Complementarity problems in linear complementarity systems

Abstract

Complementarity systems are described by differential and algebraic equations and inequalities similar to those appearing in the linear complementarity problem (LCP) of mathematical programming. Typical examples of such systems include mechanical systems subject to unilateral constraints, electrical networks with diodes, processes subject to relays and/or Coulomb friction and many more. For linear complementarity systems the rational complementarity problem (RCP) turns out to be crucial to solve well-posedness issues as well as to simulate these systems. In this paper, the main results can be split into two parts. In the first part it is proven that the existence and uniqueness of initial solutions to linear complementarity systems is equivalent to existence and uniqueness of solutions to the RCP, The second part is concerned with the relation between solvability of RCP and the solvability of a family of LCPs. By using the available literature on solvability of LCPs, we can establish solvability of an RCP, and, as a consequence, of linear complementarity systems. The strength of the results is demonstrated by presenting sufficient conditions for uniqueness of solutions to relay systems.