Mathematics Colloquium

Russel Luke


"Nonconvex optimization methods versus convex relaxations: the cost of certainty"

Date: Mon, November 4, 2013
Time: 17:15
Place: Research II Lecture Hall

Abstract: Numerical algorithms for nonconvex optimization models are often eschewed because the usual optimality criteria around which numerical algorithms are designed do not distinguish solutions from critical points. This issue comes into sharp relief with what has become known as the sparsity optimization problem. The current trend for solving this problem, sparked by a famous paper of Candes and Tao [2005], is to use convex relaxations. Convex relaxations have the advantage that every point satisfying the necessary optimality criteria is also a solution to the relaxed optimization problem. This certainty comes at the cost of imposing difficult-to-verify restrictions on the affine constraints in order to guarantee the correspondence of solutions to the relaxed problem to solutions to the original problem. Moreover, convex relaxations can lead to a tremendous increase in the dimensionality of the problem. We present a different nonconvex approach; one with the advantage that the available algorithms are simple to apply, (locally) linearly convergent, and the problem formulation stays close in spirit if not in fact to the original problem, thus avoiding the curse of dimensionality. We also provide conditions under which fundamental algorithms applied to the nonconvex model are globally convergent.

We hope to convince our audience that convexity is not a categorical imperative for provablly convergent numerical algorithms.