Multistep Hybrid Extragradient Method for Triple Hierarchical Variational Inequalities

We consider a triple hierarchical variational inequality problem (THVIP), that is, a variational inequality problem defined over the set of solutions of another variational inequality problem which is defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Moreover, we propose a multistep hybrid extragradient method to compute the approximate solutions of the THVIP and present the convergence analysis of the sequence generated by the proposed method. We also derive a solution method for solving a system of hierarchical variational inequalities (SHVI), that is, a system of variational inequalities defined over the intersection of the fixed point set of a strict pseudocontractive mapping and the solution set of the classical variational inequality problem. Under very mild conditions, it is proven that the sequence generated by the proposed method converges strongly to a unique solution of the SHVI

Triple Hierarchical Variational Inequalities with Constraints of Mixed Equilibria, Variational Inequalities, Convex Minimization, and Hierarchical Fixed Point Problems

We introduce and analyze a hybrid iterative algorithm by virtue of Korpelevich's extragradient method, viscosity approximation method, hybrid steepest-descent method, and averaged mapping approach to the gradient-projection algorithm. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inequality problems (VIPs), the solution set of general system of variational inequalities (GSVI), and the set of minimizers of convex minimization problem (CMP), which is just a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm to solve a hierarchical fixed point problem with constraints of finitely many GMEPs, finitely many VIPs, GSVI, and CMP. The results obtained in this paper improve and extend the corresponding results announced by many others

On the role of the coefficients in the strong convergence of a general type Mann iterative scheme

Let H be a Hilbert space. Let $(W_{n})_{n\in\mathbb{N}}$ be a suitable family of mappings. Let S be a nonexpansive mapping and D be a strongly monotone operator. We study the convergence of the general scheme $x_{n+1}=W_{n}(\alpha_{n}Sx_{n}+(1-\alpha_{n})(I-\mu_{n}D)x_{n})$ in dependence on the coefficients $(\alpha_{n})_{n\in\mathbb{N}}$ , $(\mu_{n})_{n\in\mathbb{N}}$

A Regularized Gradient Projection Method for the Minimization Problem

We investigate the following regularized gradient projection algorithm xn+1=Pc(I−γn(∇f+αnI))xn, n≥0. Under some different control conditions, we prove that this gradient projection algorithm strongly converges to the minimum norm solution of the minimization problem minx∈Cf(x)

Trilevel and Multilevel Optimization using Monotone Operator Theory

We consider rather a general class of multi-level optimization problems, where a convex objective function is to be minimized subject to constraints of optimality of nested convex optimization problems. As a special case, we consider a trilevel optimization problem, where the objective of the two lower layers consists of a sum of a smooth and a non-smooth term.~Based on fixed-point theory and related arguments, we present a natural first-order algorithm and analyze its convergence and rates of convergence in several regimes of parameters