Cutting Planes and Elementary Closures for Non-linear Integer Programming

Cutting Planes and Elementary Closures for Non-linear Integer Programming Juan Pablo Vielma University of Pittsburgh joint work with D. Dadush and S. S. Dey S. Modaresi and M. Kılınç Georgia Institute of Technology University of Pittsburgh NSF CMMI-1030662 and ONR N000141110724 DIMACS, November, 2011 Rutgers University, New Jersey

Cutting Planes for Integer Programming Valid Inequalities for the convex hull of integer feasible solutions. 50+ Years of development for Linear Integer Programming. Used to get tighter Linear Programming Relaxations. Crucial for state of the art solvers. 2/24

Convex Non-Linear Integer Programming Problems with convex continuous relaxation. Many applications, results and algorithms available. Cutting planes significantly less developed. Need new tools: linear results strongly rely on rationals. 3/24

Two Classic Cutting Planes Chvátal-Gomory Cuts (Gomory 68, Chvátal 73): AKA Gomory Fractional Cut Simple, but yield pure cutting plane algorithm, Blossom s for Matching and Comb s for TSP. Split Cuts (Cook, Kannan and Shrijver 1990): AKA MIG (Gomory 1960) and MIR (Nemhauser and Wolsey 1988) Yield Flow Cover Cuts and modern IP solvers. 4 /24

Outline Chvátal-Gomory Cuts for Non-Linear IP: Polyhedrality of the Chvátal-Gomory Closure Split Cuts for Non-Linear IP: Closed form Expressions. Finite Generation v/s Polyhedrality of Split Closure Other Results and Open Questions 5/24

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CG Cuts and Support Function 7/24

CG Cuts and Support Function Valid for H Z n 7/24

CG Cuts and Support Function Valid for H Z n Valid for C Z n CG Cut 7/24

CG Closure = Add all CG Cuts 8/24

CG Closure = Add all CG Cuts Polyhedral? 8/24

CG Closure = Add all CG Cuts Polyhedral? C = a Z n {x R n : a, x σ C (a)} 8/24

Polyhedrality of CG Closure Not always a polyhedron 9/24

Polyhedrality of CG Closure Not always a polyhedron Shrijver 1980: Theorem: If C is a rational polyhedron then is too. (Constructive Proof) 9/24

Polyhedrality of CG Closure Not always a polyhedron Shrijver 1980: Theorem: If C is a rational polyhedron then is too. (Constructive Proof) Question: What about for non-rational polytopes? 9/24

CG Closure is Finitely Generated Theorem (Dadush, Dey, V. 2011): If C is a compact convex set then there exists a finite such that: Corollary: is a rational polytope. In particular answers Shrijver s question. Also answered by Dunkel and Schulz 2011. 10/24

Corollary: Stability of CG Closure 11/24

Proof Sketch of Theorem For strictly convex sets without integral points in boundary Proof Outline: Step 1: Create finite s.t.. Separate points in boundary Compactness argument Step 2: Show only missed finite number of cuts 12/24

Separate points in 6 5 4 3 2 1 0-1 -3-2 -1 0 1 2 3 4 5 13/24

Separate points in 13/24

Compactness Argument K := bd(c) S u := {x : a u,x > σ C (a u )} 3 2 1 0 0 1 2 3 14/24

Compactness Argument K := bd(c) S u := {x : a u,x > σ C (a u )} compact 3 2 1 0 0 1 2 3 14/24

Compactness Argument K := bd(c) S u := {x : a u,x > σ C (a u )} compact 3 2 1 S 1 = m i=1 a ui 0 0 1 2 3 14/24

Step 2 : Separate 3 2 1 0 0 1 2 3 15/24

Step 2 : Separate 3 ε > 0 εb n + v C v V 2 1 0 0 1 2 3 15/24

Step 2 : Separate 3 ε > 0 εb n + v C v V a 1 ε 2 σ C (a) σ C (a) 1 σ v+εb n(a) 1 = v, a + εa 1 v, a 1 0 0 1 2 3 15/24

Step 2 : Separate 3 ε > 0 εb n + v C v V a 1 ε 2 σ C (a) σ C (a) 1 σ v+εb n(a) 1 = v, a + εa 1 v, a S 2 =(1/ε)B Z n 1 0 0 1 2 3 15/24

Main Tool: Lifting Cuts for Faces Part 1: Kill Irrationality: Kronecker s approx. Part 2: Lift inside Dirichlet s approx. 16/24

Split Cuts 17/24

Split Cuts Split Disjunctions and Split Cuts 18/24

Split Cuts Split Disjunctions and Split Cuts Split Disjunction 18/24

Split Cuts Split Disjunctions and Split Cuts Split Disjunction Split Cuts 18/24

Split Cuts Known Facts for Rational Polyhedra Formulas for simplicial cones: MIG (Gomory 1960) and MIR (Nemhauser and Wolsey 1988) 19/24

Split Cuts Known Facts for Rational Polyhedra Formulas for simplicial cones: MIG (Gomory 1960) and MIR (Nemhauser and Wolsey 1988) Split Closure : Rational Polyhedron (Cook, Kannan and Shrijver 1990) Constructive Proofs: Dash, Günlük and Lodi 2007; V. 2007. 19/24

Split Cuts Split Cut for Ellipsoids Dadush, Dey and V. 2011: Also see Belotti, Góez, Polik, Ralphs, Terlaky 2011 20/24

Split Cuts Split Cut for Quadratic Cones Modaresi, Kılınç, V. 2011: 21/24

Split Cuts Split Closure is Finitely Generated Theorem (Dadush, Dey, V. 2011): If C is a strictly convex set then there exists a finite such that: Does not imply polyhedrality of split closure. Split Closure is not stable 22/24

Split Cuts Split Closure Can Be Non-Polyhedral 23/24

Other Results and Open Questions CG closure is polyhedron for a class of unbounded sets: Class includes rational polyhedra = True generalization of Schrijver theorem. Open Questions: Constructive characterization of CG closure. Algorithms to separate CG/Split cuts. 24/24