This paper introduces the notion of a weighted Complementarity Problem (wCP), which consists in finding a pair of vectors $(x,s)$ belonging to the intersection of a manifold with a cone, such that their product in a certain algebra, $x\circ s$, equals a given weight vector $w$. When $w$ is the zero vector, then wCP reduces to a Complementarity Problem (CP). The motivation for introducing the more general notion of a wCP lies in the fact that several equilibrium problems in economics can be formulated in a natural way as wCP. Moreover, those formulations lend themselves to the development of highly efficient algorithms for solving the corresponding equilibrium problems. For example, Fisher's competitive market equilibrium model can be formulated as a wCP that can be efficiently solved by interior-point methods. Moreover, it is shown that the Quadratic Programming and Weighted Centering problem, which generalizes the notion of a Linear Programming and Weighted Centering problem recently proposed by Anstreicher, can be formulated as a special linear monotone wCP. The main contribution of the paper is to introduce and analyze two interior-point methods for solving general monotone linear wCPs.
Technical Report, Department of Mathematics and Statistics, University of Maryland Baltimore County, 1000 Hilltop Circle, Baltimore, MD 22150, May, 2012