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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 * C and ξF(x * ) of the generalized variational inequality such that

ξ,y-x * 0,yC(1)

where C is a nonempty closed convex set in n , F is a multi-valued mapping from C into n with nonempty values, and ·,· and · denote the inner product and norm in 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

ζ,y-x0,yC,ζF(y),xS·(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:

If F:C2 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).

If F is also Lipschitz continuous with modulus L>0 and if there exist positive constants c,δ such that dist(𝐱,S)cr μ (𝐱,ξ), (𝐱,ξ)P(δ), then there is a constant α>0 such that for sufficiently large i one has: dist(x i ,s)(1/(αi+dist -2 (x 0 ,s))) 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:
65K15Numerical methods for variational inequalities and related problems
49J40Variational methods including variational inequalities
49M25Discrete approximations in calculus of variations
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