Lifting for conic mixed-integer programming

Abstract

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 lowerdimensional 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.

Cutting-Planes for Optimization of Convex Functions over Nonconvex Sets

“…The extension of lifting to the nonlinear or continuous setting is not new: see [17], [27], [34], and [7]. Also see [16], which lifts "tangent" inequalities to approximate multilinear functions.…”

“…The construction that culminates in Definition 2.5 is a generalization of the classical lifting construction in mixed-integer programming (see [44], [27], [34], [7]). The LFO inequality at y uses the local structure of P to strengthen the linearization inequality (2); the strengthening is only local, however the LFO inequality is (globally) valid.…”

A Conic Integer Programming Approach to Stochastic Joint Location-Inventory Problems

“…366-381, © 2012 INFORMS conic quadratic mixed-integer programs and Çezik and Iyengar (2005) give convex quadratic cuts for mixed 0-1 conic programs. Atamtürk and Narayanan (2011) propose lifting methods for conic mixed-integer programming. Atamtürk and Narayanan (2009) propose cover-type inequalities for submodular knapsack sets and Atamtürk and Narayanan (2008) introduce polymatroid inequalities that can help with solving special structured conic quadratic programs efficiently.…”

A strong conic quadratic reformulation for machine-job assignment with controllable processing times

“…Atamtürk and Narayanan [10] give nonlinear conic mixed-integer rounding cuts for conic mixed-integer programming. Atamtürk and Narayanan [11] describe lifting techniques for conic integer programming. Whereas these papers develop cuts for general conic mixed-integer programs, in this study we exploit the structure of a certain objective function in order to derive strong conic formulations.…”