58 research outputs found
Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces
No full textAbstractIn this paper, we introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. Then, we prove a strong convergence theorem which is connected with Combettes and Hirstoaga's result [P.L. Combettes, S.A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005) 117–136] and Wittmann's result [R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. 58 (1992) 486–491]. Using this result, we obtain two corollaries which improve and extend their results
Attouch–Théra duality revisited: Paramonotonicity and operator splitting
No full textAbstractThe problem of finding the zeros of the sum of two maximally monotone operators is of fundamental importance in optimization and variational analysis. In this paper, we systematically study Attouch–Théra duality for this problem. We provide new results related to Passty’s parallel sum, to Eckstein and Svaiter’s extended solution set, and to Combettes’ fixed point description of the set of primal solutions. Furthermore, paramonotonicity is revealed to be a key property because it allows for the recovery of all primal solutions given just one arbitrary dual solution. As an application, we generalize the best approximation results by Bauschke, Combettes and Luke [H.H. Bauschke, P.L. Combettes, D.R. Luke, A strongly convergent reflection method for finding the projection onto the intersection of two closed convex sets in a Hilbert space, Journal of Approximation Theory 141 (2006) 63–69] from normal cone operators to paramonotone operators. Our results are illustrated through numerous examples
A parallel constraint disintegration and approximation scheme for quadratic signal recovery
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