Complexity of convex optimization using geometry-based measures and a reference point

Our concern lies in solving the following convex optimization problem:where P is a closed convex subset of the n-dimensional vector space X. We bound the complexity of computing an almost-optimal solution of G_P in terms of natural geometry-based measures of the feasible region and the level-set of almost-optimal solutions, relative to a given reference point x^r that might be close to the feasible region and / or the almost-optimal level set. This contrasts with other complexity bounds for convex optimization that rely on data-based condition numbers or algebraic measures, and that do not take into account any a priori reference point information.Mathematics Subject Classification (2000):90C, 90C05, 90C60This research has been partially supported through the Singapore-MIT Alliance. Portions of this research were undertaken when the author was a Visiting Scientist at Delft University of Technology.Received: 1, October 2001