The Chvátal-Gomory Closure of a Strictly Convex Body is a Rational Polyhedron
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1 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
2 Outline Introduction Proof: Step Step Intersection with Rational Polyhedra Example of Non-Polyhedral Closure. Conclusions and Future Work /
3 Introduction CG Cuts for Rational Polyhedra /
4 Introduction CG Cuts for Rational Polyhedra /
5 Introduction CG Cuts for Rational Polyhedra /
6 Introduction CG Cuts for Rational Polyhedra /
7 Introduction CG Cuts for General Convex Sets /
8 Introduction CG Cuts for General Convex Sets /
9 Introduction CG Cuts for General Convex Sets /
10 Introduction CG Cuts for General Convex Sets /
11 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? /
12 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 /
13 Proof Proof Outline: Generation Procedure Step : Construct a finite set such that 7/
14 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/
15 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/
16 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/
17 Proof: Step Separate non-integral points in /
18 Proof: Step Separate non-integral points in /
19 Proof: Step Separate non-integral points in /
20 Proof: Step Separate non-integral points in /
21 Proof: Step Separate non-integral points in 9/
22 Proof: Step Separate non-integral points in 9/
23 Proof: Step Separate non-integral points in 9/
24 Proof: Step Separate non-integral points in 9/
25 Proof: Step Separate neighborhood of integers /
26 Proof: Step Separate neighborhood of integers /
27 Proof: Step Separate neighborhood of integers /
28 Proof: Step Separate neighborhood of integers Similar to non-integer separation + compactness argument /
29 Proof: Step Compactness Argument /
30 Proof: Step Compactness Argument /
31 Proof: Step Compactness Argument /
32 Proof: Step Compactness Argument /
33 Proof: Step Compactness Argument /
34 Proof: Step Compactness Argument compact /
35 Proof: Step Compactness Argument /
36 Proof: Step Compactness Argument /
37 Proof: Step Compactness Argument /
38 Proof: Step Compactness Argument /
39 Proof: Step Compactness Argument /
40 Proof: Step Compactness Argument /
41 Proof: Step Compactness Argument /
42 Proof: Step Step : Separate /
43 Proof: Step Step : Separate /
44 Proof: Step Step : Separate ε > εb n +v C v V /
45 Proof: Step Step : Separate ε > εb n +v C v V /
46 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 /
47 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 /
48 Strictly Convex Rational Polyhedron Example: Ellipsoid and Halfspace P polyhedron, F face of P CGC(F)=CGC(C) F (Schrijver, 98) /
49 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 /
50 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) /
51 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 ) /
52 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 /
53 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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