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TITLE: Using eigenvalue techniques to obtain better bounds for convex objective, non-convex optimization problems
SPEAKER: Dr. Daniel Bienstock
ABSTRACT:
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.
