On Maingé’s Approach for Hierarchical Optimization Problems

Abstract

This paper aims at investigating an iterative method for solving a system of variational inequalities with fixed-point set constraints. Our scheme can be regarded as a more general variant of the algorithm proposed by Maingé. Strong convergence results are established in the setting of Hilbert spaces. We propose an alternative analysis that allows us to relax some assumption imposed in his paper for convergence of the considered method. As a complementary result, we show how to adapt these processes to the case when the constraints involve operators belonging to the class of hemi-contractive mappings; this goes beyond the scope of Maingé's result.

Some Existence and Convergence Theorems for Solving a System of Hierarchical Optimization Problems

“…Our result and method allow us to prove the existence and approximation of solutions to the system of hierarchical variational inequality problems (1.4). The results presented in this article extend and improve the results of Li [25], Maingé [26], Kraikaew and Saejung [27], and many authors.…”

“…In 2011, Kraikaew and Saejung [27] introduced bi-level hierarchical optimization problems [24], i.e, find (x ⇤ , y ⇤ ) 2 F (T 1 ) ⇥ F (T 2 ) such that for given positive real numbers ⇢ and ⌘, the following inequalities hold:…”

Existence and Algorithm for the Systems of Hierarchical Variational Inclusion Problems

“…Our analysis and method allow us to prove the existence and approximation of solutions to the system of a hierarchical variational inclusion problem (5). The results presented in this paper extend and improve the results of Chang et al [15], Maingé [17], Kraikaew and Saejung [18], and some authors.…”

Strong Convergence Results for Hierarchical Circularly Iterative Method about Hierarchical Circularly Optimization

“…For prove our result, we give the following lemma. In [11], the existence and uniqueness of solutions of some related hierarchical optimization problems had been discussed.…”

“…The concept of variational inequalities plays an important role in structural analysis, mechanics and economics. Recently, the hierarchical variational inequalities and hierarchical iterative sequence problems have attached many authors' attention (see [2][3][4][5][6][7][8][9][10][11]).…”