Structure of Valid Inequalities for Mixed Integer Conic Programs Fatma Kılın¸ c-Karzan Tepper School of Business Carnegie Mellon University 18 th Combinatorial Optimization Workshop Aussois, France January 6-10, 2014 F. Kılın¸ c-Karzan (CMU) Structure of Valid Inequalities for MICPs 1 / 34
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. Kılın¸ c-Karzan (CMU) Structure of Valid Inequalities for MICPs 2 / 34
Mixed Integer Conic Programming Mixed Integer Linear Program c T x min s.t. Ax ≥ b x ∈ Z d × R n − d F. Kılın¸ c-Karzan (CMU) Structure of Valid Inequalities for MICPs 3 / 34
Mixed Integer Conic Programming Mixed Integer Linear Program c T x min Ax − b ∈ R m s.t. + x ∈ Z d × R n − d min c T F. Kılın¸ c-Karzan (CMU) Structure of Valid Inequalities for MICPs 3 / 34
Mixed Integer Conic Programming Mixed Integer Linear Program Mixed Integer Convex Program c T x min c T x min Ax − b ∈ R m s.t. s.t. x ∈ Q + x ∈ Z d × R n − d x ∈ Z d × R n − d where Q is a closed convex set min c T min c T Q F. Kılın¸ c-Karzan (CMU) Structure of Valid Inequalities for MICPs 3 / 34
Mixed Integer Conic Programming Mixed Integer Linear Program Mixed Integer Conic Program c T x c T x min min Ax − b ∈ R m s.t. s.t. Ax − b ∈ K + x ∈ Z d × R n − d x ∈ Z d × R n − d where K is a convex cone min c T x 3 x 1 0 x 2 F. Kılın¸ c-Karzan (CMU) Structure of Valid Inequalities for MICPs 4 / 34
Mixed Integer Conic Programming Mixed Integer Linear Program Mixed Integer Conic Program c T x c T x min min Ax − b ∈ R m s.t. s.t. Ax − b ∈ K + x ∈ Z d × R n − d x ∈ Z d × R n − d where K is a convex cone min c T Nonnegative orthant + = { y ∈ R m : y i ≥ 0 ∀ i } R m Lorentz cone L m = { y ∈ R m : y m ≥ � y 2 2 + . . . + y 2 m − 1 } Positive semidefinite cone + = { X ∈ R m × m : a T Xa ≥ 0 ∀ a ∈ R m } S m F. Kılın¸ c-Karzan (CMU) Structure of Valid Inequalities for MICPs 4 / 34
Mixed Integer Conic Programming Mixed Integer Linear Program Mixed Integer Conic Program c T x c T x min min Ax − b ∈ R m s.t. s.t. Ax − b ∈ K + x ∈ Z d × R n − d x ∈ Z d × R n − d where K is a convex cone min c T min c T F. Kılın¸ c-Karzan (CMU) Structure of Valid Inequalities for MICPs 4 / 34
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. Kılın¸ c-Karzan (CMU) Structure of Valid Inequalities for MICPs 5 / 34
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. Kılın¸ c-Karzan (CMU) Structure of Valid Inequalities for MICPs 5 / 34
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. Kılın¸ c-Karzan (CMU) Structure of Valid Inequalities for MICPs 5 / 34
Is it possible to develop a framework to establish strength of valid linear inequalities for MICPs? F. Kılın¸ c-Karzan (CMU) Structure of Valid Inequalities for MICPs 6 / 34
Problem Setting We are interested in the convex hull of the following set: S ( A , K , B ) = { x ∈ E : Ax ∈ B , x ∈ K} where 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 + := { x ∈ R n : x i ≥ 0 ∀ i } Nonnegative orthant, R n � Lorentz cone, L n := { x ∈ R n : x n ≥ x 2 2 + . . . + x 2 n − 1 } + := { X ∈ R n × n : a T Xa ≥ 0 ∀ a ∈ R n } Positive semidefinite cone, S n F. Kılın¸ c-Karzan (CMU) Structure of Valid Inequalities for MICPs 7 / 34
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. Kılın¸ c-Karzan (CMU) Structure of Valid Inequalities for MICPs 8 / 34
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 ): + × Z q : Wy + Hv − b ∈ K} { ( y , v ) ∈ R k where K ⊂ E is a full-dimensional, closed, convex, pointed cone. F. Kılın¸ c-Karzan (CMU) Structure of Valid Inequalities for MICPs 8 / 34
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 ): + × Z q : Wy + Hv − b ∈ K} { ( y , v ) ∈ R k where K ⊂ E is a full-dimensional, closed, convex, pointed cone. Define � y � � � and B = b − H Z q , x = , A = W , − Id , 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. Kılın¸ c-Karzan (CMU) Structure of Valid Inequalities for MICPs 8 / 34
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 ): + × Z q : Wy + Hv − b ∈ K} { ( y , v ) ∈ R k where K ⊂ E is a full-dimensional, closed, convex, pointed cone. Define � y � � � and B = b − H Z q , x = , A = W , − Id , 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. Kılın¸ c-Karzan (CMU) Structure of Valid Inequalities for MICPs 8 / 34
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. Kılın¸ c-Karzan (CMU) Structure of Valid Inequalities for MICPs 9 / 34
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. Kılın¸ c-Karzan (CMU) Structure of Valid Inequalities for MICPs 9 / 34
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. Kılın¸ c-Karzan (CMU) Structure of Valid Inequalities for MICPs 9 / 34
Goal S ( A , K , B ) = { x ∈ E : Ax ∈ B , x ∈ K} We are interested in conv ( S ( A , K , B )) F. Kılın¸ c-Karzan (CMU) Structure of Valid Inequalities for MICPs 10 / 34
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. Kılın¸ c-Karzan (CMU) Structure of Valid Inequalities for MICPs 10 / 34
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