A class of gap functions for variational inequalities
Let \(K\) be a closed convex set in \(R^ n\) and \(T : K \to R^ n\) be a continuous operator. The problem of finding \(u \in K\) such that \(\langle Tu,v - u \rangle \geq 0\) for all \(v \in K\), is known as the variational inequality problem (VIP). A function \(g : K \to R \cup \{-\infty, \infty\}\) is a gap (merit) function for (VIP) if (i) \(g\) is restricted in sign on \(K\) and (ii) \(g(u) = 0\) if and only if \(u \in \Omega\), where \(\Omega\) is the solution set of (VIP). In this interesting paper, the authors consider and study a new class of gap functions. They also discuss some algorithmic equivalence results relating their work to that of \textit{S. Dafermos} [Math. Program. 26, 40- 47 (1983; Zbl 0506.65026)], \textit{G. Cohen} [J. Optimization Theory Appl. 59, No. 2, 325-334 (1988; Zbl 0653.90062)] and \textit{M. A. Noor} [ibid. 73, No. 2, 409-413 (1992; Zbl 0794.49009)]. Using the gap functions, a general descent framework has been developed for finding the approximate solution of variational inequalities. The results presented in this paper represent an improvement of the previously known results in this area. Remark: In a recent paper, the reviewer [Some nonlinear variational inequalities, Tamkang J. Math. Vol. 26, No. 2 (1995)] has proved that the gap functions discussed in this paper can be derived by using the auxiliary principle technique.
scientific article; zbMATH DE number 617331
scientific article