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We study perturbations of a hybrid steepest descent method for locating common fixed points of an arbitrary pool { T λ } of nonexpansive mappings. The difficulty of handling a possibly uncountable family is settled down to dealing with a countable family of auxiliary mappings { S n } associated to { T λ } , in the sense that the approximate fixed points of { S n } provide the common fixed points of { T λ }. Algorithms with strong convergence for solving the associated variational inequality problems are presented. Applications to convex minimization problems and convex feasibility problems are provided, together with numerical examples for comparisons of our algorithms with the existing ones. [ABSTRACT FROM AUTHOR]

First-order methods such as proximal gradient, which use Forward–Backward Splitting techniques have proved to be very effective in solving nonsmooth convex minimization problem, which is useful in solving various practical problems in different fields such as machine learning and image processing. In this paper, we propose few new forward–backward splitting algorithms, which consume less number of iterations to converge to an optimum. In addition, we derive convergence rates for the proposed formulations and show that the speed of convergence of these algorithms is significantly better than the traditional forward–backward algorithm. To demonstrate the practical applicability, we apply them to two real-world problems of machine learning and image processing. The first issue deals with the regression on high-dimensional datasets, whereas the second one is the image deblurring problem. Numerical experiments have been conducted on several publicly available real datasets to verify the obtained theoretical results. Results demonstrate the superiority of our algorithms in terms of accuracy, the number of iterations required to converge and the rate of convergence against the classical first-order methods. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study the generalized-D-gap function for an equilibrium problem. We give sufficient conditions under which a critical point of the generalized-D-gap function is a solution of the equilibrium problem, and develop a descent scheme for locating these critical points. Using the generalized-D-gap function, we construct an error bound for the equilibrium problem. We support our approach with concrete examples. [ABSTRACT FROM AUTHOR]