Error-tolerant Multisecant Method for Nonlinearly Constrained Optimization

TL;DR

This paper introduces an error-tolerant, matrix-free multisecant method for nonlinearly constrained optimization that effectively handles data inaccuracies and nonconvex problems, offering a promising alternative to traditional gradient-based algorithms.

Contribution

It develops a novel multisecant quasi-Newton algorithm that is robust to data errors and incorporates preconditioning and regularization for nonconvex problems.

Findings

Demonstrates robustness to data inaccuracies

Shows improved performance with preconditioning

Effective on nonconvex constrained problems

Abstract

We present a derivative-based algorithm for nonlinearly constrained optimization problems that is tolerant of inaccuracies in the data. The algorithm solves a semi-smooth set of nonlinear equations that are equivalent to the first-order optimality conditions, and it is matrix-free in the sense that it does not require the explicit Lagrangian Hessian or Jacobian of the constraints. The solution method is quasi-Newton, but rather than approximating only the Hessian or constraint Jacobian, the Jacobian of the entire nonlinear set of equations is approximated using a multisecant method. We show how preconditioning can be incorporated into the multisecant update in order to improve the performance of the method. For nonconvex problems, we propose a simple modification of the secant conditions to regularize the Hessian. Numerical experiments suggest that the algorithm is a promising alternative to conventional gradient-based algorithms, particularly when errors are present in the data.