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<p>Dargahi et al. [J. Optim. Theory Appl. 200: 394-403, 2024] have analyzed the spectral secant equation for the Davidon-Fletcher-Powell (DFP) quasi-Newton updating formula. In this paper, we extend this method to obtain an accelerated derivative-free memoryless DFP method, combined with a hyperplane projection technique and an inertial extrapolation step, for solving unconstrained nonlinear equations. More specifically, for the pre-specified spectral parameter of the modified memoryless DFP formula, we first analyze its spectral condition number based on the eigenvalues of the DFP formula. By incorporating an Oren-Luenberger-like self-scaling parameter [Manage. Sci. 20(5): 845-862, 1973], we then propose a new hybrid adaptive formula for the pre-specified spectral parameter. The global convergence of the proposed method is established under a weaker, non-restrictive monotonicity condition on the underlying mapping. Unlike typical methods, its theoretical proof does not rely on monotonicity or pseudo-monotonicity assumptions. Additionally, we provide both asymptotic and non-asymptotic global convergence rates measured by iteration complexity. Numerical experiments on unconstrained nonlinear equations demonstrate the effectiveness of the proposed method. Moreover, its potential application to sparse signal restoration problems is illustrated.</p>

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An accelerated derivative-free memoryless Davidon-Fletcher-Powell method and its iteration-complexity analysis

  • 期刊论文

  • 发表日期：

    2025年9月25日

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  • Pengjie Liu,
  • Jiayi Li,
  • Hu Shao,
  • Ting Wu

摘要

Dargahi et al. [J. Optim. Theory Appl. 200: 394-403, 2024] have analyzed the spectral secant equation for the Davidon-Fletcher-Powell (DFP) quasi-Newton updating formula. In this paper, we extend this method to obtain an accelerated derivative-free memoryless DFP method, combined with a hyperplane projection technique and an inertial extrapolation step, for solving unconstrained nonlinear equations. More specifically, for the pre-specified spectral parameter of the modified memoryless DFP formula, we first analyze its spectral condition number based on the eigenvalues of the DFP formula. By incorporating an Oren-Luenberger-like self-scaling parameter [Manage. Sci. 20(5): 845-862, 1973], we then propose a new hybrid adaptive formula for the pre-specified spectral parameter. The global convergence of the proposed method is established under a weaker, non-restrictive monotonicity condition on the underlying mapping. Unlike typical methods, its theoretical proof does not rely on monotonicity or pseudo-monotonicity assumptions. Additionally, we provide both asymptotic and non-asymptotic global convergence rates measured by iteration complexity. Numerical experiments on unconstrained nonlinear equations demonstrate the effectiveness of the proposed method. Moreover, its potential application to sparse signal restoration problems is illustrated.