Substructure exploitation of a nonsmooth Newton method for large-scale optimal control problems with full discretization

Abstract

We investigate the application of full discretization and a nonsmooth Newton method to large-scale optimal control problems. Based on a first discretize, then optimize approach, we discretize the state and control variables in time following a collocation method. Then, a nonsmooth Newton method combined with a line search globalization strategy is used to find a solution to the resulting finite-dimensional nonlinear optimization problem. In order to reduce the computational effort of solving the linear systems that arise from the application of the nonsmooth Newton method, we propose a structure exploitation strategy that results in a sparse banded matrix. We propose as well a substructure exploitation strategy based on a block LU decomposition. The different exploitation strategies combined with the use of appropriate linear solvers are demonstrated and compared for a quadratic 2D heat equation control problem discretized with the method of lines, and the approach that proved to be the most efficient is applied to a nonlinear version of the problem.