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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 $\xi \in F(x^*)$ of the generalized variational inequality such that $$\langle \xi, y-x^* \rangle \geq 0, \forall y \in C\tag{1}$$ where $C$ is a nonempty closed convex set in $\mathbb{R}^n$, $F$ is a multi-valued mapping from $C$ into $\mathbb{R}^n$ with nonempty values, and $\langle .,.\rangle$ and $\|.\|$ denote the inner product and norm in $\mathbb{R}^n$, 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 $$\langle \zeta, y-x\rangle \geq 0, \forall y \in C, \zeta \in F(y), \forall x \in S.\tag{2}$$ (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: {\parindent6mm \item{1)} If $F:C \rightarrow 2^{R_n}$ 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 $\{x_i\}$ converging to a solution of (1). \item{2)} If $F$ is also Lipschitz continuous with modulus $L > 0$ and if there exist positive constants $ \langle c, \delta \rangle $ such that $dist (\bold{x}, S) \leq c \|r_\mu (\bold{x}, \xi)\|$, $\forall (\bold{x}, \xi) \in P(\delta)$, then there is a constant $\alpha > 0$ such that for sufficiently large $i$ one has: $dist(x_i, s) \leq (1/(\alpha i + dist^{-2}(x_0, s)))^{1/2}$.\par} 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.

65K15Numerical methods for variational inequalities and related problems
49J40Variational methods including variational inequalities
49M25Discrete approximations in calculus of variations
Full Text: DOI
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