Convergence of Successive Linear Programming Algorithms for Noisy Functions

TL;DR

This paper extends convergence analysis of successive linear programming algorithms to noisy, composite optimization problems, demonstrating their robustness and providing practical strategies for stabilization in noisy environments.

Contribution

It generalizes existing convergence results to noisy composite functions and offers computational insights for practical stabilization parameter selection.

Findings

Convergence to a critical region is achieved despite noise in function and gradient evaluations.

The analysis applies to image reconstruction and constrained optimization problems.

Strategies for choosing stabilization parameters are proposed and illustrated.

Abstract

Gradient-based methods have been highly successful for solving a variety of both unconstrained and constrained nonlinear optimization problems. In real-world applications, such as optimal control or machine learning, the necessary function and derivative information may be corrupted by noise, however. Sun and Nocedal have recently proposed a remedy for smooth unconstrained problems by means of a stabilization of the acceptance criterion for computed iterates, which leads to convergence of the iterates of a trust-region method to a region of criticality, Sun and Nocedal (2022). We extend their analysis to the successive linear programming algorithm, Byrd et al. (2023a,2023b), for unconstrained optimization problems with objectives that can be characterized as the composition of a polyhedral function with a smooth function, where the latter and its gradient may be corrupted by noise. This gives the flexibility to cover, for example, (sub)problems arising image reconstruction or constrained optimization algorithms. We provide computational examples that illustrate the findings and point to possible strategies for practical determination of the stabilization parameter that balances the size of the critical region with a relaxation of the acceptance criterion (or descent property) of the algorithm.