Perspective Relaxation of Minlps with Indicator Variables

We study mixed integer nonlinear programs (MINLP) that are driven by a collection of indicator variables where each indicator variable controls a subset of the decision variables. An indicator variable, when it is “turned off”, forces some of the decision variables to assume a fixed value, and, when it is “turned on”, forces them to belong to a convex set. Most of the integer variables in known MINLP problems are of this type. We first study a mixed integer set defined by a single separable quadratic constraint and a collection of variable upper and lower bound constraints. This is an interesting set that appears as a substructure in many applications. We present the convex hull description of this set. We then extend this to produce an explicit characterization of the convex hull of the union of a point and a bounded convex set defined by analytic functions. Further, we show that for many classes of problems, the convex hull can be expressed via conic quadratic constraints, and thus relaxations can be solved via second-order cone programming. Our work is closely related with the earlier work of Ceria and Soares (1996) as well as recent work by Frangioni and Gentile (2006) and, Aktürk, Atamtürk and Gürel (2007). Finally, we apply our results to develop tight formulations of mixed integer nonlinear programs in which the nonlinear functions are separable and convex and in which indicator variables play an important role. In particular, we present strong computational results with two applications – quadratic facility location and network design with congestion – that show the power of the reformulation technique.