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A large number of nonlinear and optimization problems can be reduced to altering point problems. This paper aims to introduce the parallel normal S-iteration technique and study its convergence rates for solving such problems in infinite-dimensional Hilbert spaces under practical assumptions. We place particular emphasis on the parallel splitting method for the sum of two maximal monotone operators and that can apply for solving a class of convex composite minimization problems. Moreover, we present applications of our iterative methods to some nonlinear problems, such as a system of variational inequalities and a system of inclusion problems. Finally, to demonstrate the applicability of the altering point technique, the performances of our proposed parallel normal S-iteration methods are presented through numerical experiments in signal recovery problems. [ABSTRACT FROM AUTHOR]

Halpern's iteration method, discovered by Halpern in 1967, is an iterative algorithm for finding fixed points of a nonexpansive mapping in Hilbert and Banach spaces. Since many optimization problems can be cast into fixed point problems of nonexpansive mappings, Halpern's method plays an important role in optimization methods. This paper discusses recent advances in convergence and rate of convergence results of Halpern's method, and applications in optimization problems, including variational inequalities, monotone inclusions, Douglas-Rachford splitting method, and minimax problems. [ABSTRACT FROM AUTHOR]

A matrix approach to approximating solutions of variational inequalities in Hilbert spaces is introduced. This approach uses two matrices: one for iteration process and the other for regularization. Ergodicity and convergence (both weak and strong) are studied. Our methods combine new or well-known iterative methods (such as the original Mann’s method) with regularized processes involved regular matrices in the sense of Toeplitz. [ABSTRACT FROM PUBLISHER]

Abstract: We use contractions to regularize a class of monotone variational inequalities, where the monotone operators are complements of nonexpansive mappings and the solutions are sought in the set of fixed points of another nonexpansive mapping. Such variational inequalities include monotone inclusions and convex optimization problems to be solved over the fixed point sets of nonexpansive mappings. Both implicit and explicit schemes are shown to be strongly convergent. An application in hierarchical minimization is included. [Copyright &y& Elsevier]

Abstract: A recent trend in the iterative methods for constructing fixed points of nonlinear mappings is to use the viscosity approximation technique. The advantage of this technique is that one can find a particular solution to the associated problems, and in most cases this particular solution solves some variational inequality. In this paper, we try to extend this technique to find a particular common fixed point of a finite family of asymptotically nonexpansive mappings in a Banach space which is reflexive and has a weakly continuous duality map. Both implicit and explicit viscosity approximation schemes are proposed and their strong convergence to a solution to a variational inequality is proved. [Copyright &y& Elsevier]

A Mann-type hybrid steepest-descent method for solving the variational inequality 〈F(u*), v - u*〉 ≥ 0, v ∈ C is proposed, where F is a Lipschitzian and strong monotone operator in a real Hilbert space H and C is the intersection of the fixed point sets of finitely many non-expansive mappings in H. This method combines the well-known Mann's fixed point method with the hybrid steepest-descent method. Strong convergence theorems for this method are established, which extend and improve certain corresponding results in recent literature, for instance, Yamada (The hybrid steepest-descent method for variational inequality problems over the intersection of the fixed-point sets of nonexpansive mappings, in Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, D. Butnariu, Y. Censor, and S. Reich, eds., North-Holland, Amsterdam, Holland, 2001, pp. 473-504), Xu and Kim (Convergence of hybrid steepest-descent methods for variational inequalities, J. Optim. Theor. Appl. 119 (2003), pp. 185-201), and Zeng, Wong and Yao (Convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities, J. Optim. Theor. Appl. 132 (2007), pp. 51-69). [ABSTRACT FROM AUTHOR]

Abstract: Let H be a real Hilbert space. Consider on H a nonexpansive mapping T with a fixed point, a contraction f with coefficient , and a strongly positive linear bounded operator A with coefficient . Let . It is proved that the sequence generated by the iterative method converges strongly to a fixed point which solves the variational inequality for . [Copyright &y& Elsevier]

Abstract: Consider the variational inequality (VI) of finding a point such that where are nonexpansive self-mappings of a closed convex subset of a Hilbert space, and is the set of fixed points of . Assume that the solution set of this VI is nonempty. This paper introduces two schemes, one implicit and one explicit, that can be used to find the minimum-norm solution of VI (∗); namely, the unique solution to the quadratic minimization problem: . [Copyright &y& Elsevier]

Abstract: We introduce an iterative method for finding a common element of the set of solutions of an equilibrium problem and of the set of fixed points of a finite family of nonexpansive mappings in a Hilbert space. We prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality, which is the optimality condition for a minimization problem. [Copyright &y& Elsevier]