Projection Algorithms for Variational Inclusions

We present a projection algorithm for finding a solution of a variational inclusion problem in a real Hilbert space. Furthermore, we prove that the proposed iterative algorithm converges strongly to a solution of the variational inclusion problem which also solves some variational inequality

New Perturbed Proximal Point Algorithms for Set-valued Quasi Variational Inclusions

In this paper, by using some new and innovative techniques, some perturbed iterative algorithms for solving generalized set-valued variational inclusions are suggested and analyzed. Since the generalized set-valued variational inclusions include many variational inclusions , variational inequalities and set-valued operator equation studied by others in recent years, the results obtained in this paper continue to hold for them and represent a significant refinement and improvement of the previously known results in this area

Dynamical systems and forward-backward algorithms associated with the sum of a convex subdifferential and a monotone cocoercive operator

In a Hilbert framework, we introduce continuous and discrete dynamical systems which aim at solving inclusions governed by structured monotone operators $A=\partial\Phi+B$, where $\partial\Phi$ is the subdifferential of a convex lower semicontinuous function $\Phi$, and $B$ is a monotone cocoercive operator. We first consider the extension to this setting of the regularized Newton dynamic with two potentials. Then, we revisit some related dynamical systems, namely the semigroup of contractions generated by $A$, and the continuous gradient projection dynamic. By a Lyapunov analysis, we show the convergence properties of the orbits of these systems. The time discretization of these dynamics gives various forward-backward splitting methods (some new) for solving structured monotone inclusions involving non-potential terms. The convergence of these algorithms is obtained under classical step size limitation. Perspectives are given in the field of numerical splitting methods for optimization, and multi-criteria decision processes.Comment: 25 page