Projections Onto Convex Sets (POCS) Based Optimization by Lifting

TL;DR

This paper introduces a novel POCS-based optimization method that lifts the problem dimension by one, enabling globally optimal solutions for various convex and some non-convex functions through iterative projections.

Contribution

The paper presents a new lifting-based POCS framework that extends the applicability of projection methods to a broader class of optimization problems, including non-convex cases.

Findings

Provides globally optimal solutions for total-variation and l1 cost functions.

Handles non-convex lp, p<1 functions using supporting hyperplanes.

Demonstrates effectiveness through experimental results.

Abstract

Two new optimization techniques based on projections onto convex space (POCS) framework for solving convex and some non-convex optimization problems are presented. The dimension of the minimization problem is lifted by one and sets corresponding to the cost function are defined. If the cost function is a convex function in R^N the corresponding set is a convex set in R^(N+1). The iterative optimization approach starts with an arbitrary initial estimate in R^(N+1) and an orthogonal projection is performed onto one of the sets in a sequential manner at each step of the optimization problem. The method provides globally optimal solutions in total-variation, filtered variation, l1, and entropic cost functions. It is also experimentally observed that cost functions based on lp, p<1 can be handled by using the supporting hyperplane concept.