Algorithms for a class of bilevel programs involving pseudomonotone variational inequalities
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ALGORITHMS FOR A CLASS OF BILEVEL PROGRAMS INVOLVING PSEUDOMONOTONE VARIATIONAL INEQUALITIES Bui Van Dinh · Le Dung Muu
Received: 4 May 2012 / Revised: 29 May 2013 / Accepted: 31 May 2013 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2013
Abstract We propose algorithms for finding the projection of a given point onto the solution set of the pseudomonotone variational inequality problem. This problem arises in the Tikhonov regularization method for pseudomonotone variational inequality. Since the solution set of the variational inequality is not given explicitly, the available methods of mathematical programming and variational inequalities cannot be applied directly. Keywords Bilevel variational inequality · Pseudomonotonicity · Projection method · Armijo linesearch · Convergence Mathematics Subject Classification (2010) 49M37 · 90C26 · 65K15 1 Introduction Variational inequality (VI) is a fundamental topic in applied mathematics. VIs are used for formulating and solving various problems arising in mathematical physics, economics, engineering and other fields. Theory, methods and applications of VIs can be found in some comprehensive books and monographs (see e.g. [8, 9, 15, 16]). Mathematical programs with variational inequality constraints can be considered as one of the further development directions of variational inequality [18]. Recently, these problems have received much attention of researchers due to their vast applications. In this paper, we are concerned with a special case of VIs with variational inequality constraints. Namely, we consider the bilevel variational inequality problem (BVI): min{x − x g : x ∈ S}, (1.1) where x g ∈ C and S = {u ∈ C : F (u), y − u ≥ 0 ∀y ∈ C}
B.V. Dinh Le Quy Don Technical University, Hanoi, Vietnam e-mail: [email protected]
B
L.D. Muu ( ) Institute of Mathematics, VAST, Hanoi, Vietnam e-mail: [email protected]
B.V. DINH, L.D. MUU
i.e., S is the solution set of the variational inequality VI(C, F ) defined as Find x ∗ ∈ C such that F x ∗ , y − x ∗ ≥ 0 ∀y ∈ C.
(1.2)
Throughout the paper, we suppose that C is a nonempty closed convex subset in the Euclidean space Rn and F : Rn → Rn . We call problem (1.1) the upper problem and (1.2) the lower one. It should be noticed that the solution set S of the lower problem (1.2) is convex whenever F is pseudomonotone on C. However, the main difficulty is that, even if the constrained set S is convex, it is not given explicitly as in a standard mathematical programming problem, and therefore the available methods of convex optimization and variational inequality cannot be applied directly to problem (1.1). In the literature, there exist several solution methods that can be used to solve bilevel variational inequality problem (1.1) (see e.g. [1, 3, 6, 7, 14, 19, 20, 22, 28, 30] and the references cited therein): penalty function methods, regularization methods and hybrid fixed point-projection methods. Most of these methods can be used only
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