Numerical results for a globalized active-set Newton method for mixed complementarity problems

Abstract

Abstract. We discuss a globalization scheme for a class of active-set Newton methods for solving the mixed complementarity problem (MCP), which was proposed by the authors in [3]. Mathematical subject classification: 90C30, 65K05.

A Class of Active-Set Newton Methods for Mixed ComplementarityProblems

“…For example, this behavior is observed also for the problem badfree from the MCPLIB collection: once x k is close to the solutionx = (0, 0, 0.5, 0.5, 1), GNM/AS produces x k+1 =x. At the same time, for x k close tox, a degenerate Λ k is computed, and SNM/FB fails; see [5].…”

“…We emphasize that the overall goal of the presented approach is not to improve the SNM (or any other algorithm) when it works efficiently but rather to safeguard fast local convergence in irregular cases for which other methods do not work well. The results of [5] indicate that this can be achieved without paying a high price over iterations of the globally convergent hybrid SNM/FB method. We now describe some possible scenarios of the local behavior of SNM/FB and GNM/AS by applying them to some small test problems with various combinations of satisfied and violated regularity properties of the solution that have been discussed above.…”

“…We have also implemented a globalization of our local algorithm, in the spirit of hybrid versions of SNM/FB methods for complementarity problems based on the Ψ F B reformulation (e.g., the "General Line Search Algorithm" of [22]). Details of this implementation and numerical results on the MCPLIB test problems collection (the newer version of [6]) can be found in the technical report [5]. Before describing our local experiments, we give some general conclusions for the globalized version of [5].…”

“…Details of this implementation and numerical results on the MCPLIB test problems collection (the newer version of [6]) can be found in the technical report [5]. Before describing our local experiments, we give some general conclusions for the globalized version of [5]. The option of switching to our active-set step never harms too much the global algorithm, though we certainly have to pay some extra price for computing this step at those iterations at which it is eventually rejected (there are safeguards and heuristics that can be used to avoid computing the active-set step too early; see [5]).…”

“…Before describing our local experiments, we give some general conclusions for the globalized version of [5]. The option of switching to our active-set step never harms too much the global algorithm, though we certainly have to pay some extra price for computing this step at those iterations at which it is eventually rejected (there are safeguards and heuristics that can be used to avoid computing the active-set step too early; see [5]). We emphasize that the overall goal of the presented approach is not to improve the SNM (or any other algorithm) when it works efficiently but rather to safeguard fast local convergence in irregular cases for which other methods do not work well.…”