Penalty methods for a class of non-Lipschitz optimization problems

TL;DR

This paper develops a theoretical framework for penalty methods applied to non-Lipschitz, nonconvex optimization problems with applications in data science, and demonstrates their effectiveness in finding sparse solutions.

Contribution

It provides new insights into exact penalty parameters and proposes a convergent penalty algorithm for complex non-Lipschitz problems.

Findings

Proved existence of exact penalty parameters for local minimizers and stationary points.

Developed a penalty method with convergence guarantees to KKT points.

Numerical results show the method efficiently finds sparse solutions.

Abstract

We consider a class of constrained optimization problems with a possibly nonconvex non-Lipschitz objective and a convex feasible set being the intersection of a polyhedron and a possibly degenerate ellipsoid. Such problems have a wide range of applications in data science, where the objective is used for inducing sparsity in the solutions while the constraint set models the noise tolerance and incorporates other prior information for data fitting. To solve this class of constrained optimization problems, a common approach is the penalty method. However, there is little theory on exact penalization for problems with nonconvex and non-Lipschitz objective functions. In this paper, we study the existence of exact penalty parameters regarding local minimizers, stationary points and $\epsilon$-minimizers under suitable assumptions. Moreover, we discuss a penalty method whose subproblems are solved via a nonmonotone proximal gradient method with a suitable update scheme for the penalty parameters, and prove the convergence of the algorithm to a KKT point of the constrained problem. Preliminary numerical results demonstrate the efficiency of the penalty method for finding sparse solutions of underdetermined linear systems.