A study on two-metric projection methods

TL;DR

This paper analyzes the global convergence and complexity of the two-metric projection method for bound-constrained and -norm minimization problems, providing theoretical guarantees and numerical validation.

Contribution

It extends the two-metric projection method to nonconvex problems with complexity guarantees and generalizes it for -norm minimization.

Findings

Established global complexity guarantees for nonconvex problems.

Generalized the method for -norm minimization.

Validated theoretical results with numerical experiments.

Abstract

The two-metric projection method is a simple yet elegant algorithm proposed by Bertsekas in 1984 to address bound/box-constrained optimization problems. The algorithm's low per-iteration cost and potential for using Hessian information makes it a favourable computation method for this problem class. However, its global convergence guarantee is not studied in the nonconvex regime. In our work, we first investigate the global complexity of such a method for finding first-order stationary solution. After properly scaling each step, we equip the algorithm with competitive complexity guarantees. Furthermore, we generalize the two-metric projection method for solving $\ell_1$-norm minimization and discuss its properties via theoretical statements and numerical experiments.