Lifting for conic mixed-integer programming

Lifting is a procedure for deriving valid inequalities for mixed-integer sets from valid inequalities for suitable restrictions of those sets. Lifting has been shown to be very effective in developing strong valid inequalities for linear integer programming and it has been successfully used to solve such problems with branch-and-cut algorithms. Here we generalize the theory of lifting to conic integer programming, i.e., integer programs with conic constraints. We show how to derive conic valid inequalities for a conic integer program from conic inequalities valid for its lower-dimensional restrictions. In order to simplify the computations, we also discuss sequence-independent lifting for conic integer programs. When the cones are restricted to nonnegative orthants, conic lifting reduces to the lifting for linear integer programming as one may expect.

[1]  Stephen P. Boyd,et al.  Applications of second-order cone programming , 1998 .

[2]  Alper Atamtürk,et al.  On the facets of the mixed–integer knapsack polyhedron , 2003, Math. Program..

[3]  Jean-Philippe P. Richard,et al.  Lifting inequalities: a framework for generating strong cuts for nonlinear programs , 2010, Math. Program..

[4]  Alper Atamtürk,et al.  Conic mixed-integer rounding cuts , 2009, Math. Program..

[5]  Laurence A. Wolsey.  Valid Inequalities and Superadditivity for 0-1 Integer Programs , 1977, Math. Oper. Res..

[6]  Alper Atamtürk,et al.  Cuts for Conic Mixed-Integer Programming , 2007, IPCO.

[7]  Arkadi Nemirovski,et al.  On Polyhedral Approximations of the Second-Order Cone , 2001, Math. Oper. Res..

[8]  Mehmet Tolga Çezik,et al.  Cuts for mixed 0-1 conic programming , 2005, Math. Program..

[9]  Laurence A. Wolsey,et al.  Lifting, superadditivity, mixed integer rounding and single node flow sets revisited , 2003, 4OR.

[10]  Sinan Gürel,et al.  A strong conic quadratic reformulation for machine-job assignment with controllable processing times , 2009, Oper. Res. Lett..

[11]  Laurence A. Wolsey,et al.  Technical Note - Facets and Strong Valid Inequalities for Integer Programs , 1976, Oper. Res..

[12]  Alper Atamtürk,et al.  Sequence Independent Lifting for Mixed-Integer Programming , 2004, Oper. Res..

[13]  Arkadi Nemirovski,et al.  Lectures on modern convex optimization - analysis, algorithms, and engineering applications , 2001, MPS-SIAM series on optimization.

[14]  George L. Nemhauser,et al.  Lifted inequalities for 0-1 mixed integer programming: Superlinear lifting , 2003, Math. Program..

[15]  George L. Nemhauser,et al.  Lifted inequalities for 0-1 mixed integer programming: Basic theory and algorithms , 2002, Math. Program..

[16]  Aharon Ben-Tal,et al.  Lectures on modern convex optimization , 1987 .

[17]  Manfred W. Padberg.  Technical Note - A Note on Zero-One Programming , 1975, Oper. Res..

[18]  George L. Nemhauser,et al.  A Lifted Linear Programming Branch-and-Bound Algorithm for Mixed-Integer Conic Quadratic Programs , 2008, INFORMS J. Comput..

[19]  Alper Atamtürk,et al.  The submodular knapsack polytope , 2009, Discret. Optim..