Solution Analysis I

Abstract

In this chapter, we begin an in-depth study of the VI/CP. Our first concern is the issue of the existence of a solution to the VI. There are several general approaches to obtain existence results for this problem. One is historical and has its origin in an infinite-dimensional setting. In this first approach, a basic existence theorem pertaining to the VI (K,F) with a compact convex set K and a continuous mapping F is obtained by a fixed-point theorem. From this theorem, extended results are derived by replacing the boundedness of the set K by refined conditions on F. These results can then be applied to various problem classes including the MiCP. The starting point of the alternative approach is to derive via a degree-theoretic argument an existence theorem for the VI (K, F) with a closed convex set K and a continuous map F under a key degree condition on the pair. This condition is then shown to hold for various special cases where the function F satisfies certain properties. Yet a third approach is via the demonstration that an equivalent (constrained or unconstrained) optimization problem of the VI/CP based on a certain merit function has an optimal solution, which can be shown to solve the VI/CP in question under additional conditions. In this and the next chapter, we do not consider the third approach, which is covered indirectly in the two algorithmic Chapters 9 and 10.