Strong convergence of relaxed hybrid steepest-descent methods for triple hierarchical constrained optimization

Abstract

Up to now, a large number of practical problems such as signal processing and network resource allocation have been formulated as the monotone variational inequality over the fixed point set of a nonexpansive mapping, and iterative algorithms for solving these problems have been proposed. The purpose of this article is to investigate a monotone variational inequality with variational inequality constraint over the fixed point set of one or finitely many nonexpansive mappings, which is called the triple-hierarchical constrained optimization. Two relaxed hybrid steepest-descent algorithms for solving the triple-hierarchical constrained optimization are proposed. Strong convergence for them is proven. Applications of these results to constrained generalized pseudoinverse are included.

Composite relaxed extragradient method for triple hierarchical variational inequalities with constraints of systems of variational inequalities

“…Under some suitable conditions, the strong convergence of the iteration sequences generated by the algorithm is established. Our results improve and extend the corresponding results announced by some others, e.g., Iiduka [14], Zeng et al [33], Anh et al [1], and Yao et al [27].…”

“…Very recently, some authors continued the study of Iiduka's THVIP (i.e., Problem 1.3 and its variant and extension; see e.g., [6,33]). …”

Hybrid steepest-descent viscosity methods for triple hierarchical variational inequalities with constraints of mixed equilibria and bilevel variational inequalities

“…More precisely, it is referred as a triple hierarchical variational inequality problem (THVIP); see Ceng et al [6]. Very recently, some authors continued the study of Iiduka's THVIP (i.e., Problem 1.4) and its variant and extension; see e.g., [3,6,29]. Step 0.…”

On the triple hierarchical variational inequalities with constraints of mixed equilibria, variational inclusions and systems of generalized equilibria

“…In addition, the authors also gave the geometric convergence rate estimate for approximate solutions. Also, various types of iterative algorithms for solving variational inclusions have been further studied and developed; for more details, refer to [14,22,27,29,32] and the references therein.…”

“…Since the original problem is a variational inequality, in this paper, we call it a triple hierarchical variational inequality (THVI). Subsequently, Zeng, Wong and Yao [22] introduced and considered the following triple hierarchical variational inequality (THVI): …”