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A double projection algorithm for multi-valued variational inequalities and a unified framework of the method. (English) Zbl 1223.65047

The goal of this paper is to find the solution ${x}^{*}\in C$ and $\xi \in F\left({x}^{*}\right)$ of the generalized variational inequality such that

$〈\xi ,y-{x}^{*}〉\ge 0,\forall y\in C\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $C$ is a nonempty closed convex set in ${ℝ}^{n}$, $F$ is a multi-valued mapping from $C$ into ${ℝ}^{n}$ with nonempty values, and $〈·,·〉$ and $\parallel ·\parallel$ denote the inner product and norm in ${ℝ}^{n}$, respectively.

The authors assume that the solution set $S$ of the problem (1) is nonempty and $F$ is continuous on $C$ with nonempty compact convex values satisfies the inequality

$〈\zeta ,y-x〉\ge 0,\forall y\in C,\zeta \in F\left(y\right),\forall x\in S·\phantom{\rule{2.em}{0ex}}\left(2\right)$

(Here property (2) holds if $F$ is pseudomonotone on C in the sense of Karamardian. In particular, if is monotone, then (2) holds.) Main result:

If $F:C\to {2}^{{R}_{n}}$ is continuous with nonempty compact convex values on $C$ and the condition (2) holds, then either the proposed algorithm terminates in a finite number of iterations or generates an infinite sequence $\left\{{x}_{i}\right\}$ converging to a solution of (1).

If $F$ is also Lipschitz continuous with modulus $L>0$ and if there exist positive constants $〈c,\delta 〉$ such that $dist\left(𝐱,S\right)\le c\parallel {r}_{\mu }\left(𝐱,\xi \right)\parallel$, $\forall \left(𝐱,\xi \right)\in P\left(\delta \right)$, then there is a constant $\alpha >0$ such that for sufficiently large $i$ one has: $dist\left({x}_{i},s\right)\le {\left(1/\left(\alpha i+dis{t}^{-2}\left({x}_{0},s\right)\right)\right)}^{1/2}$.

The details of the algorithm are presented and several preliminary results for convergence analysis are proved. The authors give a unified framework of a projection-type algorithm for multi-valued variational inequalities. At the end the results of some numerical experiments for the proposed algorithm are presented.

##### MSC:
 65K15 Numerical methods for variational inequalities and related problems 49J40 Variational methods including variational inequalities 49M25 Discrete approximations in calculus of variations