β, 1) x L n { and inf µ x : µ M(β, 1) } c 2,0 µ x L n and inf µ { µ x : µ M(β, } 1) c 2,0 F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 18 / 27

51 Deriving a Nonlinear (but Convex) Valid Inequality when K = L n Assume c 1,0 c 2,0 β 2 = 1 Consider the set of undominated v.i. µ T x µ 0 for given β 1 = β > 0 and β 2 = 1, i.e., µ 0 = min{c 1,0, c 2,0 } = c 2,0 and µ M(β, 1):={µ R n : α 1, α 2 bd L n s.t. µ = α 1 + βc 1 = α 2 + c 2 } = {µ R n : µ (β c 1 c 2 ) µ n (βc 1,n c 2,n )= M2 }, µ β c 1 2 =µ n βc 1,n where M := (β 2 c c 2 2 2) (β 2 c 2 1,n c 2 2,n). x conv(c 1 C 2 ) x L n and µ x c 2,0 µ M(β, 1) x L n { and inf µ x : µ M(β, 1) } c 2,0 µ { x L n and inf µ x : µ } M(β, 1) c 2,0 µ {βc 1 ρ + M2 ( τ : ρ+τ x L n and max ρ,τ ) β c 1 c 2 =x, ρ L } c n βc 1,n + c 2,0 2,n F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 18 / 27

52 Convex Disjunctive Cut for K = L n Theorem Disjunction on second-order cone, L n = {x R n : x n x 2 }: either c T 1 x c 1,0 or c T 2 x c 2,0 with c 1,0 c 2,0 For any β > 0 s.t. βc 1,0 c 2,0 and βc 1 c 2 / ±int(l n ), then the following inequality is valid for conv(c 1 C 2 ): 2c 2,0 (βc 1 + c 2 ) x ((βc 1 c 2 ) x) 2 + N(β) (xn 2 x 2 2 ) where N(β) := β c 1 c (βc 1,n c 2,n ) 2. F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 19 / 27

53 Convex Disjunctive Cut for K = L n Theorem Disjunction on second-order cone, L n = {x R n : x n x 2 }: either c T 1 x c 1,0 or c T 2 x c 2,0 with c 1,0 c 2,0 For any β > 0 s.t. βc 1,0 c 2,0 and βc 1 c 2 / ±int(l n ), then the following inequality is valid for conv(c 1 C 2 ): 2c 2,0 (βc 1 + c 2 ) x ((βc 1 c 2 ) x) 2 + N(β) (xn 2 x 2 2 ) where N(β) := β c 1 c (βc 1,n c 2,n ) 2. From its construction, this inequality exactly captures all undominated linear v.i. µ T x µ 0 corresponding to β 1 = β and β 2 = 1, e.g., µ 0 = c 2,0 and µ M(β, 1) F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 19 / 27

54 Convex Disjunctive Cut for K = L n Theorem Disjunction on second-order cone, L n = {x R n : x n x 2 }: either c T 1 x c 1,0 or c T 2 x c 2,0 with c 1,0 c 2,0 For any β > 0 s.t. βc 1,0 c 2,0 and βc 1 c 2 / ±int(l n ), then the following inequality is valid for conv(c 1 C 2 ): 2c 2,0 (βc 1 + c 2 ) x ((βc 1 c 2 ) x) 2 + N(β) (xn 2 x 2 2 ) where N(β) := β c 1 c (βc 1,n c 2,n ) 2. From its construction, this inequality exactly captures all undominated linear v.i. µ T x µ 0 corresponding to β 1 = β and β 2 = 1, e.g., µ 0 = c 2,0 and µ M(β, 1) is valid and convex F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 19 / 27

55 Convex Disjunctive Cut for K = L n Theorem Disjunction on second-order cone, L n = {x R n : x n x 2 }: either c T 1 x c 1,0 or c T 2 x c 2,0 with c 1,0 c 2,0 For any β > 0 s.t. βc 1,0 c 2,0 and βc 1 c 2 / ±int(l n ), then the following inequality is valid for conv(c 1 C 2 ): 2c 2,0 (βc 1 + c 2 ) x ((βc 1 c 2 ) x) 2 + N(β) (xn 2 x 2 2 ) where N(β) := β c 1 c (βc 1,n c 2,n ) 2. From its construction, this inequality exactly captures all undominated linear v.i. µ T x µ 0 corresponding to β 1 = β and β 2 = 1, e.g., µ 0 = c 2,0 and µ M(β, 1) is valid and convex reduces to the linear inequality βc 1 x c 2,0 in L n when βc 1 c 2 ± bd L n F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 19 / 27

56 Structure of Convex Valid Inequalities 2c 2,0 (βc 1 + c 2 ) x ((βc 1 c 2 ) x) 2 + N(β) (x 2 n x 2 2 ) Any further structure than convexity? F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 20 / 27

57 Structure of Convex Valid Inequalities 2c 2,0 (βc 1 + c 2 ) x ((βc 1 c 2 ) x) 2 + N(β) (x 2 n x 2 2 ) Proposition Any further structure than convexity? An equivalent conic quadratic form given by ( ) N(β)x + 2(c2 β c x c 2,0 ) 1 c 2 L n βc 1,n + c 2,n is valid whenever a symmetry condition, e.g., 2c 2,0 + (βc 1 + c 2 ) x ((βc 1 c 2 ) x) 2 + N(β) (xn 2 x 2 ) holds for all x conv(c 1 C 2 ). F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 20 / 27

58 Structure of Convex Valid Inequalities 2c 2,0 (βc 1 + c 2 ) x ((βc 1 c 2 ) x) 2 + N(β) (x 2 n x 2 2 ) Any further structure than convexity? ( N(β)x + 2(c2 x c 2,0 ) ) β c 1 c 2 L n βc 1,n + c 2,n An equivalent conic quadratic form is valid, e.g., symmetry condition holds, when C 1 C 2 =, i.e., a proper split disjunction, or {x L n : βc 1 x c 2,0, c 2 x c 2,0}={x L n : βc 1 x = c 2,0, c 2 x = c 2,0} F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 20 / 27

59 When does a Single Convex Inequality Suffice? A parametric family of convex inequalities: For any β > 0 s.t. βc 1,0 c 2,0 and βc 1 c 2 / ±int(l n ), 2c 2,0 (βc 1 + c 2 ) x is valid for conv(c 1 C 2 ). ((βc 1 c 2 ) x) 2 + N(β) (x 2 n x 2 2 ) F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 21 / 27

60 When does a Single Convex Inequality Suffice? A parametric family of convex inequalities: For any β > 0 s.t. βc 1,0 c 2,0 and βc 1 c 2 / ±int(l n ), 2c 2,0 (βc 1 + c 2 ) x is valid for conv(c 1 C 2 ). ((βc 1 c 2 ) x) 2 + N(β) (x 2 n x 2 2 ) Theorem In certain cases such as c 1 L n or c 2 L n, or c 1,0 = c 2,0 {±1} and conv(c 1 C 2 ) is closed, e.g., in the case of split disjunctions, it is sufficient (for conv(c 1 C 2 )) to consider only one inequality with β = 1. F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 21 / 27

61 Example: Nice Case Disjunction: x 3 1 or x 1 + x 3 1 conv(c 1 C 2 ) = { } x L 3 : 2 (x 1 + 2x 3 ) x3 2 x 2 2 F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 22 / 27

62 Example: Nice Case Disjunction: x 3 1 or x 1 + x 3 1 conv(c 1 C 2 ) = { } x L 3 : 2 (x 1 + 2x 3 ) x3 2 x 2 2 F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 22 / 27

63 Not So Nice Case There are cases when we need infinitely many convex inequalities from this family. Recessive directions, and conv(c 1 C 2 ) being non-closed, play a key role in these cases. F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 23 / 27

64 Not So Nice Case There are cases when we need infinitely many convex inequalities from this family. Recessive directions, and conv(c 1 C 2 ) being non-closed, play a key role in these cases. We can still give expressions for a single inequality describing conv(c 1 C 2 ), but it is really nasty looking... F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 23 / 27

65 Example: Nasty Case Disjunction: x 3 1 or x 2 0 conv(c 1 C 2 ) = {x L 3 : x 2 1, 1 + x 1 x 3 } 1 max{0, x 2 } 2 F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 24 / 27

66 Final Remarks Introduce and study the properties of K-minimal and K-sublinear inequalities for conic MIPs F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 25 / 27

67 Final Remarks Introduce and study the properties of K-minimal and K-sublinear inequalities for conic MIPs For two-term disjunctions on Lorentz cone, L n Derive explicit expressions for disjunctive conic cuts Cover most of the recent results on conic MIR, split, and two-term disjunctive inequalities for Lorentz cones (i.e., Belotti et al. 11, Andersen & Jensen 13, Modaresi et al. 13) Extends to elementary split disjunctions on p-order cones F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 25 / 27

68 Final Remarks Introduce and study the properties of K-minimal and K-sublinear inequalities for conic MIPs For two-term disjunctions on Lorentz cone, L n Derive explicit expressions for disjunctive conic cuts Cover most of the recent results on conic MIR, split, and two-term disjunctive inequalities for Lorentz cones (i.e., Belotti et al. 11, Andersen & Jensen 13, Modaresi et al. 13) Extends to elementary split disjunctions on p-order cones More intuitive and elegant derivations leading to new insights When we can have valid conic inequalities in the original space When a single convex inequality will suffice F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 25 / 27

69 Final Remarks Introduce and study the properties of K-minimal and K-sublinear inequalities for conic MIPs For two-term disjunctions on Lorentz cone, L n Derive explicit expressions for disjunctive conic cuts Cover most of the recent results on conic MIR, split, and two-term disjunctive inequalities for Lorentz cones (i.e., Belotti et al. 11, Andersen & Jensen 13, Modaresi et al. 13) Extends to elementary split disjunctions on p-order cones More intuitive and elegant derivations leading to new insights When we can have valid conic inequalities in the original space When a single convex inequality will suffice Extends to disjunctions on cross-sections of the Lorentz cone (Joint work with S. Yıldız and G. Cornuéjols) F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 25 / 27

70 Thank you! F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 26 / 27

71 References Kılınç-Karzan, F., On Minimal Valid Inequalities for Mixed Integer Conic Programs. GSIA Working Paper Number: 2013-E20, GSIA, Carnegie Mellon University, Pittsburgh, PA, June Kılınç-Karzan, F., Yıldız, S., Two-Term Disjunctions on the Second-Order Cone. April A shorter version is published as part of Proceedings of 17 th IPCO Conference. F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 27 / 27

72 A Simple Example for Insufficiency of Cut Generating Functions 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 in Mixed Integer Conic Sets 28 / 27

73 A Simple Example for Insufficiency of Cut Generating Functions 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 in Mixed Integer Conic Sets 28 / 27

74 A Simple Example for Insufficiency of Cut Generating Functions 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 in Mixed Integer Conic Sets 28 / 27

75 A Simple Example for Insufficiency of Cut Generating Functions 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( ). [Sharp contrast to the strong conic IP dual result of Moran, Dey, & Vielma 12. ] F. Kılınç-Karzan (CMU) Structure in Mixed Integer Conic Sets 28 / 27