GAMS/PATH User Guide Version 4.3

Chapter 1
Complementarity
A fundamental problem of mathematics is to find a solution to a square system
of nonlinear equations. Two generalizations of nonlinear equations have
been developed, a constrained nonlinear system which incorporates bounds
on the variables, and the complementarity problem. This document is primarily
concerned with the complementarity problem.
The complementarity problem adds a combinatorial twist to the classic
square system of nonlinear equations, thus enabling a broader range of situations
to be modeled. In its simplest form, the combinatorial problem is to
choose from 2n inequalities a subset of n that will be satisfied as equations.
These problems arise in a variety of disciplines including engineering and
economics [20] where we might want to compute Wardropian and Walrasian
equilibria, and optimization where we can model the first order optimality
conditions for nonlinear programs [29, 30]. Other examples, such as bimatrix
games [31] and options pricing [27], abound.
Our development of complementarity is done by example. We begin by
looking at the optimality conditions for a transportation problem and some
extensions leading to the nonlinear complementarity problem. We then discuss
a Walrasian equilibrium model and use it to motivate the more general
mixed complementarity problem. We conclude this chapter with information
on solving the models using the PATH solver and interpreting the results.
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