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Relaxed alternating CQ algorithms for the split equality problem in Hilbert spaces.

Hai Yu1, Fenghui Wang1

  • 1Department of Mathematics, Luoyang Normal University, Luoyang, P.R. China.

Journal of Inequalities and Applications
|March 7, 2019
PubMed
Summary
This summary is machine-generated.

Researchers developed new algorithms to solve the split equality problem (SEP) in Hilbert spaces. These methods, including relaxed alternating algorithms, demonstrate efficiency and convergence, answering a key research question.

Keywords:
Relaxed alternating methodSplit equality problemSplit feasibility problem

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Area of Science:

  • Optimization
  • Functional Analysis
  • Numerical Analysis

Background:

  • The split equality problem (SEP) is a significant problem in mathematical analysis.
  • Solving SEP is crucial for various applications in applied mathematics and engineering.
  • Existing methods for SEP have limitations, motivating the search for new approaches.

Purpose of the Study:

  • To propose novel algorithms for solving the split equality problem (SEP) in Hilbert spaces.
  • To investigate the convergence properties of these new algorithms.
  • To provide an affirmative answer to a research question posed by Moudafi.

Main Methods:

  • Converting the split equality problem into a coupled fixed-point equation.
  • Developing a new iterative algorithm for the general SEP.
  • Proposing two relaxed alternating algorithms for SEP when convex sets are level sets of convex functionals.
  • Utilizing a novel technique to prove strong convergence.

Main Results:

  • The proposed new algorithm effectively solves the SEP.
  • The first relaxed alternating algorithm is proven to be weakly convergent.
  • The second relaxed alternating algorithm is proven to be strongly convergent.
  • Numerical results indicate the efficiency of the developed algorithms.

Conclusions:

  • The study successfully introduced efficient algorithms for the split equality problem.
  • The strong convergence of the second relaxed algorithm provides a significant advancement.
  • The findings offer a new perspective and tools for solving SEP in Hilbert spaces.