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Related Experiment Videos

Inertial proximal alternating minimization for nonconvex and nonsmooth problems.

Yaxuan Zhang1, Songnian He1

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

Journal of Inequalities and Applications
|October 14, 2017
PubMed
Summary

This study introduces a new proximal alternating minimization algorithm with inertial effects for solving complex optimization problems involving nonconvex nonsmooth functions. The algorithm ensures convergence to critical points by utilizing a key function that guarantees sufficient decrease properties.

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Area of Science:

  • Optimization Theory
  • Nonconvex Analysis
  • Nonsmooth Optimization

Background:

  • Minimization problems involving nonconvex and nonsmooth functions are prevalent in various scientific and engineering fields.
  • Existing algorithms often struggle with convergence guarantees for such complex objective functions.
  • The choice of a smooth auxiliary function (R) offers flexibility in problem formulation.

Purpose of the Study:

  • To develop and analyze a novel proximal alternating minimization algorithm for a specific class of nonconvex nonsmooth minimization problems.
  • To incorporate an inertial effect into the algorithm to potentially accelerate convergence.
  • To establish strong convergence guarantees for the generated sequences to critical points.

Main Methods:

  • A proximal alternating minimization algorithm with an inertial effect is proposed.
Keywords:
Kurdyka-Lojasiewicz inequalityconvergenceinertialnonconvex nonsmooth optimizationproximal alternating minimization

Related Experiment Videos

  • A key function (H) is constructed to ensure a sufficient decrease property for the iterates.
  • The Kurdyka-Lojasiewicz inequality is employed to prove the convergence of the algorithm.

    • Main Results:

    • The proposed algorithm generates sequences with a sufficient decrease property.
    • Under the Kurdyka-Lojasiewicz condition on the key function H, strong convergence to a critical point is proven.
    • The algorithm effectively handles the minimization of the sum of two nonconvex nonsmooth functions plus a chosen smooth function.

    Conclusions:

    • The developed algorithm provides a robust method for tackling challenging nonconvex nonsmooth optimization problems.
    • The incorporation of inertial effects and the use of the Kurdyka-Lojasiewicz inequality are crucial for establishing convergence.
    • This work contributes to the theoretical understanding and practical solution of complex optimization tasks in applied mathematics and related disciplines.