Convex envelope results and strong formulations for a class of mixed-integer programs

In this article the authors present a novel technique for deriving the convex envelope of certain nonconvex fixed-charge functions of the type that arise in several related applications that have been considered in the literature. One common attribute of these problems is that they involve choosing levels for the undertaking of several activities. Two or more activities share a common resource, and a fixed charge is incurred when any of these activities is undertaken at a positive level. The authors consider nonconvex programming formulations for these problems in which the fixed charges are expressed in the form of concave functions. With the use of the developed convex envelope results, they show that the convex envelope relaxations of the nonconvex formulations lead to the linear programming relaxations of the strong IP/MIP formulations of these problems. Moreover, the present technique for deriving convex envelopes offers a useful construct that could be exploited in other related contexts as well.