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We introduce and analyze a hybrid steepest-descent algorithm by combining Korpelevich’s extragradient method, the steepest-descent method, and the averaged mapping approach to the gradient-projection algorithm. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to the unique solution of a triple hierarchical constrained optimization problem (THCOP) over the common fixed point set of finitely many nonexpansive mappings, with constraints of finitely many generalized mixed equilibrium problems (GMEPs), finitely many variational inclusions, and a convex minimization problem (CMP) in a real Hilbert space.

Let

Let

In 1976, Korpelevich [

Let

In [

for each

Given a positive number

Let

On the other hand, let

Let

In this paper, we introduce and study the following triple hierarchical constrained optimization problem (THCOP) with constraints of the CMP (

Motivated and inspired by the above facts, we introduce and analyze a hybrid iterative algorithm via Korpelevich’s extragradient method, the steepest-descent method, and the gradient-projection algorithm obtained by the averaged mapping approach. It is proven that under mild conditions, the proposed algorithm converges strongly to a unique element of

Throughout this paper, we assume that

A mapping

monotone if

It is obvious that if

The metric projection from

Some important properties of projections are gathered in the following proposition.

For given

Next we list some elementary conclusions for the mixed equilibrium problem where

Assume that

for each

In the following, we recall some facts and tools in a real Hilbert space

Let

Let

if

A mapping

Let

If

Let

If

If

The composite of finitely many averaged mappings is averaged. That is, if each of the mappings

If the mappings

Let

Let

Let

Let

Let

Recall that a Banach space

Finally, recall that a set-valued mapping

Let

There holds the resolvent identity

For

Based on Huang [

Consequently,

Let

Let

Let

In this section, we will introduce and analyze a hybrid steepest-descent algorithm for finding a solution of the THCOP (

The following is to state and prove the main result in this paper.

Let

Let

We divide the rest of the proof into several steps.

Indeed, utilizing (

Indeed, utilizing (

Furthermore, since

Indeed, from

Furthermore, by Proposition