A Projected Variable Smoothing for Weakly Convex Optimization and Supremum Functions

TL;DR

This paper introduces a projected variable smoothing algorithm for weakly convex optimization, providing complexity bounds, and explores the Moreau envelope and proximity operators for supremum functions, with applications to robust optimization and LASSO.

Contribution

It proposes a novel smoothing algorithm with complexity analysis and characterizes proximity operators for supremum of weakly convex functions, extending optimization tools.

Findings

Complexity bound of $\ ilde{O}(\epsilon^{-3})$ for the smoothing algorithm.

Explicit formulas for proximity operators in key cases.

Numerical results demonstrating effectiveness on max dispersion problem.

Abstract

In this paper, we address two main topics. First, we study the problem of minimizing the sum of a smooth function and the composition of a weakly convex function with a linear operator on a closed vector subspace. For this problem, we propose a projected variable smoothing algorithm and establish a complexity bound of $\mathcal{O}(\epsilon^{-3})$ to achieve an $\epsilon$-approximate solution. Second, we investigate the Moreau envelope and the proximity operator of functions defined as the supremum of weakly convex functions, and we compute the proximity operator in two important cases. In addition, we apply the proposed algorithm for solving a distributionally robust optimization problem, the LASSO with linear constraints, and the max dispersion problem. We illustrate numerical results for the max dispersion problem.