OPTIMIZATION APPROACHES TO VARIATIONAL INEQUALITY PROBLEMS

The variational inequality problem (VIP) is to find a point $x\in S$ such that ($F(x), $ $y-x\rangle\geq 0 $ for all $y\in S $, where $S $ is a nonempty closed convex subset of $\Re^{n}, $ $F $ is a continuous mapping from $\Re^{n} $ into itself, and ( $\cdot, $ $\cdot\rangle $ denotes the inner product in $\Re^{n} $. An important special class of the VIP is the complementarity problem $(\mathrm{C}\mathrm{P}) $ , which is to find a point $x $ such that $F(x)\geq 0, $ $x\geq 0, $ $\langle F(x),x\rangle=0 $. The VIP and the CP have been widely used to formulate various equilibrium problems that arise in engineering, economics and operations research. Recently much effort has been made to reformulate the VIP and the CP as an equivalent optimization problem. Such reformulations turn out to be useful not only in designing a globally convergent algorithm for solving the VIP or the CP but also in analyzing the rate of convergence of an iterative method for solving those problems. This paper surveys recent developments in merit functions used to formulate equivalent optimization problems for the VIP and $\mathrm{C}\mathrm{P} $