Steepest Descent Preconditioning for Nonlinear GMRES Optimization

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.