An active signature method for constrained abs-linear minimization

In this paper we consider the solution of optimization tasks with a piecewise linear objective function and piecewise linear constraints. First, we state optimality conditions for that class of problems using the abs-linearization approach and prove that they can be verified in polynomial time. Subsequently, we propose an algorithm called Constrained Active Signature Method that … Read more

Nonconvex piecewise linear functions: Advanced formulations and simple modeling tools

We present novel mixed-integer programming (MIP) formulations for (nonconvex) piecewise linear functions. Leveraging recent advances in the systematic construction of MIP formulations for disjunctive sets, we derive new formulations for univariate functions using a geometric approach, and for bivariate functions using a combinatorial approach. All formulations derived are small (logarithmic in the number of piecewise … Read more

A combinatorial approach for small and strong formulations of disjunctive constraints

We present a framework for constructing small, strong mixed-integer formulations for disjunctive constraints. Our approach is a generalization of the logarithmically-sized formulations of Vielma and Nemhauser for SOS2 constraints, and we offer a complete characterization of its expressive power. We apply the framework to a variety of disjunctive constraints, producing novel, small, and strong formulations … Read more

Analysis of MILP Techniques for the Pooling Problem

The $pq$-relaxation for the pooling problem can be constructed by applying McCormick envelopes for each of the bilinear terms appearing in the so-called $pq$-formulation of the pooling problem. This relaxation can be strengthened by using piecewise-linear functions that over- and under-estimate each bilinear term. The resulting relaxation can be written as a mixed integer linear … Read more