Strongly Convergent Algorithms for Variational Inequality Problem Over the Set of Solutions the Equilibrium Problems
This chapter deals with a variational inequality problem over the set of solutions the equilibrium problem or over the set of solutions the system of equilibrium problems in a real Hilbert space. Several new iterative algorithms are proposed. Strong conve
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Strongly Convergent Algorithms for Variational Inequality Problem Over the Set of Solutions the Equilibrium Problems Vladimir V. Semenov
Abstract This chapter deals with a variational inequality problem over the set of solutions the equilibrium problem or over the set of solutions the system of equilibrium problems in a real Hilbert space. Several new iterative algorithms are proposed. Strong convergence theorems for algorithms are proved. The convergence of iterative algorithms with the presence of computational errors without traditional summability conditions also studied. To this aim, we use new Mainge’s techniques for analysis non–Fejerian iterative processes (Set–Valued Analysis. 16, 899–912, 2008).
10.1 Introduction Throughout, H is a real Hilbert space with inner product (·, ·) and induced norm ·. We denote the strongly convergence and the weak convergence of (xn ) to x ∈ H by xn → x and xn x, respectively. For operator A : H → H, set M ⊆ H, and bifunction F : H × H → R ∪ {+∞} we denote by VI(A, M) and EP(F, M) sets {x ∈ M : (Ax, y − x) ≥ 0 ∀ y ∈ M} and {x ∈ M : F(x, y) ≥ 0 ∀ y ∈ M}, respectively. In this chapter, we are interested in the approximate solvability of problems find x ∈ VI(A, EP(F, C)),
(10.1)
find x ∈ VI(A, EP(F1 , C1 ) ∩ EP(F2 , C2 )).
(10.2)
and,
V. V. Semenov (B) Department of Computational Mathematics, Taras Shevchenko National University of Kyiv, Volodimirska str., 64, Kyiv 03601, Ukraine e-mail: [email protected] M. Z. Zgurovsky and V. A. Sadovnichiy (eds.), Continuous and Distributed Systems, Solid Mechanics and Its Applications 211, DOI: 10.1007/978-3-319-03146-0_10, © Springer International Publishing Switzerland 2014
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V. V. Semenov
Problems of form (10.1) or (10.2) are referred as variational inequality over the set of solutions the equilibrium problem or over the set of solutions the system of equilibrium problems. This problems have found applications in a wide array of disciplines, including mechanics, economics, partial differential equations, information theory, approximation theory, signal and image processing, game theory, optimal transport theory, probability and statistics, and machine learning. About the computational aspects mainly, see [5, 6, 12, 13, 15, 18, 22, 24]. Our main objective is to devise iterative algorithms for solving (10.1) and (10.2) and to analyze their asymptotic behavior. We’ll use Mainge’s techniques for analysis non–Fejerian iterative processes [11]. We are continuing our research published in [1, 7, 10, 12, 16, 17, 20, 22]. For solving the problem (10.1), let us assume that set C ⊆ H, bifunction F : C × C → R, and operator A : H → H all satisfy the following set of standard properties: (A1) (A2) (A3) (A4) (A5) (A6) (A7)
C ⊆ H is a nonempty closed convex set; F(x, x) = 0, for all x ∈ C; F(x, y) + F(y, x) ≤ 0, for all x, y ∈ C (monotonicity); for each x ∈ C, the fuctional F(x, ·) is convex and lower semicontinuous; for each x, y, z ∈ C, lim supt→ + 0 F(x + t(z − x), y) ≤ F(x, y); EP(F, C) = ∅; A : H → H is a l–strongly mo
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