COMPLEXITY OF CONVEX OPTIMIZATION USING GEOMETRY-BASED MEASURES AND A REFERENCE POINT ^

Our concern lies in solving the following convex optimization prob-lem: Gp: minimize ^ c^x s.t. Ax = b X eP, 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 Gp in terms of natural geometry-based measures of the feasible re-gion and the level-set of almost-optimal solutions, relative to a given reference point x ^ 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 ref-erence point information