Our concern lies in solving the following convex problem: GP: minimizex cTx s.t. Ax = b x ∈ P, 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 region and the level-set of almost-optimal solutions, relative to a given reference point xr 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.
