Prox-PEP: A Proximal Partial Exact Penalty Algorithm for Weakly Convex Stochastic Nonlinear Programming

TL;DR

Prox-PEP is a proximal algorithm designed for weakly convex stochastic nonlinear programming, employing quadratic subproblems and an exact penalty approach to efficiently handle nonlinear constraints.

Contribution

The paper introduces Prox-PEP, a novel proximal method with quadratic subproblems and a dynamic penalty strategy, providing comprehensive convergence and complexity guarantees.

Findings

Achieves an $ ilde{O}(T^{-1/4})$ expected convergence rate for stationarity.

Provides high-probability bounds with $ ilde{O}(T^{-1/8})$ for stationarity and $ ilde{O}(T^{-1/4})$ for constraints.

Guarantees strong convexity of subproblems via second-order approximation matrices.

Abstract

This paper considers stochastic optimization problems with weakly convex objective and constraint functions. We propose Prox-PEP, a proximal method equipped with quadratic subproblems. To handle nonlinear equality constraints, we employ an exact penalty approach, transforming them into inequality constraints with auxiliary slack variables. At each iteration, we construct quadratic approximations for both the objective and the constraint functions to facilitate efficient subproblem computation. By carefully designing the second-order approximation matrices, the subproblem constructed via the augmented Lagrangian function is strictly guaranteed to be strongly convex. Furthermore, we adopt a dynamic strategy for the equality penalty parameter: it monotonically increases up to a predefined threshold and remains constant thereafter. Building upon this algorithmic framework, we establish comprehensive asymptotic complexities. We prove that Prox-PEP achieves an $\mathcal{O}(T^{-1/4})$ average expected oracle complexity for $\epsilon$-KKT stationarity, specifically bounding the squared norm of the gradient of the Moreau envelope of the Lagrangian function, alongside constraint violations and complementarity conditions. Additionally, under standard light-tailed martingale noise assumptions, we derive an $\mathcal{O}(T^{-1/8})$ high-probability convergence bound for the norm of the gradient of the Lagrangian's Moreau envelope, as well as $\mathcal{O}(T^{-1/4})$ high-probability bounds for both constraint violations and complementarity conditions.