Inexact restoration and adaptive mesh refinement for optimal control

Abstract

A new adaptive mesh refinement algorithm is proposed for solving Eulerdiscretization of state- and control-constrained optimal controlproblems. Our approach is designed to reduce the computational effortby applying the inexact restoration (IR) method, a numerical methodfor nonlinear programming problems, in an innovative way. The initialiterations of our algorithm start with a coarse mesh, which typicallyinvolves far fewer discretization points than the fine mesh over whichwe aim to obtain a solution. The coarse mesh is then refinedadaptively, by using the sufficient conditions of convergence of theIR method. The resulting adaptive mesh refinement algorithm isconvergent to a fine mesh solution, by virtue of convergence of the IRmethod. We illustrate the algorithm on a computationally challengingconstrained optimal control problem involving a container crane.Numerical experiments demonstrate that significant computationalsavings can be achieved by the new adaptive mesh refinement algorithmover the fixed-mesh algorithm. Conceivably owing to the small numberof variables at start, the adaptive mesh refinement algorithm appearsto be more robust as well, i.e., it can find solutions with a muchwider range of initial guesses, compared to the fixed-mesh algorithm.