The Chvátal-Gomory Closure of a Strictly Convex Body is a Rational Polyhedron

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Transcription:

The Chvátal-Gomory Closure of a Strictly Convex Body is a Rational Polyhedron Juan Pablo Vielma Joint work with Daniel Dadush and Santanu S. Dey July, Atlanta, GA

Outline Introduction Proof: Step Step Intersection with Rational Polyhedra Example of Non-Polyhedral Closure. Conclusions and Future Work /

Introduction CG Cuts for Rational Polyhedra /

Introduction CG Cuts for Rational Polyhedra /

Introduction CG Cuts for Rational Polyhedra /

Introduction CG Cuts for Rational Polyhedra /

Introduction CG Cuts for General Convex Sets - - - - /

Introduction CG Cuts for General Convex Sets - - - - /

Introduction CG Cuts for General Convex Sets - - - - /

Introduction CG Cuts for General Convex Sets - - - - /

Introduction CG Closure of a Convex Set CG Closure: Is CG closure a polyhedron? Finite set s.t. Yes, for rational polyhedra (Schrijver, 98) What about other convex sets? /

Introduction What we know for Convex Bodies There exists k s.t. (Chvátal, 97) Also for unbounded rational polyhedra ( Schrijver, 98). Result does not imply polyhedrality of /

Proof Proof Outline: Generation Procedure Step : Construct a finite set such that 7/

Proof Proof Outline: Generation Procedure Step : Construct a finite set such that Step : Construct a finite set such that CG cuts from separate all points in all of 7/

Proof Proof Outline: Generation Procedure Step : Construct a finite set such that Step : Construct a finite set such that CG cuts from separate all points in all of 7/

Proof: Step Outline of Step Step : Construct a finite set such that and (a) Separate non-integral points in. (b) Separate neighborhood of integral points in. (c) Compactness argument to construct finite. 8/

Proof: Step Separate non-integral points in - - - - 9/

Proof: Step Separate non-integral points in - - - - 9/

Proof: Step Separate non-integral points in - - - - 9/

Proof: Step Separate non-integral points in - - - - 9/

Proof: Step Separate non-integral points in 9/

Proof: Step Separate non-integral points in 9/

Proof: Step Separate non-integral points in 9/

Proof: Step Separate non-integral points in 9/

Proof: Step Separate neighborhood of integers - - - - /

Proof: Step Separate neighborhood of integers - - - - /

Proof: Step Separate neighborhood of integers - - - - /

Proof: Step Separate neighborhood of integers Similar to non-integer separation + compactness argument - - - - /

Proof: Step Compactness Argument - - - - /

Proof: Step Compactness Argument - - - - /

Proof: Step Compactness Argument - - - - /

Proof: Step Compactness Argument - - - - /

Proof: Step Compactness Argument - - - - /

Proof: Step Compactness Argument compact - - - - /

Proof: Step Compactness Argument - - - - /

Proof: Step Compactness Argument - - - - /

Proof: Step Compactness Argument - - - - /

Proof: Step Compactness Argument - - - - /

Proof: Step Compactness Argument - - - - /

Proof: Step Compactness Argument - - - - /

Proof: Step Compactness Argument - - - - /

Proof: Step Step : Separate - - - - /

Proof: Step Step : Separate - - - - /

Proof: Step Step : Separate ε > εb n +v C v V - - - - /

Proof: Step Step : Separate ε > εb n +v C v V - - - - /

Proof: Step Step : Separate ε > εb n +v C v V a ε σ C (a) σ C (a) σ v+εb n(a) = v,a+εa v,a - - - - /

Proof: Step Step : Separate ε > εb n +v C v V a ε σ C (a) σ C (a) σ v+εb n(a) = v,a+εa v,a S =(/ε)b Z n - - - - /

Strictly Convex Rational Polyhedron Example: Ellipsoid and Halfspace P polyhedron, F face of P CGC(F)=CGC(C) F (Schrijver, 98) - - - - /

Strictly Convex Rational Polyhedron Example: Ellipsoid and Halfspace P polyhedron, F face of P CGC(F)=CGC(C) F (Schrijver, 98) We can generalize it. - - - - /

Strictly Convex Rational Polyhedron Example: Ellipsoid and Halfspace P polyhedron, F face of P CGC(F)=CGC(C) F (Schrijver, 98) We can generalize it. a,x σ F (a) - - - - /

Strictly Convex Rational Polyhedron Example: Ellipsoid and Halfspace P polyhedron, F face of P CGC(F)=CGC(C) F (Schrijver, 98) We can generalize it. a,x σ F (a) a,x σ C (a ) - - - - /

Example of Non-Polyhedral Closure Split Closure of an Ellipsoid Pure Integer Case: C = {x R : A(x c) } A =, c=(/,/,/) T / / Two split cuts: x x x x /

Conclusions and Future Work Non-Constructive because of compactness argument in step. Current work: General compact convex sets including nonrational polytopes ( Almost done). Split closure is finitely generated. Open Problems: Simpler Proof (Circle in?). R Constructive/Algorithmic proof. /

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