Implicit and explicit iterative algorithms for hierarchical variational inequality in uniformly smooth Banach spaces

Area of Science:

• Optimization Theory
• Functional Analysis
• Numerical Analysis

Background:

• Hierarchical variational inequality problems are fundamental in applied mathematics.
• Solving these problems in general Banach spaces presents significant theoretical challenges.
• Existing algorithms often lack guaranteed convergence or applicability to broad classes of problems.

Purpose of the Study:

• To develop and analyze novel iterative algorithms for solving hierarchical variational inequalities.
• To address the challenge of constrained optimization within a system of variational inequalities.
• To establish strong convergence of the proposed algorithms to a unique solution in a specific Banach space setting.

Main Methods:

• Introduction of implicit and explicit iterative schemes.
• Analysis of algorithm convergence in uniformly convex and 2-uniformly smooth Banach spaces.
• Utilizing fixed-point iterative techniques and variational inequality theory.

Main Results:

• Demonstrated strong convergence of the proposed iterative algorithms.
• Established the existence and uniqueness of a solution to the hierarchical variational inequality problem.
• Provided a unified framework applicable to a general system of variational inequalities.

Conclusions:

• The developed algorithms offer an effective approach for solving hierarchical variational inequalities.
• The findings extend and improve upon previous results in the field.
• This work contributes to the advancement of optimization techniques in functional analysis.