Linear Monotone Complementarity and Associated Vector Fields

Overview

We develop the theoretical tools necessary for the algorithms to follow. The logarithmic penalty technique allows the introduction of the central path. In the case of linear or quadratic optimization, the optimality system and the central path are suitably cast into the framework of linear monotone complementarity problems.

The analysis of linear monotone complementarity problems starts with some results of global nature, involving the partition of variables, standard and canonical forms. Then comes a discussion on the magnitude of the variables in a neighborhood of the central path. We introduce two families of vector fields associated with the central path: the affine and centralization directions. The magnitude of the components of these fields are analyzed in detail. We also discuss the convergence of the differential system obtained by a convex combination of the affine and centralization directions.

Since the results stated here are motivated by the analysis of algorithms presented afterwards, a quick reading is sufficient for a first step. Besides, a large part of the technical difficulties are due to the modified field theory, useful for problems without strict complementarity. A reader interested mainly by linear optimization (where strict complementarity always holds) can therefore skip this part .