The variational inequality problem (VIP) is considered here. We present a general algorithmic scheme which employs projections onto hyperplanes that separate balls from the feasible set of the VIP instead of projections onto the feasible set itself. Our algorithmic scheme includes the classical projection method and Fukushima's subgradient projection method as special cases.

In this short survey, I revisit the role of the proximal point method in large scale optimization. I focus on three recent examples: a proximally guided subgradient method for weakly convex stochastic approximation, the prox-linear algorithm for minimizing compositions of convex functions and smooth maps, and Catalyst generic acceleration for regularized Empirical Risk Minimization.

In this paper, we introduce projective inertial parallel subgradient extragradient-line algorithm for solving variational inequalites of L-Lipschitz continuous and monotone mappings which L is unknown. We prove a strong convergence result under some mild conditions in Hilbert space. also present numerical examples Euclidean space R3 compared with Parallel-Viscosity-Type Subgradient Extragradien...

This paper considers stochastic subgradient mirror-descent method for solving constrained convex minimization problems. In particular, a stochastic subgradient mirror-descent method with weighted iterate-averaging is investigated and its per-iterate convergence rate is analyzed. The novel part of the approach is in the choice of weights that are used to construct the averages. Through the use o...

The problem of minimizing the sum of nonsmooth, convex objective functions defined on a real Hilbert space over the intersection of fixed point sets of nonexpansive mappings, onto which the projections cannot be efficiently computed, is considered. The use of proximal point algorithms that use the proximity operators of the objective functions and incremental optimization techniques is proposed...

We study two projection algorithms for solving the Variational Inequality Problem (VIP) in Hilbert space. One algorithm is a modi…ed subgradient extragradient method in which an additional projection onto the intersection of two half-spaces is employed. Another algorithm is based on the shrinking projection method. We establish strong convergence theorems for both algorithms.

A Correction to this paper has been published: https://doi.org/10.1007/s11075-021-01202-w

in this paper, a boussinesq equation is solved by using the adomian's decomposition method, modified adomian's decomposition method and homotopy analysis method. the approximate solution of this equation is calculated in the form of series which its components are computed by applying a recursive relation. the existence and uniqueness of the solution and the convergence of the propose...