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Modified projection algorithms for solving the split equality problems.

Qiao-Li Dong1, Songnian He1

  • 1College of Science, Civil Aviation University of China, Tianjin 300300, China.

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Summary
This summary is machine-generated.

This study enhances the CQ algorithm for the split equality problem (SEP) with adaptive stepsizes and practical projection methods. The modified algorithm demonstrates improved efficiency and convergence for solving complex mathematical problems.

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Area of Science:

  • Applied Mathematics
  • Optimization Theory
  • Numerical Analysis

Background:

  • The split equality problem (SEP) is a significant problem in applied mathematics with wide-ranging applications.
  • The CQ algorithm, proposed by Byrne and Moudafi (2013), is a known method for solving the SEP.
  • Existing methods may lack efficiency or practicality in certain scenarios.

Purpose of the Study:

  • To propose a modified CQ algorithm with adaptive stepsize computation for solving the split equality problem.
  • To introduce a more practical relaxation scheme using projections onto half-spaces.
  • To analyze the weak convergence properties of the proposed algorithms.

Main Methods:

  • Modification of the CQ algorithm to include adaptive stepsize calculation.
  • Incorporation of an additional projection step onto two half-spaces in each iteration.
  • Development of a relaxation scheme utilizing projections onto half-spaces for enhanced practicality.

Main Results:

  • The proposed modified CQ algorithm computes the stepsize adaptively.
  • The algorithm incorporates an additional projection step, enhancing its performance.
  • A relaxation scheme is introduced, offering a more practical approach to solving the SEP.
  • Weak convergence is established for both the modified CQ algorithm and the relaxation scheme.

Conclusions:

  • The modified CQ algorithm and the proposed relaxation scheme offer effective and practical solutions for the split equality problem.
  • The adaptive stepsize and half-space projection methods contribute to improved algorithmic performance.
  • Further analysis confirms the weak convergence of the presented algorithms, validating their utility in applied mathematics.