Valid Inequalities for Separable Concave Constraints with Indicator Variables

Abstract

We study valid inequalities for a set relevant for optimization models that have both binary indicator variables, which indicate positivity of associated continuous variables, and separable concave constraints. Such models reduce to a mixed-integer linear program (MILP) when the concave constraints are ignored, and to a nonconvex global optimization problem when the binary restrictions are ignored. In algorithms to solve such problems to global optimality, relaxations are traditionally obtained by using valid inequalities for the MILP ignoring the concave constraints, and by independently relaxing each concave constraint using the secant obtained from the bounds of the associated variable. We propose a technique to obtain valid inequalities that are based on both the MILP and the concave constraints. We begin by analyzing a low-dimensional set that contains a single binary indicator variable, a single concave constraint, and three continuous variables. Using this analysis, for the canonical Single Node Flow Set (SNFS), we demonstrate how to “tilt” a given valid inequality for the SNFS to obtain additional valid inequalities that account for separable concave functions of the arc flows. We present computational results demonstrating the utility of the new inequalities on a fixed plus concave cost transportation problem. To our knowledge, this is one of the first works that simultaneously convexifies both nonconvex functions and binary variables to strengthen the relaxations of practical mixed integer nonlinear programs.