Abstract
We propose a projection-type method for multivalued variational inequality. The iteration sequence generated by the algorithm is proven to be globally convergent to a solution, provided that the multivalued mapping is continuous with nonempty compact convex values. Moreover, we present a necessary and sufficient condition on the nonemptiness of the solution set. Preliminary computational experience is also reported.
Highlights
We consider the following multivalued variational inequality, denoted by GVI(T, K) to find x∗ ∈ K and w∗ ∈ T(x∗) such that⟨w∗, y − x∗⟩ ≥ 0, ∀y ∈ K, (1)where K is a nonempty closed convex set in Rn, T is a multivalued mapping from K into Rn with nonempty values, and ⟨⋅, ⋅⟩ and ‖ ⋅ ‖ denote the inner product and norm in Rn, respectively.Projection-type algorithms have been extensively studied in the literature; see [1–8] and the references therein
Throughout this paper, we assume that the solution set S of the problem (1) is nonempty and T is continuous on K with nonempty compact convex values satisfying the following property:
We show that Algorithm 2 is well defined and implementable
Summary
We propose a projection-type method for multivalued variational inequality. The iteration sequence generated by the algorithm is proven to be globally convergent to a solution, provided that the multivalued mapping is continuous with nonempty compact convex values. We present a necessary and sufficient condition on the nonemptiness of the solution set.
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