Improving the distance reduction step in the von Neumann algorithm

Abstract

A known first order method to find a feasible solution to a conic problem is an adapted von Neumann algorithm. We improve the distance reduction step there by projecting onto the convex hull of previously generated points using a primal active set quadratic programming (QP) algorithm. The convergence theory is improved when the QPs are as large as possible. For problems in R^2, we analyze our algorithm by epigraphs and the monotonicity of subdifferentials. Logically, the larger the set to project onto, the better the performance per iteration, and this is indeed seen in our numerical experiments.