Periodically preconditioned conjugate gradient-restoration algorithm for optimal control - The direct approach

Abstract

In this paper, a periodically preconditioned conjugate gradient-restoration algorithm is developed for a broad class of optimal control problems. The particular class of problems addressed not only has differential constraints and terminal constraints, but also nondifferential constraints which must be satisfied over the entire interval of integration. The development of an effective and efficient conjugate gradient-restoration algorithm for this class of problems is important since numerous control problems arise in this form or can be brought to this form through the use of various transformations and/or the use of slack variables. The direct approach is used to convert the infinite dimensional control problem to a finitedimensional problem and the preconditioning theory used, and the periodically preconditioned algorithm developed, are based on the resulting mathematical programming problem. Numerical examples are provided which highlight the convergence rate enhancements that periodic preconditioning can provide for the conjugate gradient-restoration algorithm.