On regularity conditions for complementarity problems

In the context of complementarity problems, various concepts of solution regularity are known, each of them playing a certain role in the related theoretical and algorithmic developments. Despite the existence of rich literature on this subject, it appears that the exact relations between some of these regularity concepts remained unknown. In this note, we not only summarize the existing results on the subject but also establish the missing relations filling all the gaps in the current understanding of how different regularity concepts relate to each other. In particular, we demonstrate that strong regularity is in fact equivalent to nonsingularity of all matrices in the natural outer estimates of the generalized Jacobians of the most widely used residual mappings for complementarity problems. On the other hand, we show that CD-regularity of the natural residual mapping does not imply even BD-regularity of the Fischer–Burmeister residual mapping. As a result, we provide the complete picture of relations between the most important regularity conditions for mixed complementarity problems, with a special emphasis on those conditions used to justify the related numerical methods. A special attention is paid to the particular cases of a nonlinear complementarity problem and of a Karush–Kuhn–Tucker system.