Split Cuts for Convex Nonlinear Mixed Integer Programming
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1 Split Cuts for Convex Nonlinear Mixed 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 and ONR N Integer Programming Workshop, March 2012 Valparaiso, Chile
2 Outline Introduction Split Cut Formulas Split Closure Conclusions 2/94
3 Introduction Split Disjunctions and Split Cuts 3/94
4 Introduction Split Disjunctions and Split Cuts 3/94
5 Introduction Split Disjunctions and Split Cuts Split Disjunction 3/94
6 Introduction Split Disjunctions and Split Cuts Split Disjunction 3/94
7 Introduction Split Disjunctions and Split Cuts Split Disjunction 3/94
8 Introduction Split Disjunctions and Split Cuts Split Disjunction 3/94
9 Introduction Split Disjunctions and Split Cuts Split Disjunction 3/94
10 Introduction Split Disjunctions and Split Cuts Split Disjunction 3/94
11 Introduction Split Disjunctions and Split Cuts Split Disjunction Split Cuts 3/94
12 Introduction Split Disjunctions and Split Cuts Split Disjunction Split Cuts 3/94
13 Introduction 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 /94
14 Split Cuts for Simplicial Cones Formulas: (MIG: Gomory 1960 and MIR: Nemhauser and Wolsey 1988) 5/94
15 Split Cuts for Simplicial Cones Formulas: (MIG: Gomory 1960 and MIR: Nemhauser and Wolsey 1988) (e.g. V. 2007) 5/94
16 Split Cuts for Quadratic Cones Formulas: (Modaresi, Kılınç, V. 2011) (also see Atamturk and Narayanan 2010 for elementary integer splits) 6/94
17 Split Cuts for Quadratic Cones Formulas: (Modaresi, Kılınç, V. 2011) (also see Atamturk and Narayanan 2010 for elementary integer splits) 6/94
18 Split Cuts for Quadratic Cones Formulas: (Modaresi, Kılınç, V. 2011) (also see Atamturk and Narayanan 2010 for elementary integer splits) 6/94
19 Split Cuts for Quadratic Cones Formulas: (Modaresi, Kılınç, V. 2011) (also see Atamturk and Narayanan 2010 for elementary integer splits) 6/94
20 Split Cuts for Quadratic Cones Formulas: (Modaresi, Kılınç, V. 2011) (also see Atamturk and Narayanan 2010 for elementary integer splits) 6/94
21 Split Cuts for Quadratic Cones Formulas: (Modaresi, Kılınç, V. 2011) (also see Atamturk and Narayanan 2010 for elementary integer splits) 6/94
22 Split Cuts for Quadratic Cones Formulas: (Modaresi, Kılınç, V. 2011) (also see Atamturk and Narayanan 2010 for elementary integer splits) 6/94
23 Split Cuts for Quadratic Cones Formulas: (Modaresi, Kılınç, V. 2011) (also see Atamturk and Narayanan 2010 for elementary integer splits) 6/94
24 Conic MIR Atamturk and Narayanan /94
25 Conic MIR Atamturk and Narayanan 2010 Extended Formulation: 7/94
26 Conic MIR Atamturk and Narayanan 2010 Extended Formulation: Linear Part 7/94
27 Conic MIR Atamturk and Narayanan 2010 Extended Formulation: Linear Part Nonlinear Part 7/94
28 Conic MIR Atamturk and Narayanan 2010 Extended Formulation: Linear Part Nonlinear Part Aggregate: 7/94
29 Conic MIR Atamturk and Narayanan 2010 Extended Formulation: Linear Part Nonlinear Part Aggregate: Conic MIR: 7/94
30 Conic MIR and Nonlinear Split Cut Modaresi, Kılınç, V Extended Formulation: Linear Part Nonlinear Part 8/94
31 Conic MIR and Nonlinear Split Cut Modaresi, Kılınç, V Extended Formulation: Linear Part Nonlinear Part 8/94
32 Conic MIR and Nonlinear Split Cut Modaresi, Kılınç, V Extended Formulation: Linear Part Nonlinear Part Conic MIR = Split cuts for linear part Nonlinear split cut 8/94
33 Split Cuts for Ellipsoids Formulas: (Dadush, Dey and V. 2011) (also see Belotti, Góez, Polik, Ralphs, Terlaky 2011) 9/94
34 Split Cuts for P-Order Cones Formulas: (Modaresi, Kılınç, V. 2011) Elementary splits: 10/94
35 Split Closure 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 11/94
36 Split Closure Split Closure Can Be Non-Polyhedral 12/94
37 Split Closure Split Closure Can Be Non-Polyhedral 12/94
38 Summary and Open Questions Formulas for nonlinear split cuts Quadratic cones, ellipsoids and others. Strong ties to conic MIR Split closure: Finitely generated, not polyhedral Future: More formulas Computation More general/constructive split closure 13/94
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