Between Mathematical Programming and Systems Theory: Linear Complementarity Systems

Abstract

Complementarity systems arise from the interconnection of an input-output system (of the type well known in mathematical systems theory) with a set of complementarity conditions (of the type well known in mathematical programming IntroductionInequalities have played a rather minor role in the powerful development of systems theory that has taken place since about 1960. In contrast to this, they are central to the field of mathematical programming that has likewise seen major advances in the past decades. Of course, systems theory is concerned with differential equations; mixing these with inequalities means giving up the smoothness properties that form the basis of much of the theory of dynamical systems. Technological innovation, however, pushes toward the consideration of systems of a mixed continuous/discrete nature, which are likely to be described by systems of differential equations as well as algebraic equations and inequalities. In fact there are many situations in which there are good reasons to consider dynamics in conjunction with inequalities; think for instance of saturation effects in control systems, unilateral constraints in robotics, piecewise linear dynamics, and so on.Among the many systems of equations and inequalities that one may imagine, the ones that are in so-called complementarity form enjoy particular attention in mathematical programming. More specifically the linear complementarity problem (LCP) has been the subject of extensive study because of its wide range of applications; see the book by Cottle et al. [5] for a comprehensive treatment. The LCP may be formulated as follows:given a vector q E JRk and a matrix M of size k x k, find vectors y and u