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Related Experiment Video

Updated: Mar 31, 2026

An R-Based Landscape Validation of a Competing Risk Model
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An R-Based Landscape Validation of a Competing Risk Model

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A Modified BFGS Formula Using a Trust Region Model for Nonsmooth Convex Minimizations.

Zengru Cui1, Gonglin Yuan2, Zhou Sheng1

  • 1Guangxi Colleges and Universities Key Laboratory of Mathematics and Its Applications, College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China.

Plos One
|October 27, 2015
PubMed
Summary

This study introduces a novel BFGS method for nonsmooth convex optimization. The approach utilizes Moreau-Yosida regularization and a trust region model, efficiently solving complex minimization problems.

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Area of Science:

  • Optimization Theory
  • Numerical Analysis
  • Mathematical Programming

Background:

  • Nonsmooth convex minimization problems present significant challenges in various scientific and engineering fields.
  • Traditional methods often struggle with the lack of differentiability, requiring specialized techniques.
  • The Moreau-Yosida regularization offers a smoothing approach to handle nonsmooth functions.

Purpose of the Study:

  • To develop an efficient and globally convergent algorithm for nonsmooth unconstrained convex minimization.
  • To modify the BFGS (Broyden–Fletcher–Goldfarb–Shanno) formula for improved performance in nonsmooth settings.
  • To leverage trust region models and a new secant equation for enhanced computational efficiency.

Main Methods:

  • The proposed method employs the Moreau-Yosida regularization to approximate nonsmooth functions with smooth ones.
  • A modified BFGS update formula is utilized, incorporating a new secant equation.
  • A trust region model guides the iterative steps, utilizing function and gradient information to compute the Hessian.
  • The Hessian matrix is updated via the BFGS formula, avoiding direct computation of second-order derivatives.

Main Results:

  • The algorithm demonstrates global convergence to an optimal solution under suitable conditions.
  • Numerical experiments confirm the algorithm's effectiveness in solving nonsmooth unconstrained convex problems.
  • The use of the BFGS formula for Hessian updates reduces computational workload and time.

Conclusions:

  • The modified BFGS algorithm with Moreau-Yosida regularization provides an effective solution for nonsmooth convex optimization.
  • The trust region approach combined with the BFGS update enhances computational efficiency.
  • This method offers a robust tool for tackling challenging minimization problems in applied mathematics and beyond.