An Exact Penalty Approach for Equality Constrained Optimization over a Convex Set
Nachuan Xiao, Tianyun Tang, Shiwei Wang, Kim-Chuan Toh
TL;DR
This work develops an exact penalty framework, the constraint dissolving approach, for solving equality-constrained optimization problems over a closed convex set $\mathcal{X}$. By introducing a locally Lipschitz mapping $\mathcal{A}$ and a penalty term, the method forms a differentiable problem $h(x)=f(\mathcal{A}(x))+\tfrac{\beta}{2}\|c(x)\|^2$ over $\mathcal{X}$ whose stationary points correspond to those of the original problem under mild conditions. The authors prove local (and under an error-bound condition global) equivalence of first-/second-order stationary points, SOSC, and strict complementarity between the original problem and the penalty formulation, and provide a systematic way to construct $\mathcal{A}$ via a projective mapping $Q$ for common convex sets. They also present extensive numerical experiments showing substantial computational gains by solving the penalty problem with standard solvers over $\mathcal{X}$, thereby validating the practical potential of the approach for tasks such as nonnegative sparse PCA, nonconvex QP with ball constraints, and fair PCA.
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.