Iterated Linear Optimization

TL;DR

This paper presents a fixed point iteration method based on linear optimization over convex sets, proving convergence and characterizing fixed points, with applications to semidefinite programming relaxations.

Contribution

It introduces a novel fixed point iteration process for convex sets, including elliptopes, with algebraic characterization of fixed points and practical applications.

Findings

Convergence to fixed points is always guaranteed.

Attractive fixed points of elliptopes are exactly their vertices.

The method can be used for rounding solutions in semidefinite programming.

Abstract

We introduce a fixed point iteration process built on optimization of a linear function over a compact domain. We prove the process always converges to a fixed point and explore the set of fixed points in various convex sets. In particular, we consider elliptopes and derive an algebraic characterization of their fixed points. We show that the attractive fixed points of an elliptope are exactly its vertices. Finally, we discuss how fixed point iteration can be used for rounding the solution of a semidefinite programming relaxation.