A generalized projection-based scheme for solving convex constrained optimization problems

TL;DR

This paper introduces a flexible projection-based algorithm for convex constrained optimization that iteratively transforms the problem into feasibility problems, allowing the use of various projection methods without requiring exact projections.

Contribution

The paper proposes a novel generalized projection scheme that transforms convex optimization into a sequence of feasibility problems, incorporating superiorization and demonstrating practical effectiveness.

Findings

Effective for convex quadratic problems

Applicable to real-life medical optimization tasks

Flexible with various projection methods

Abstract

In this paper we present a new algorithmic realization of a projection-based scheme for general convex constrained optimization problem. The general idea is to transform the original optimization problem to a sequence of feasibility problems by iteratively constraining the objective function from above until the feasibility problem is inconsistent. For each of the feasibility problems one may apply any of the existing projection methods for solving it. In particular, the scheme allows the use of subgradient projections and does not require exact projections onto the constraints sets as in existing similar methods. We also apply the newly introduced concept of superiorization to optimization formulation and compare its performance to our scheme. We provide some numerical results for convex quadratic test problems as well as for real-life optimization problems coming from medical treatment planning.