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Some convergently three-term trust region conjugate gradient algorithms under gradient function non-Lipschitz

Wujie Hu1, Jinzhao Wu1, Gonglin Yuan2

  • 1School of Electrical Engineering, Guangxi University, Nanning, Guangxi, People's Republic of China.

Scientific Reports
|May 13, 2024
PubMed
Summary
This summary is machine-generated.

Two new three-term trust region conjugate gradient algorithms demonstrate global convergence for non-Lipschitz functions. These algorithms show robust performance in image restoration and solving unconstrained problems.

Keywords:
Conjugate gradientDescent propertyGlobal convergenceGradient function non-Lipschitz continuityTrust region property

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Area of Science:

  • Optimization Algorithms
  • Numerical Analysis
  • Image Processing

Background:

  • Non-Lipschitz continuous gradient functions pose challenges for standard optimization algorithms.
  • Existing methods often require additional conditions for convergence, limiting their applicability.
  • Trust region and conjugate gradient methods are fundamental in optimization and image restoration.

Purpose of the Study:

  • Introduce two novel three-term trust region conjugate gradient algorithms: TT-TR-WP and TT-TR-CG.
  • Analyze their convergence properties under non-Lipschitz conditions.
  • Evaluate their numerical performance against classical algorithms.

Main Methods:

  • Development of two three-term trust region conjugate gradient algorithms.
  • Theoretical analysis of sufficient descent and trust region properties.
  • Global convergence analysis for non-Lipschitz functions.
  • Numerical comparison using image restoration (grayscale and color) and large-scale unconstrained problems.

Main Results:

  • The proposed algorithms, TT-TR-WP and TT-TR-CG, achieve global convergence without additional conditions.
  • In grayscale image restoration, TT-TR-CG was 2.33 times slower than TT-TR-WP, while other algorithms were slower.
  • Both algorithms performed well on color image restoration tasks.
  • TT-TR-WP and TT-TR-CG showed competitive results in unconstrained problems, with TT-TR-WP offering wide applicability and TT-TR-CG demonstrating strong robustness.

Conclusions:

  • The novel algorithms exhibit superior applicability and robustness compared to baseline methods.
  • TT-TR-WP and TT-TR-CG represent significant advancements in handling non-Lipschitz optimization problems.
  • The algorithms are effective for both image restoration and solving large-scale unconstrained optimization problems.