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Structure of Valid Inequalities for Mixed Integer Conic Programs

Fatma Kılın¸ c-Karzan

Tepper School of Business Carnegie Mellon University

18th Combinatorial Optimization Workshop Aussois, France January 6-10, 2014

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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

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Mixed Integer Conic Programming

Mixed Integer Linear Program min cTx s.t. Ax ≥ b x ∈ Zd × Rn−d

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Mixed Integer Conic Programming

Mixed Integer Linear Program min cTx s.t. Ax − b ∈ Rm

+

x ∈ Zd × Rn−d

min cT

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c-Karzan (CMU) Structure of Valid Inequalities for MICPs 3 / 34

Mixed Integer Conic Programming

Mixed Integer Linear Program min cTx s.t. Ax − b ∈ Rm

+

x ∈ Zd × Rn−d

min cT

Mixed Integer Convex Program min cTx s.t. x ∈ Q x ∈ Zd × Rn−d where Q is a closed convex set

min cT

Q

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c-Karzan (CMU) Structure of Valid Inequalities for MICPs 3 / 34

Mixed Integer Conic Programming

Mixed Integer Linear Program min cTx s.t. Ax − b ∈ Rm

+

x ∈ Zd × Rn−d

min cT

Mixed Integer Conic Program min cTx s.t. Ax − b ∈ K x ∈ Zd × Rn−d where K is a convex cone x1 x2 x3

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c-Karzan (CMU) Structure of Valid Inequalities for MICPs 4 / 34

Mixed Integer Conic Programming

Mixed Integer Linear Program min cTx s.t. Ax − b ∈ Rm

+

x ∈ Zd × Rn−d

min cT

Mixed Integer Conic Program min cTx s.t. Ax − b ∈ K x ∈ Zd × Rn−d where K is a convex cone

Nonnegative orthant

Rm

+ = {y ∈ Rm : yi ≥ 0 ∀i}

Lorentz cone

Lm = {y ∈ Rm : ym ≥

• y2

2 + . . . + y2 m−1}

Positive semidefinite cone

Sm

+ = {X ∈ Rm×m : aT Xa ≥ 0 ∀a ∈ Rm}

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c-Karzan (CMU) Structure of Valid Inequalities for MICPs 4 / 34

Mixed Integer Conic Programming

Mixed Integer Linear Program min cTx s.t. Ax − b ∈ Rm

+

x ∈ Zd × Rn−d

min cT

Mixed Integer Conic Program min cTx s.t. Ax − b ∈ K x ∈ Zd × Rn−d where K is a convex cone

min cT

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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

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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

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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

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Is it possible to develop a framework to establish strength of valid linear inequalities for MICPs?

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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 Rm B ⊂ Rm is a given set of points (can be finite or infinite) K ⊂ E is a full-dimensional, closed, convex and pointed cone

Nonnegative orthant, Rn

+ := {x ∈ Rn : xi ≥ 0 ∀i}

Lorentz cone, Ln := {x ∈ Rn : xn ≥

• x2

2 + . . . + x2 n−1 }

Positive semidefinite cone, Sn

+ := {X ∈ Rn×n : aTXa ≥ 0 ∀a ∈ Rn}

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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):

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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):

{(y, v) ∈ Rk

+ × Zq : Wy + Hv − b ∈ K}

where K ⊂ E is a full-dimensional, closed, convex, pointed cone.

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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):

{(y, v) ∈ Rk

+ × Zq : Wy + Hv − b ∈ K}

where K ⊂ E is a full-dimensional, closed, convex, pointed cone. Define x = y z

• ,

A =

• W ,

−Id

• ,

and B = b − HZq, where Id is the identity map in E. Then we arrive at S(A, K′, B) = {x ∈ (Rk × E) : Ax ∈ B, x ∈ (Rk

+ × K)

• :=K′

}

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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):

{(y, v) ∈ Rk

+ × Zq : Wy + Hv − b ∈ K}

where K ⊂ E is a full-dimensional, closed, convex, pointed cone. Define x = y z

• ,

A =

• W ,

−Id

• ,

and B = b − HZq, where Id is the identity map in E. Then we arrive at S(A, K′, B) = {x ∈ (Rk × E) : Ax ∈ B, x ∈ (Rk

+ × K)

• :=K′

} Many more examples involving modeling disjunctions, complementarity relations, Gomory’s Corner Polyhedron (1969), etc...

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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 Rm

A∗ denotes the conjugate linear map from Rm to E Ker(A) := {x ∈ E : Ax = 0} Im(A) := {Ax : x ∈ E}

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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 Rm

A∗ denotes the conjugate linear map from Rm to E Ker(A) := {x ∈ E : Ax = 0} Im(A) := {Ax : x ∈ E}

B ⊂ Rm 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} .

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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 Rm

A∗ denotes the conjugate linear map from Rm to E Ker(A) := {x ∈ E : Ax = 0} Im(A) := {Ax : x ∈ E}

B ⊂ Rm 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

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Goal S(A, K, B) = {x ∈ E : Ax ∈ B, x ∈ K} We are interested in conv(S(A, K, B))

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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 ≥ η0 for S(A, K, B):

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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 ≥ η0 for S(A, K, B): C(A, K, B) = convex cone of all linear valid inequalities for S(A, K, B) = {(µ; η0) : µ ∈ E, µ = 0, −∞ < η0 ≤ inf

x∈S(A,K,B)µ, x

• :=µ0

} 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))

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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)

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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. C(A, K, B) Cone implied inequalities

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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; η0) ∈ C(A, K, B) is a K-minimal valid inequality if for all ρ such that ρ K∗ µ and ρ = µ, we have ρ0 < η0.

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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; η0) ∈ C(A, K, B) is a K-minimal valid inequality if for all ρ such that ρ K∗ µ and ρ = µ, we have ρ0 < η0. Cm(A, K, B) = cone of K-minimal valid inequalities

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c-Karzan (CMU) Structure of Valid Inequalities for MICPs 11 / 34

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.) Cm(A, K, B) = cone of K-minimal valid inequalities C(A, K, B) Cm(A, K, B) Cone implied inequalities

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K-minimal Inequalities

Definition An inequality (µ; η0) ∈ C(A, K, B) is a K-minimal valid inequality if for all ρ such that ρ K∗ µ and ρ = µ, we have ρ0 < η0. ⇒ A K-minimal v.i. (µ; η0) cannot be written as a sum of a cone implied inequality and another valid inequality.

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K-minimal Inequalities

Definition An inequality (µ; η0) ∈ C(A, K, B) is a K-minimal valid inequality if for all ρ such that ρ K∗ µ and ρ = µ, we have ρ0 < η0. ⇒ A K-minimal v.i. (µ; η0) 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.

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K-minimal Inequalities

Definition An inequality (µ; η0) ∈ C(A, K, B) is a K-minimal valid inequality if for all ρ such that ρ K∗ µ and ρ = µ, we have ρ0 < η0. ⇒ A K-minimal v.i. (µ; η0) 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}.

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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).

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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) Cm(A, K, B) Cone implied inequalities

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c-Karzan (CMU) Structure of Valid Inequalities for MICPs 13 / 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., Cm(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.

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Necessary Conditions for K-minimality

What can we say about µ for a v.i. (µ; η0) ∈ Cm(A, K, B)? Proposition If (µ; η0) ∈ Cm(A, K, B), then for all linear maps Z : K → K s.t. AZ ∗ ≡ A, we have µ − Zµ ∈ K∗ \ {0}.

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c-Karzan (CMU) Structure of Valid Inequalities for MICPs 14 / 34

Necessary Conditions for K-minimality

Proposition If (µ; η0) ∈ Cm(A, K, B), then for all linear maps Z : K → K s.t. AZ ∗ ≡ A, we have µ − Zµ ∈ K∗ \ {0}. More on FK := {(Z : E → E) : Z is linear, and Z ∗v ∈ K ∀v ∈ K} FK is the cone of K − K positive maps. They also appear in applications of robust optimization, quantum physics, ...

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Necessary Conditions for K-minimality

Proposition If (µ; η0) ∈ Cm(A, K, B), then for all linear maps Z : K → K s.t. AZ ∗ ≡ A, we have µ − Zµ ∈ K∗ \ {0}. More on FK := {(Z : E → E) : Z is linear, and Z ∗v ∈ K ∀v ∈ K} When K = Rn

+, FK = {Z ∈ Rn×n : Zij ≥ 0 ∀i, j}

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Necessary Conditions for K-minimality

Proposition If (µ; η0) ∈ Cm(A, K, B), then for all linear maps Z : K → K s.t. AZ ∗ ≡ A, we have µ − Zµ ∈ K∗ \ {0}. More on FK := {(Z : E → E) : Z is linear, and Z ∗v ∈ K ∀v ∈ K} When K = Rn

+, FK = {Z ∈ Rn×n : Zij ≥ 0 ∀i, j}

When K = Ln, FK has a semidefinite programming representation

(Hildebrand 2006, 2008).

• In particular when Z = abT with a, b ∈ Ln, then Z ∈ FLn.
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c-Karzan (CMU) Structure of Valid Inequalities for MICPs 14 / 34

Necessary Conditions for K-minimality

Proposition If (µ; η0) ∈ Cm(A, K, B), then for all linear maps Z : K → K s.t. AZ ∗ ≡ A, we have µ − Zµ ∈ K∗ \ {0}. More on FK := {(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 Sn

+ to itself is an NP-hard problem.

(Ben Tal & Nemirovski, 1998)

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Valid Inequalities

C(A, K, B) Cone implied inequalities Cm(A, K, B)

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Valid Inequalities

C(A, K, B) Cone implied inequalities Cm(A, K, B)

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K-sublinear Inequalities

Definition (µ; η0) is a K-sublinear v.i. if it satisfies (A.1) 0 ≤ µ, u for all u s.t. Au = 0 and α, vu + v ∈ K ∀v ∈ Ext(K) holds for some α ∈ Ext(K∗), (A.2) η0 ≤ µ, x for all x ∈ S(A, K, B).

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K-sublinear Inequalities

Definition (µ; η0) is a K-sublinear v.i. if it satisfies (A.1) 0 ≤ µ, u for all u s.t. Au = 0 and α, vu + v ∈ K ∀v ∈ Ext(K) holds for some α ∈ Ext(K∗), (A.2) η0 ≤ 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.

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c-Karzan (CMU) Structure of Valid Inequalities for MICPs 16 / 34

K-sublinear Inequalities

Definition (µ; η0) is a K-sublinear v.i. if it satisfies (A.1) 0 ≤ µ, u for all u s.t. Au = 0 and α, vu + v ∈ K ∀v ∈ Ext(K) holds for some α ∈ Ext(K∗), (A.2) η0 ≤ µ, x for all x ∈ S(A, K, B). Condition (A.1) implies (A.0) µ ∈ Im(A∗) + K∗

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c-Karzan (CMU) Structure of Valid Inequalities for MICPs 16 / 34

K-sublinear Inequalities

Definition (µ; η0) is a K-sublinear v.i. if it satisfies (A.1) 0 ≤ µ, u for all u s.t. Au = 0 and α, vu + v ∈ K ∀v ∈ Ext(K) holds for some α ∈ Ext(K∗), (A.2) η0 ≤ µ, x for all x ∈ S(A, K, B). Condition (A.1) implies (A.0) µ ∈ Im(A∗) + K∗ Proposition Any valid inequality (µ; η0) ∈ C(A, K, B) satisfies condition (A.0).

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K-sublinear Inequalities

Theorem Cm(A, K, B) ⊆ Ca(A, K, B). C(A, K, B) Cone implied inequalities Cm(A, K, B) Ca(A, K, B)

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K-sublinear Inequalities

Theorem Cm(A, K, B) ⊆ Ca(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).

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K-sublinear Inequalities

C(A, K, B) Cone implied inequalities Cm(A, K, B) Ca(A, K, B) This is all fine but the definition of K-sublinearity is much less apparent, how can we hope to verify it?

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Relations to Support Functions

Consider any v.i. (µ; η0) and let Dµ = {λ ∈ Rm : µ − A∗λ ∈ K∗}. Remark Let σD(·) be the support function of Dµ, i.e., σD(h) = sup

λ

{h, λ : λ ∈ Dµ}. For all v.i. (µ; η0), we have Dµ = ∅, σD(0) = 0, and σD(Ad) ≤ µ, d for all d ∈ K.

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Necessary Conditions for K-sublinearity

Remark For any µ ∈ Im(A∗) + K∗, ⇒ Dµ = ∅ ⇒ σD(Az) ≤ µ, z holds ∀z ∈ K; and ⇒ for any η0 ≤ infb∈B σD(b), we have (µ; η0) ∈ C(A, K, B).

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Necessary Conditions for K-sublinearity

Remark For any µ ∈ Im(A∗) + K∗, ⇒ Dµ = ∅ ⇒ σD(Az) ≤ µ, z holds ∀z ∈ K; and ⇒ for any η0 ≤ infb∈B σD(b), we have (µ; η0) ∈ 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 σD(Az) = µ, z for all z ∈ K. For any (µ; η0) ∈ C(A, K, B), if S(A, K, B) is closed, then there exists z ∈ Ext(K) s.t. σD(Az) = µ, z.

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Sufficient Condition for K-sublinearity

Proposition [Sufficient Condition for K-Sublinearity] Let (µ; η0) ∈ C(A, K, B) and suppose that ∃xi ∈ Ext(K) s.t. σD(Axi) = µ, xi for all i ∈ I and

i∈I xi ∈ int(K),

then (µ; η0) ∈ Ca(A, K, B).

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Refinement of K-sublinearity for K = Rn

+

Refinement of the condition (A.1) for K-sublinearity leads to (A.0) 0 ≤ µ, u for all u ∈ Rn

+ such that Au = 0, and

(A.1i) for all i = 1, . . . , n, µi ≤ µ, u for all u ∈ Rn

+ such that Au = Ai

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Refinement of K-sublinearity for K = Rn

+

Refinement of the condition (A.1) for K-sublinearity leads to (A.0) 0 ≤ µ, u for all u ∈ Rn

+ such that Au = 0, and

(A.1i) for all i = 1, . . . , n, µi ≤ µ, u for all u ∈ Rn

+ such that Au = Ai

⇒ Hence for K = Rn

+, 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.

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Refinement of K-sublinearity for K = Rn

+

Refinement of the condition (A.1) for K-sublinearity leads to (A.0) 0 ≤ µ, u for all u ∈ Rn

+ such that Au = 0, and

(A.1i) for all i = 1, . . . , n, µi ≤ µ, u for all u ∈ Rn

+ such that Au = Ai

⇒ Hence for K = Rn

+, 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 Rn

+-Sublinearity]

Let K = Rn

+. For any (µ; η0) ∈ Ca(A, K, B), we have σD(Az) = µ, z for all

z ∈ Ext(K), i.e., σD(Ai) = µi for all i = 1, . . . , n where Ai is the ith column of the matrix A.

• F. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 22 / 34

Sufficient Condition for K-sublinearity

Proposition [Sufficient Condition for K-Sublinearity] Let (µ; η0) ∈ C(A, K, B) and suppose that ∃xi ∈ Ext(K) s.t. σD(Axi) = µ, xi for all i ∈ I and

i∈I xi ∈ int(K),

then (µ; η0) ∈ Ca(A, K, B). ⇒ Complete characterization of Rn

+-sublinear inequalities:

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c-Karzan (CMU) Structure of Valid Inequalities for MICPs 23 / 34

Sufficient Condition for K-sublinearity

Proposition [Sufficient Condition for K-Sublinearity] Let (µ; η0) ∈ C(A, K, B) and suppose that ∃xi ∈ Ext(K) s.t. σD(Axi) = µ, xi for all i ∈ I and

i∈I xi ∈ int(K),

then (µ; η0) ∈ Ca(A, K, B). ⇒ Complete characterization of Rn

+-sublinear inequalities:

All Rn

+-sublinear inequalities are generated by sublinear (subadditive and

positively homogeneous, in fact also piecewise linear and convex) functions, i.e.,

support functions σDµ(·) of Dµ.

• This recovers a number of results from Johnson ’81, and Conforti et al.’13.
• F. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 23 / 34

Sufficient Condition for K-sublinearity

Proposition [Sufficient Condition for K-Sublinearity] Let (µ; η0) ∈ C(A, K, B) and suppose that ∃xi ∈ Ext(K) s.t. σD(Axi) = µ, xi for all i ∈ I and

i∈I xi ∈ int(K),

then (µ; η0) ∈ Ca(A, K, B). ⇒ Complete characterization of Rn

+-sublinear inequalities:

All Rn

+-sublinear inequalities are generated by sublinear (subadditive and

positively homogeneous, in fact also piecewise linear and convex) functions, i.e.,

support functions σDµ(·) of Dµ.

• 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. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 23 / 34

Sufficient Condition for K-minimality

Theorem [Sufficient Condition for K-Minimality] Suppose that Cm(A, K, B) = ∅. Consider any (µ; η0) ∈ Ca(A, K, B) s.t. η0 = infb∈B σD(b) and ∃xi ∈ K s.t.

i xi ∈ int(K), Axi = bi with bi ∈ B satisfying

σD(bi) = η0 and µ, xi = η0, then (µ; η0) ∈ Cm(A, K, B).

• F. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 24 / 34

Sufficient Condition for K-minimality

Theorem [Sufficient Condition for K-Minimality] Suppose that Cm(A, K, B) = ∅. Consider any (µ; η0) ∈ Ca(A, K, B) s.t. η0 = infb∈B σD(b) and ∃xi ∈ K s.t.

i xi ∈ int(K), Axi = bi with bi ∈ B satisfying

σD(bi) = η0 and µ, xi = η0, then (µ; η0) ∈ Cm(A, K, B). For general cones K other than Rn

+, unfortunately there is a gap between

the current necessary condition and the sufficient condition for K-minimality.

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c-Karzan (CMU) Structure of Valid Inequalities for MICPs 24 / 34

A Simple Example K = L3, A = [1, 0, 0] and B = {−1, 1} . , i.e., S(A, K, B) = {x ∈ R3 : x1 ∈ {−1, 1}, x3 ≥

• x2

1 + x2 2}

x1 (0, 0) x2 x3 conv(S(A, K, B)) = {x ∈ R3 : −1 ≤ x1 ≤ 1, x3 ≥

• 1 + x2

2}

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c-Karzan (CMU) Structure of Valid Inequalities for MICPs 25 / 34

A Simple Example K = L3, A = [1, 0, 0] and B = {−1, 1} . , i.e., S(A, K, B) = {x ∈ R3 : x1 ∈ {−1, 1}, x3 ≥

• x2

1 + x2 2}

conv(S(A, K, B)) = {x ∈ R3 : −1 ≤ x1 ≤ 1, x3 ≥

• 1 + x2

2}

K-minimal inequalities are: (a) µ(+) = (1; 0; 0) with η(+) = −1 and µ(−) = (−1; 0; 0) with η(−) = −1; (b) µ(t) = (0; t; √ t2 + 1) with η(t) = 1 for all t ∈ R.

• F. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 25 / 34

A Simple Example K = L3, A = [1, 0, 0] and B = {−1, 1} . , i.e., S(A, K, B) = {x ∈ R3 : x1 ∈ {−1, 1}, x3 ≥

• x2

1 + x2 2}

conv(S(A, K, B)) = {x ∈ R3 : −1 ≤ x1 ≤ 1, x3 ≥

• 1 + x2

2}

K-minimal inequalities are: (a) µ(+) = (1; 0; 0) with η(+) = −1 and µ(−) = (−1; 0; 0) with η(−) = −1; (b) µ(t) = (0; t; √ t2 + 1) with η(t) = 1 for all t ∈ R. (these can be expressed as a single conic inequality x3 ≥

• 1 + x2

2.)

• F. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 25 / 34

A Simple Example K = L3, A = [1, 0, 0] and B = {−1, 1} . , i.e., S(A, K, B) = {x ∈ R3 : x1 ∈ {−1, 1}, x3 ≥

• x2

1 + x2 2}

conv(S(A, K, B)) = {x ∈ R3 : −1 ≤ x1 ≤ 1, x3 ≥

• 1 + x2

2}

K-minimal inequalities are: (a) µ(+) = (1; 0; 0) with η(+) = −1 and µ(−) = (−1; 0; 0) with η(−) = −1; (b) µ(t) = (0; t; √ t2 + 1) with η(t) = 1 for all t ∈ R. (these can be expressed as a single conic inequality x3 ≥

• 1 + x2

2.)

Linear inequalities in (b) cannot be generated by any cut generating function ρ(·), i.e., ρ(Ai) = µ(t)

i

is not possible for any function ρ(·).

• F. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 25 / 34

A Simple Example K = L3, A = [1, 0, 0] and B = {−1, 1} . , i.e., S(A, K, B) = {x ∈ R3 : x1 ∈ {−1, 1}, x3 ≥

• x2

1 + x2 2}

conv(S(A, K, B)) = {x ∈ R3 : −1 ≤ x1 ≤ 1, x3 ≥

• 1 + x2

2}

K-minimal inequalities are: (a) µ(+) = (1; 0; 0) with η(+) = −1 and µ(−) = (−1; 0; 0) with η(−) = −1; (b) µ(t) = (0; t; √ t2 + 1) with η(t) = 1 for all t ∈ R. (these can be expressed as a single conic inequality x3 ≥

• 1 + x2

2.)

Linear inequalities in (b) cannot be generated by any cut generating function ρ(·), i.e., ρ(Ai) = 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, xi is integer for all i ∈ I}.

[Sharp contrast to the results of Dey, Moran & Vielma ’12. ]

• F. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 25 / 34

Can we derive conic valid inequalities for S(A, K, B) using this framework?

• F. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 26 / 34

Disjunctive Cuts for Lorentz Cone, Ln

Start with a simple set for x, i.e., x ∈ K = Ln Consider a two-term disjunction of form either πT

1 x ≥ π1,0 or πT 2 x ≥ π2,0 must hold.

• F. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 27 / 34

Disjunctive Cuts for Lorentz Cone, Ln

Start with a simple set for x, i.e., x ∈ K = Ln Consider a two-term disjunction of form either πT

1 x ≥ π1,0 or πT 2 x ≥ π2,0 must hold.

Let Si := {x : πT

i x ≥ πi,0, x ∈ K}.

• F. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 27 / 34

Disjunctive Cuts for Lorentz Cone, Ln

Start with a simple set for x, i.e., x ∈ K = Ln Consider a two-term disjunction of form either πT

1 x ≥ π1,0 or πT 2 x ≥ π2,0 must hold.

Let Si := {x : πT

i x ≥ πi,0, x ∈ K}.

S1 S2

Without loss of generality assume that π1,0, π2,0 ∈ {0, ±1} and S1 = ∅ and S2 = ∅.

• F. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 27 / 34

Disjunctive Cuts for Lorentz Cone, Ln

Start with a simple set for x, i.e., x ∈ K = Ln Consider a two-term disjunction of form either πT

1 x ≥ π1,0 or πT 2 x ≥ π2,0 must hold.

Let Si := {x : πT

i x ≥ πi,0, x ∈ K}.

By setting A = πT

1

πT

2

• , and B =

π1,0 + R+ R R π2,0 + R+

• we arrive back at

S(A, K, B) = {x ∈ Rn : Ax ∈ B, x ∈ K} and S(A, K, B) = S1 ∪ S2.

• F. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 27 / 34

Disjunctive Cuts for Lorentz Cone, Ln

The cases of S1 ⊆ S2 or S2 ⊆ S1 are not interesting, so we assume Assumption The disjunction πT

1 x ≥ π1,0 and πT 2 x ≥ π2,0 satisfy

{β ∈ Rn

+ : βπ1,0 ≥ π2,0, π2 − βπ1 ∈ Ln} = ∅, and

{β ∈ Rn

+ : βπ2,0 ≥ π1,0, π1 − βπ2 ∈ Ln} = ∅.

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c-Karzan (CMU) Structure of Valid Inequalities for MICPs 28 / 34

Disjunctive Cuts for Lorentz Cone, Ln Convex nonlinear valid inequalities for Ln via a disjunctive argument [Sketch of derivation]: Given a disjunction πT

1 x ≥ π1,0 and πT 2 x ≥ π2,0 with π1,0, π2,0 ∈ {0, ±1}

Characterize the structure of linear Ln-minimal valid inequalities

• F. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 29 / 34

Disjunctive Cuts for Lorentz Cone, Ln Convex nonlinear valid inequalities for Ln via a disjunctive argument [Sketch of derivation]: Given a disjunction πT

1 x ≥ π1,0 and πT 2 x ≥ π2,0 with π1,0, π2,0 ∈ {0, ±1}

Characterize the structure of linear Ln-minimal valid inequalities Proposition [K.-K., Yıldız] For all Ln-minimal valid linear inequalities µ⊤x ≥ µ0 for conv(S1 ∪ S2) there exists α1, α2 ∈ bd(Ln), and β1, β2 ∈ (R+ \ {0}) s.t. µ = α1 + β1π1, µ = α2 + β2π2, min{π1,0β1, π2,0β2} = µ0 = min{π1,0, π2,0}, and at least one of β1 and β2 is equal to 1.

• F. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 29 / 34

Disjunctive Cuts for Lorentz Cone, Ln Convex nonlinear valid inequalities for Ln via a disjunctive argument [Sketch of derivation]: Given a disjunction πT

1 x ≥ π1,0 and πT 2 x ≥ π2,0 with π1,0, π2,0 ∈ {0, ±1}

Characterize the structure of linear Ln-minimal valid inequalities Based on their characterization, e.g., (β1, β2) values, group all of the linear Ln-minimal valid linear inequalities via an optimization problem over µ

• F. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 29 / 34

Disjunctive Cuts for Lorentz Cone, Ln Convex nonlinear valid inequalities for Ln via a disjunctive argument [Sketch of derivation]: Given a disjunction πT

1 x ≥ π1,0 and πT 2 x ≥ π2,0 with π1,0, π2,0 ∈ {0, ±1}

Characterize the structure of linear Ln-minimal valid inequalities Based on their characterization, e.g., (β1, β2) values, group all of the linear Ln-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. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 29 / 34

Disjunctive Cuts for Lorentz Cone, Ln Convex nonlinear valid inequalities for Ln via a disjunctive argument [Sketch of derivation]: Given a disjunction πT

1 x ≥ π1,0 and πT 2 x ≥ π2,0 with π1,0, π2,0 ∈ {0, ±1}

Characterize the structure of linear Ln-minimal valid inequalities Based on their characterization, e.g., (β1, β2) values, group all of the linear Ln-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. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 29 / 34

Disjunctive Cuts for Lorentz Cone, Ln Disjunction: either πT

1 x ≥ π1,0 or πT 2 x ≥ π2,0

Theorem [K.-K., Yıldız] Let σ = min{π1,0, π2,0}. For any β > 0 such that βπ1 − π2 / ∈ ±int(Ln), the following convex inequality is valid for conv(S1 ∪ S2): 2σ − (βπ1 + π2)⊤x ≤

• ((βπ1 − π2)⊤x)2 + N(β) ∗ (x2

n −

x2

2)

where N(β) := β π1 − π22

2 − (βπ1,n − π2,n)2, and

exactly captures all linear v.i. corresponding to β1 = β and β2 = 1.

• F. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 30 / 34

Disjunctive Cuts for Lorentz Cone, Ln 2σ − (βπ1 + π2)⊤x ≤

• ((βπ1 − π2)⊤x)2 + N(β) ∗ (x2

n −

x2

2)

Theorem [K.-K., Yıldız] In certain cases such as conv(S1 ∪ S2) is closed, Splits, i.e., π1 = −απ2 for some α > 0, and π1,0 = π2,0 = σ with σ = 1 it is sufficient (for conv(S1 ∪ S2)) to consider only one inequality with β = 1.

• F. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 31 / 34

Disjunctive Cuts for Lorentz Cone, Ln 2σ − (βπ1 + π2)⊤x ≤

• ((βπ1 − π2)⊤x)2 + N(β) ∗ (x2

n −

x2

2)

Theorem [K.-K., Yıldız] In certain cases such as conv(S1 ∪ S2) is closed, Splits, i.e., π1 = −απ2 for some α > 0, and π1,0 = π2,0 = σ with σ = 1 it is sufficient (for conv(S1 ∪ S2)) 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)

N(1) ( π1 − π2)

• 2

≤

• xn + 2(π⊤

1 x − σ)

N(1) (π1,n − π2,n)

• F. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 31 / 34

Disjunctive Cuts for Lorentz Cone, Ln Disjunction: x3 ≥ 1

• r

2x1 + 2x3 ≥ 1 conv(S1 ∪ S2) =

• x ∈ L3 : 2 − (2x1 + 3x2) ≤
• (−2x1 − x3)2 + 3 (x2

3 − x2 1 − x2 2)

• F. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 32 / 34

Disjunctive Cuts for Lorentz Cone, Ln Disjunction: x3 ≥ 1

• r

2x1 + 2x3 ≥ 1 conv(S1 ∪ S2) =

• x ∈ L3 : 2 − (2x1 + 3x2) ≤
• (−2x1 − x3)2 + 3 (x2

3 − x2 1 − x2 2)

• F. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 32 / 34

Disjunctive Cuts for Lorentz Cone, Ln Disjunction: −x3 ≥ −1

• r

−x2 ≥ 0 conv(S1 ∪ S2) =

• x ∈ L3 : x2 ≤ 1, 1 + |x1| − x3 ≤
• 1 − max{0, x2}2
• F. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 32 / 34

Final remarks

Introduce a unifying framework for defining K-minimal and K-sublinear inequalities for conic MIPs

• F. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 33 / 34

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. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 33 / 34

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 = Rn

+

(i.e., Johnson ’81 and Conforti et al.’13)

• F. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 33 / 34

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 = Rn

+

(i.e., Johnson ’81 and Conforti et al.’13)

For K = Ln, 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. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 33 / 34

Thank you!

• F. Kılın¸

c-Karzan (CMU) Structure of Valid Inequalities for MICPs 34 / 34