The parallel sum of nonlinear monotone operators
Non-Linear Analysis
Mathematical Programming: Series A and B
Mathematical Programming: Series A and B
A Weak-to-Strong Convergence Principle for Fejé-Monotone Methods in Hilbert Spaces
Mathematics of Operations Research
A Generalized Proximal Point Algorithm for the Variational Inequality Problem in a Hilbert Space
SIAM Journal on Optimization
Dualization of Generalized Equations of Maximal Monotone Type
SIAM Journal on Optimization
Finding best approximation pairs relative to two closed convex sets in Hilbert spaces
Journal of Approximation Theory
A family of projective splitting methods for the sum of two maximal monotone operators
Mathematical Programming: Series A and B
Fitzpatrick Functions and Continuous Linear Monotone Operators
SIAM Journal on Optimization
General Projective Splitting Methods for Sums of Maximal Monotone Operators
SIAM Journal on Control and Optimization
Convex Analysis and Monotone Operator Theory in Hilbert Spaces
Convex Analysis and Monotone Operator Theory in Hilbert Spaces
On Weak Convergence of the Douglas-Rachford Method
SIAM Journal on Control and Optimization
A Monotone+Skew Splitting Model for Composite Monotone Inclusions in Duality
SIAM Journal on Optimization
Hi-index | 0.00 |
The 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-Thera 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.