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<p>Ill-posed and ill-conditioned problems are pervasive in the modeling of real-world phenomena across a wide range of scientific and engineering disciplines, posing substantial challenges for numerical optimization. This work introduces a unified and efficient framework to address such problems using three enhanced optimization algorithms: Conjugate Gradient (CG), Broyden–Fletcher–Goldfarb–Shanno (BFGS), and a globalized Limited-memory BFGS (LBFGS), each integrated with an effective regularization strategy. The proposed methods are evaluated on benchmark problems with varying levels of ill-conditioning and additive noise. Particular emphasis is placed on analyzing performance, stability, and convergence behavior as problem severity increases. A comprehensive comparative study is conducted to assess the sensitivity of each method to observational noise and initial conditions. The framework is further validated on a representative real-world ill-posed problem, demonstrating robustness and practical effectiveness. Numerical results confirm that the developed methods provide reliable and efficient tools for solving ill-posed optimization problems under conditions of noise and uncertainty.</p>

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A unified framework for solving ill-posed problems using regularized conjugate gradient, Broyden–Fletcher–Goldfarb–Shanno (BFGS), and limited-memory BFGS

  • 期刊论文

  • 发表日期：

    2025年8月19日

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  • Mohamed Ziouane,
  • Mourad Nachaoui,
  • Amine Laghrib

摘要

Ill-posed and ill-conditioned problems are pervasive in the modeling of real-world phenomena across a wide range of scientific and engineering disciplines, posing substantial challenges for numerical optimization. This work introduces a unified and efficient framework to address such problems using three enhanced optimization algorithms: Conjugate Gradient (CG), Broyden–Fletcher–Goldfarb–Shanno (BFGS), and a globalized Limited-memory BFGS (LBFGS), each integrated with an effective regularization strategy. The proposed methods are evaluated on benchmark problems with varying levels of ill-conditioning and additive noise. Particular emphasis is placed on analyzing performance, stability, and convergence behavior as problem severity increases. A comprehensive comparative study is conducted to assess the sensitivity of each method to observational noise and initial conditions. The framework is further validated on a representative real-world ill-posed problem, demonstrating robustness and practical effectiveness. Numerical results confirm that the developed methods provide reliable and efficient tools for solving ill-posed optimization problems under conditions of noise and uncertainty.