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<p>A novel quasi-Newton method is proposed wherein the Hessian approximation is constructed using orthogonal polynomial quadrature, specifically based on Legendre polynomials. In contrast to traditional approaches that rely on second-order Taylor expansions or finite-difference approximations, the method estimates the second directional derivative through an efficient quadrature formula, providing controlled accuracy and enhanced numerical stability. Theoretical properties of the method, including convergence behavior and error bounds, are rigorously analyzed. Extensive numerical experiments on a wide range of benchmark functions are performed, demonstrating that the proposed approach achieves a favorable balance between computational efficiency and solution accuracy when compared to classical quasi-Newton methods employing scalar Hessian approximations. Additionally, the influence of the number of quadrature nodes on convergence speed and accuracy is systematically investigated.</p>

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Legendre polynomial based approximation of the Hessian for quasi-Newton optimization

  • 期刊论文

  • 发表日期：

    2025年7月26日

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  • Stefan Panic

摘要

A novel quasi-Newton method is proposed wherein the Hessian approximation is constructed using orthogonal polynomial quadrature, specifically based on Legendre polynomials. In contrast to traditional approaches that rely on second-order Taylor expansions or finite-difference approximations, the method estimates the second directional derivative through an efficient quadrature formula, providing controlled accuracy and enhanced numerical stability. Theoretical properties of the method, including convergence behavior and error bounds, are rigorously analyzed. Extensive numerical experiments on a wide range of benchmark functions are performed, demonstrating that the proposed approach achieves a favorable balance between computational efficiency and solution accuracy when compared to classical quasi-Newton methods employing scalar Hessian approximations. Additionally, the influence of the number of quadrature nodes on convergence speed and accuracy is systematically investigated.