Daniel Bienstock, Columbia University
Industrial Engineering and Operations Research
Consider the problem of optimizing a convex function subject to nonconvex constraints; in particular, minimizing a positive-definite quadratic subject to nonconvex constraints. The approach favored by discrete optimizers would rely on solving some (hopefully strong) convex relaxation of the problem, and then resorting to branching and/or cutting. However, when the objective is convex, this approach will fail, because even if the relaxation consists of the convex hull of the feasible region, the optimum over the relaxation will typically be infeasible, and (typically) "far away" from the feasible region, yielding a very poor estimate (lower bound) on the value of the problem.
We describe a simple technique that relies on the so-called S-Lemma and on combinatorial estimates of the distance from a point to the feasible region, to obtain fast, strong bounds on the value of interesting cases of the situation described in the above paragraph.
Professor Daniel Bienstock first joined Columbia University's Industrial Engineering and Operations Research Department in 1989. Professor Bienstock teaches courses on integer programming and optimization.
Before joining Columbia University, Professor Bienstock was involved in combinatorics and optimization research at Bellcore. He has also participated in collaborative research with Bell Laboratories (Lucent), AT&T Laboratories, Tellium, and Lincoln Laboratory on various network design problems.
Professor Bienstock's teaching and research interests include combinatorial optimization and integer programming, parallel computing and applications to networking. Professor Bienstock has published in journals such as Math Programming, SIAM, and Math of OR.