James R. Luedtke We study solution approaches for the design of reliably connected networks. Speci fically, given a network with arcs that may fail at random, the goal is to select a minimum cost subset of arcs such the probability that a connectivity requirement is satis ed is at least 1-\epsilon, where \epsilon is a risk tolerance. We consider two … Read more
Strong branching is an effective branching technique that can significantly reduce the size of the branch-and-bound tree for solving Mixed Integer Nonlinear Programming (MINLP) problems. The focus of this paper is to demonstrate how to effectively use discarded information from strong branching to strengthen relaxations of MINLP problems. Valid inequalities such as branching-based linearizations, various … Read more
We present a new approach for exactly solving chance-constrained mathematical programs having discrete distributions with nite support and random polyhedral constraints. Such problems have been notoriously difficult to solve due to nonconvexity of the feasible region, and most available methods are only able to nd provably good solutions in certain very special cases. Our approach … Read more
We describe a computationally effective method for generating disjunctive inequalities for convex mixed-integer nonlinear programs (MINLPs). The method relies on solving a sequence of cut-generating linear programs, and in the limit will generate an inequality as strong as can be produced by the cut-generating nonlinear program suggested by Stubbs and Mehrotra. Using this procedure, we … Read more
We study approaches for obtaining convex relaxations of global optimization problems containing multilinear functions. Specifically, we compare the concave and convex envelopes of these functions with the relaxations that are obtained with a standard relaxation approach, due to McCormick. The standard approach reformulates the problem to contain only bilinear terms and then relaxes each term … Read more
Dentcheva and Ruszczynski recently proposed using a stochastic dominance constraint to specify risk preferences in a stochastic program. Such a constraint requires the random outcome resulting from one’s decision to stochastically dominate a given random comparator. These ideas have been extended to problems with multiple random outcomes, using the notion of positive linear stochastic dominance. … Read more
Stochastic dominance constraints allow a decision-maker to manage risk in an optimization setting by requiring their decision to yield a random outcome which stochastically dominates a reference random outcome. We present new integer and linear programming formulations for optimization under first and second-order stochastic dominance constraints, respectively. These formulations are more compact than existing formulations, … Read more
We study approximations of optimization problems with probabilistic constraints in which the original distribution of the underlying random vector is replaced with an empirical distribution obtained from a random sample. We show that such a sample approximation problem with risk level larger than the required risk level will yield a lower bound to the true … Read more
Linear programs with joint probabilistic constraints (PCLP) are difficult to solve because the feasible region is not convex. We consider a special case of PCLP in which only the right-hand side is random and this random vector has a finite distribution. We give a mixed-integer programming formulation for this special case and study the relaxation … Read more