Abstract
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. AMS Subject Classifications: 49J40; 65K05; 47H09.
Highlights
Let H be a real Hilbert space with inner product 〈·, ·〉 and norm ∥ · ∥, let C be a nonempty closed convex subset of H and let R be the set of all real numbers
For a given nonlinear operator A : H ® H, the following classical variational inequality problem is formulated as finding a point x* Î C such that
The constraint set has been defined in [3,5] as the intersection of finite, closed, and convex subsets, C0 and Ci (i = 1,2,...,m), of a real Hilbert space, and is represented as the fixed point set of the direct product mapping composed of the metric projections onto the Cis
Summary
Let H be a real Hilbert space with inner product 〈·, ·〉 and norm ∥ · ∥, let C be a nonempty closed convex subset of H and let R be the set of all real numbers. ∞; It is proven that under Conditions (B1)-(B5), the sequence {xn}∞ n=0 generated by Algorithm II converges strongly to the unique solution of Problem II. The metric projection PC onto a given nonempty, closed, and convex set C (⊂ H), satisfies the nonexpansivity with Fix(PC) = C [[22], Theorem 3.1.4(i)], [[29], p.
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