Nonsmooth Newton’s Method: Some Structure Exploitation

Abstract

We investigate real asymmetric linear systems arising in the search direction generation in a nonsmooth Newton’s method. This applies to constrained optimisation problems via reformulation of the necessary conditions into an equivalent nonlinear and nonsmooth system of equations. We propose a strategy to exploit the problem structure. First, based on the sub-blocks of the original matrix, some variables are selected and ruled out for a posteriori recovering; then, a smaller and symmetric linear system is generated; eventually, from the solution of the latter, the remaining variables are obtained. We prove the method is applicable if the original linear system is well-posed. We propose and discuss different selection strategies. Finally, numerical examples are presented to compare this method with the direct approach without exploitation, for full and sparse matrices, in a wide range of problem size.