Steepest Descent Preconditioning for Nonlinear GMRES Optimization

TL;DR

This paper introduces steepest descent preconditioning variants for the nonlinear GMRES optimization algorithm, providing convergence analysis and demonstrating significant acceleration over standard methods on various problems.

Contribution

The paper proposes two steepest descent preconditioning methods for N-GMRES, offers a convergence proof for the line search variant, and shows their effectiveness through numerical experiments.

Findings

N-GMRES with steepest descent preconditioning accelerates convergence.

Performance is competitive with nonlinear conjugate gradient and BFGS methods.

Preconditioning enhances N-GMRES's potential for nonlinear optimization.

Abstract

Steepest descent preconditioning is considered for the recently proposed nonlinear generalized minimal residual (N-GMRES) optimization algorithm for unconstrained nonlinear optimization. Two steepest descent preconditioning variants are proposed. The first employs a line search, while the second employs a predefined small step. A simple global convergence proof is provided for the N-GMRES optimization algorithm with the first steepest descent preconditioner (with line search), under mild standard conditions on the objective function and the line search processes. Steepest descent preconditioning for N-GMRES optimization is also motivated by relating it to standard non-preconditioned GMRES for linear systems in the case of a quadratic optimization problem with symmetric positive definite operator. Numerical tests on a variety of model problems show that the N-GMRES optimization algorithm is able to very significantly accelerate convergence of stand-alone steepest descent optimization. Moreover, performance of steepest-descent preconditioned N-GMRES is shown to be competitive with standard nonlinear conjugate gradient and limited-memory Broyden-Fletcher-Goldfarb-Shanno methods for the model problems considered. These results serve to theoretically and numerically establish steepest-descent preconditioned N-GMRES as a general optimization method for unconstrained nonlinear optimization, with performance that appears promising compared to established techniques. In addition, it is argued that the real potential of the N-GMRES optimization framework lies in the fact that it can make use of problem-dependent nonlinear preconditioners that are more powerful than steepest descent (or, equivalently, N-GMRES can be used as a simple wrapper around any other iterative optimization process to seek acceleration of that process), and this potential is illustrated with a further application example.