Improving the distance reduction step in the von Neumann algorithm

TL;DR

This paper enhances the von Neumann algorithm for conic problems by improving the distance reduction step through projection onto convex hulls using a primal active set QP, leading to better convergence and performance.

Contribution

It introduces a novel projection method within the von Neumann algorithm using primal active set QP, with theoretical convergence improvements and empirical performance gains.

Findings

Larger projection sets improve per-iteration performance.

The algorithm's convergence is better with larger QPs.

Numerical experiments confirm improved efficiency.

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.