Abstract
We propose a new projection algorithm for generalized variational inequality with multivalued mapping. Our method is proven to be globally convergent to a solution of the variational inequality problem, provided that the multivalued mapping is continuous and pseudomonotone with nonempty compact convex values. Preliminary computational experience is also reported.
Highlights
We consider the following generalized variational inequality
To find x∗ ∈ C and ξ ∈ F x∗ such that ξ, y − x∗ ≥ 0, ∀ y ∈ C, 1.1 where C is a nonempty closed convex set in Rn, F is a multivalued mapping from C into Rn with nonempty values, and ·, · and · denote the inner product and norm in Rn, respectively
1 proved if the set C is a box and F is order monotone, the proximal point algorithm still applies for problem 1.1
Summary
We consider the following generalized variational inequality. Theory and algorithm of generalized variational inequality have been much studied in the literature 1–9. The well-known proximal point algorithm 10 requires the multivalued mapping F to be monotone. 1 proved if the set C is a box and F is order monotone, the proximal point algorithm still applies for problem 1.1. 15 proposes a projection algorithm for generalized variational inequality with pseudomonotone mapping. We introduce a different projection algorithm for generalized variational inequality. Throughout this paper, we assume that the solution set S of problem 1.1 is nonempty and F is continuous on C with nonempty compact convex values satisfying the following property: ζ, y − x ≥ 0, ∀ y ∈ C, ζ ∈ F y , ∀ x ∈ S.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
