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Regularized gradient-projection methods for finding the minimum-norm solution of the constrained convex minimization

Ming Tian1, Hui-Fang Zhang2

  • 1College of Science, Civil Aviation University of China, Tianjin, 300300 China ; Tianjin Key Laboratory for Advanced Signal Processing, Civil Aviation University of China, Tianjin, 300300 China.

Journal of Inequalities and Applications
|January 24, 2017
PubMed
Summary
This summary is machine-generated.

This study introduces an iterative algorithm for finding the minimum-norm solution in convex minimization problems within Hilbert spaces. The algorithm is proven to converge strongly, with applications to split feasibility problems.

Keywords:
minimum-normregularized gradient-projection methodthe constrained convex minimization problemvariational inequality

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Area of Science:

  • Optimization Theory
  • Functional Analysis
  • Numerical Analysis

Background:

  • Convex minimization problems are fundamental in applied mathematics and optimization.
  • Finding minimum-norm solutions is crucial for ill-posed problems and regularization.
  • Variational inequalities and split feasibility problems have wide applications.

Purpose of the Study:

  • To develop and analyze a novel iterative algorithm for solving constrained convex minimization problems.
  • To establish strong convergence theorems for the generated sequence.
  • To demonstrate the algorithm's applicability to the split feasibility problem.

Main Methods:

  • An iterative algorithm is proposed, utilizing gradient information of a convex function.
  • The convergence analysis relies on properties of Hilbert spaces and convex analysis.
  • The algorithm is designed to find solutions to variational inequalities.

Main Results:

  • The sequence generated by the algorithm converges strongly to the minimum-norm solution.
  • Conditions for strong convergence are established under specific assumptions on the function and space.
  • The algorithm is successfully applied to solve the split feasibility problem.

Conclusions:

  • The proposed iterative algorithm is effective for finding minimum-norm solutions to constrained convex minimization problems.
  • The strong convergence results provide theoretical guarantees for the algorithm's performance.
  • The application to split feasibility problems highlights the algorithm's practical utility.