Outer-Inner Approximation Projection Methods for Multivalued Variational Inequalities
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Outer-Inner Approximation Projection Methods for Multivalued Variational Inequalities Pham Ngoc Anh1 · Le Thi Hoai An2
Received: 19 August 2015 / Revised: 25 October 2015 / Accepted: 1 November 2015 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016
Abstract In this paper, we present new projection methods for solving multivalued variational inequalities on a given nonlinear convex feasible domain. The first one is an extension of the extragradient method to multivalued variational inequalities under the asymptotic optimality condition, but it must satisfy certain Lipschitz continuity conditions. To avoid this requirement, we propose linesearch procedures commonly used in variational inequalities to obtain an approximation linesearch method for solving multivalued variational inequalities. Next, basing on a family of nonempty closed convex subsets of Rn and linesearch techniques, we give inner approximation projection algorithms for solving multivalued variational inequalities and the convergence of the algorithms is established under few assumptions. Keywords Multivalued variational inequalities · Upper semicontinuous · Linesearch · Projection method Mathematics Subject Classification (2010) 65K10 · 90C25
1 Introduction Let C be a nonempty closed convex subset of Rn , and let F be a multivalued mapping from n C to 2R . Then the function F is called L-Lipschitz continuous on C, if ρ(F (x), F (y)) ≤ Lx − y
∀x, y ∈ C,
Pham Ngoc Anh
[email protected] 1
Laboratory of Applied Mathematics and Computing, PTIT, Hanoi, Vietnam
2
Laboratory of Theoretical and Applied Computer Science-LITA EA 3097, University of Lorraine, Ile du Saulcy, 57045 Metz, France
P.N. Anh, L.T.H. An
where ρ is the Hausdorff distance defined by ρ(F (x), F (y)) = max{d(F (x), F (y)), d(F (y), F (x))}, d(F (x), F (y)) =
sup
inf
w∈F (x) w ∈F (y)
w − w .
We consider the typical form of multivalued variational inequalities (shortly MV I (F, C)) as follows: Find x ∗ ∈ C and w∗ ∈ F (x ∗ ) such that w∗ , x − x ∗ ≥ 0
∀x ∈ C.
The mapping F is called the cost function. Sol(F, C) denotes the set of solutions of MV I (F, C). In a special case, the cost function F is a single-valued mapping, then Problem MV I (F, C) can be formulated as variational inequalities (shortly V I (F, C)): Find x ∗ ∈ C such that F (x ∗ ), x − x ∗ ≥ 0
∀x ∈ C.
The cost function F is called L-Lipschitz continuous on C, if F (x) − F (y) ≤ Lx − y for all x, y ∈ C. The class of multivalued variational inequalities is very large. In particular, it includes the optimization problems, the variational inequalities, the nonlinear complementarity problems, and the Nash equilibrium model in noncooperative games (see [1, 4, 6, 13, 23]). The multivalued variational inequalities have been considered by many authors and various methods have been developed for solving Problem MV I (F, C) (see [2, 5, 7, 8, 12, 17, 21]) when F is monotone on C, i.e., wx − wy , x − y ≥ 0
∀x, y ∈ C, wx ∈ F (x), w
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