Inexact Sequential Quadratic Optimization with Penalty Parameter Updates Within the QP Solve: Extended Version

TL;DR

This paper introduces an inexact SQP method with a dynamic penalty parameter update strategy, improving convergence and efficiency for large-scale nonlinear optimization problems using matrix-free subproblem solvers.

Contribution

It proposes a novel penalty parameter updating strategy within inexact SQP methods, enhancing convergence and robustness in large-scale nonlinear optimization.

Findings

The method achieves reliable convergence with inexact QP solves.

Dynamic penalty updates improve progress prediction toward optimality.

Numerical experiments demonstrate the effectiveness of the proposed approach.

Abstract

This paper focuses on the design of sequential quadratic optimization (commonly known as SQP) methods for solving large-scale nonlinear optimization problems. The most computationally demanding aspect of such an approach is the computation of the search direction during each iteration, for which we consider the use of matrix-free methods. In particular, we develop a method that requires an inexact solve of a single QP subproblem to establish the convergence of the overall SQP method. It is known that SQP methods can be plagued by poor behavior of the global convergence mechanism. To confront this issue, we propose the use of an exact penalty function with a dynamic penalty parameter updating strategy to be employed within the subproblem solver in such a way that the resulting search direction predicts progress toward both feasibility and optimality. We present our parameter updating strategy and prove that, under reasonable assumptions, the strategy does not modify the penalty parameter unnecessarily. We also discuss a matrix-free subproblem solver in which our updating strategy can be incorporated. We close the paper with a discussion of the results of numerical experiments that illustrate the benefits of our proposed techniques.