An Exact Penalty Approach for Equality Constrained Optimization over a Convex Set

TL;DR

This paper introduces an exact penalty method called the constraint dissolving approach for nonlinear constrained optimization problems, transforming them into smoother problems that preserve key optimality conditions and improve computational efficiency.

Contribution

The paper proposes a novel constraint dissolving approach that transforms nonlinear constrained problems into smooth problems, maintaining optimality conditions and enabling efficient solution methods.

Findings

The approach preserves first- and second-order optimality conditions.

The method extends globally under an error bound condition.

Numerical experiments show high efficiency and practical potential.

Abstract

In this paper, we consider the nonlinear constrained optimization problem (NCP) with constraint set $\{x \in \mathcal{X}: c(x) = 0\}$, where $\mathcal{X}$ is a closed convex subset of $\mathbb{R}^n$. We propose an exact penalty approach, named constraint dissolving approach, that transforms (NCP) into its corresponding constraint dissolving problem (CDP). The transformed problem (CDP) admits $\mathcal{X}$ as its feasible region with a locally Lipschitz smooth objective function. We prove that (NCP) and (CDP) share the same first-order stationary points, second-order stationary points, second-order sufficient condition (SOSC) points, and strong SOSC points, in a neighborhood of the feasible region. Moreover, we prove that these equivalences extend globally under a particular error bound condition. Therefore, our proposed constraint dissolving approach enables direct implementations of optimization approaches over $\mathcal{X}$ and inherits their convergence properties to solve problems that take the form of (NCP). Preliminary numerical experiments illustrate the high efficiency of directly applying existing solvers for optimization over $\mathcal{X}$ to solve (NCP) through (CDP). These numerical results further demonstrate the practical potential of our proposed constraint dissolving approach.