Nonsmooth exact penalty methods for equality-constrained optimization: complexity and implementation
Youssef Diouane, Maxence Gollier, Dominique Orban
TL;DR
The paper develops and analyzes a proximal-implementation of nonsmooth exact penalty methods for equality-constrained optimization. By solving penalized unconstrained subproblems with proximal-type solvers, it achieves convergence to approximate KKT points under MFCQ and Lipschitz smoothness, with a worst-case outer-inner complexity of $O(epsilon^{-2})$ (and $O(epsilon^{-8})$ in degenerate cases). It provides explicit proximal-operator formulations, including an efficient dual-trust-region approach for the ℓ2 penalty, and demonstrates competitive robustness and performance against augmented Lagrangian methods and SQP on small CUTEst problems. The work offers a practical, first-of-its-kind proximal-based exact penalty implementation and outlines clear avenues for extension and improvement.
Abstract
Penalty methods are a well known class of algorithms for constrained optimization. They transform a constrained problem into a sequence of unconstrained \emph{penalized} problems in the hope that approximate solutions of the latter converge to a solution of the former. If Lagrange multipliers exist, exact penalty methods ensure that the penalty parameter only need increase a finite number of times, but are typically scorned in smooth optimization for the penalized problems are not smooth. This led researchers to consider the implementation of exact penalty methods inconvenient. Recent advances in proximal methods have led to increasingly efficient solvers for nonsmooth optimization. We study a general exact penalty algorithm and use it to show that the exact $\ell_2$-penalty method for equality-constrained optimization can, in fact, be implemented efficiently by solving the penalized problem using a proximal-type algorithm. We study the convergence of our algorithm and establish a worst-case complexity bound of $\mathcal{O}(ε^{-2})$ to bring a stationarity measure below $ε> 0$ under the Mangarasian-Fromowitz constraint qualification and Lipschitz continuity of the objective gradient and constraint Jacobian. While the Lipschitz continuity of the objective gradient is not required for convergence in view of recent works, it is used in our analysis to derive the complexity bound. In a degenerate scenario where the penalty parameter grows unbounded, the complexity becomes $\mathcal{O}(ε^{-8})$, which is worse than another bound found in the literature. Finally, we report numerical experience on small-scale problems from a standard collection and compare our solver with an augmented-Lagrangian and an SQP method. Our preliminary implementation is superior to the augmented Lagrangian in terms of robustness and efficiency, and is competitive with the SQP method.