Split Cuts for Convex Nonlinear Mixed Integer Programming
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1 Split Cuts for Convex Nonlinear Mixed Integer Programming Juan Pablo Vielma Massachusetts Institute of Technology 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 International Symposium on Mathematical Programming, August 2012 Berlin, Germany
2 Outline Introduction Split Cut Formulas Summary 2/13
3 Introduction Split Disjunctions and Split Cuts 3/13
4 Introduction Split Disjunctions and Split Cuts 3/13
5 Introduction Split Disjunctions and Split Cuts Split Disjunction 3/13
6 Introduction Split Disjunctions and Split Cuts Split Disjunction 3/13
7 Introduction Split Disjunctions and Split Cuts Split Disjunction 3/13
8 Introduction Split Disjunctions and Split Cuts Split Disjunction 3/13
9 Introduction Split Disjunctions and Split Cuts Split Disjunction 3/13
10 Introduction Split Disjunctions and Split Cuts Split Disjunction 3/13
11 Introduction Split Disjunctions and Split Cuts Split Disjunction Split Cuts 3/13
12 Introduction Split Disjunctions and Split Cuts Split Disjunction Split Cuts 3/13
13 Introduction Split Cuts for Simplicial Cones Formulas: (MIG: Gomory 1960 and MIR: Nemhauser and Wolsey 1988) 4/13
14 Formulas Split Cuts for Quadratic Cones Formulas: (Modaresi, Kılınç, V. 2011) (also see Atamturk and Narayanan 2010 for elementary integer splits) 5/13
15 Formulas Split Cuts for Quadratic Cones Formulas: (Modaresi, Kılınç, V. 2011) (also see Atamturk and Narayanan 2010 for elementary integer splits) 5/13
16 Formulas Split Cuts for Quadratic Cones Formulas: (Modaresi, Kılınç, V. 2011) (also see Atamturk and Narayanan 2010 for elementary integer splits) 5/13
17 Formulas Split Cuts for Quadratic Cones Formulas: (Modaresi, Kılınç, V. 2011) (also see Atamturk and Narayanan 2010 for elementary integer splits) 5/13
18 Formulas Split Cuts for Quadratic Cones Formulas: (Modaresi, Kılınç, V. 2011) (also see Atamturk and Narayanan 2010 for elementary integer splits) 5/13
19 Formulas Split Cuts for Quadratic Cones Formulas: (Modaresi, Kılınç, V. 2011) (also see Atamturk and Narayanan 2010 for elementary integer splits) 5/13
20 Formulas Split Cuts for Quadratic Cones Formulas: (Modaresi, Kılınç, V. 2011) (also see Atamturk and Narayanan 2010 for elementary integer splits) 5/13
21 Formulas Split Cuts for Quadratic Cones Formulas: (Modaresi, Kılınç, V. 2011) (also see Atamturk and Narayanan 2010 for elementary integer splits) 5/13
22 Formulas Split Cuts for Ellipsoids Formulas: (Dadush, Dey and V. 2011) (also see Belotti, Góez, Polik, Ralphs, Terlaky 2011) 6/13
23 Formulas Split Cuts for Paraboloids Formulas: (Modaresi, Kılınç, V. 2012) C := (x, t 0 ) 2 R n R : ka(x c)k 2 2 apple t 0 C 0, 1 := {(x, t 0 ) 2 R n R : ka(x c)k 2 2 apple t 0, kb(x d)k 2 2 apple t 0 + ha, xi + b} (also see Belotti, Góez, Polik, Ralphs, Terlaky 2012 for bounded case) 7/13
24 Formulas Message: Use the Right Split Cut Cone Paraboloid Ellipsoid 8/13
25 Formulas Split Cuts for P-Order Cones Formulas: (Modaresi, Kılınç, V. 2011) Elementary splits: 9/13
26 Atamturk and Narayanan /13
27 Atamturk and Narayanan 2010 Extended Formulation: 10/13
28 Atamturk and Narayanan 2010 Extended Formulation: Linear Part 10/13
29 Atamturk and Narayanan 2010 Extended Formulation: Linear Part Nonlinear Part 10/13
30 Atamturk and Narayanan 2010 Extended Formulation: Linear Part Nonlinear Part Aggregate: 10/13
31 Atamturk and Narayanan 2010 Extended Formulation: Linear Part Nonlinear Part Aggregate: : 10/13
32 and Nonlinear Split Cut Modaresi, Kılınç, V Extended Formulation: Linear Part Nonlinear Part 11/13
33 and Nonlinear Split Cut Modaresi, Kılınç, V Extended Formulation: Linear Part Nonlinear Part 11/13
34 and Nonlinear Split Cut Modaresi, Kılınç, V Extended Formulation: Linear Part Nonlinear Part = Split cuts for linear part Nonlinear split cut 11/13
35 Nonlinear Split Cut v/s MIR Strength Single Split Cut is Stronger than MIR 12/13
36 Nonlinear Split Cut v/s MIR Strength Single Split Cut is Stronger than MIR 12/13
37 Nonlinear Split Cut v/s MIR Strength Single Split Cut is Stronger than MIR MIR Closure 12/13
38 Nonlinear Split Cut v/s MIR Strength Single Split Cut is Stronger than MIR Feasible for Nonlinear Split Closure MIR Closure 12/13
39 Nonlinear Split Cut v/s MIR Strength Single Split Cut is Stronger than MIR Feasible for Nonlinear Split Closure Nonlinear Split Cut MIR Closure 12/13
40 Summary and Open Questions Formulas for nonlinear split cuts Quadratic cones, ellipsoids and others. Strong ties to conic MIR. Future: Computation (INFORMS Phoenix). Extended Formulations (INFORMS Phoenix). More formulas. 13/13
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