Forward-reflected-backward method with variance reduction

We propose a variance reduced algorithm for solving monotone variational inequalities. Without assuming strong monotonicity, cocoercivity, or boundedness of the domain, we prove almost sure convergence of the iterates generated by the algorithm to a solution. In the monotone case, the ergodic average converges with the optimal O(1/k) rate of convergence. When strong monotonicity is assumed, the algorithm converges linearly, without requiring the knowledge of strong monotonicity constant. We finalize with extensions and applications of our results to monotone inclusions, a class of non-monotone variational inequalities and Bregman projections.

Iterative Design for the Common Solution of Monotone Inclusions and Variational Inequalities

Some new forward–backward multi-choice iterative algorithms with superposition perturbations are presented in a real Hilbert space for approximating common solution of monotone inclusions and variational inequalities. Some new ideas of constructing iterative elements can be found and strong convergence theorems are proved under mild restrictions, which extend and complement some already existing work.

Solving Composite Fixed Point Problems with Block Updates

Various strategies are available to construct iteratively a common fixed point of nonexpansive operators by activating only a block of operators at each iteration. In the more challenging class of composite fixed point problems involving operators that do not share common fixed points, current methods require the activation of all the operators at each iteration, and the question of maintaining convergence while updating only blocks of operators is open. We propose a method that achieves this goal and analyze its asymptotic behavior. Weak, strong, and linear convergence results are established by exploiting a connection with the theory of concentrating arrays. Applications to several nonlinear and nonsmooth analysis problems are presented, ranging from monotone inclusions and inconsistent feasibility problems, to variational inequalities and minimization problems arising in data science.