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<p>A novel quasi–Newton optimization method based on a low–order rational Padé approximation, for estimating curvature information along the descent direction, has been proposed. The method constructs a Padé [2/2] model of the objective function restricted to a one dimensional search line and extracts a scalar curvature surrogate that replaces the Hessian in a directional sense. This leads to an adaptive curvature scaled gradient update that requires neither Hessian evaluations nor matrix updates. It is shown that the proposed Padé based curvature estimate is a consistent approximation of the directional Rayleigh quotient of the Hessian. Global convergence to stationary points is established under standard smoothness assumptions when the method is combined with Armijo backtracking, and linear convergence is obtained under strong convexity. A detailed truncation/round-off analysis reveals that the Padé curvature estimate satisfies: <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\widehat{\gamma }_k = \lambda _k + \mathcal {O}(h^2) + \mathcal {O}\!\left( \frac{u}{h^p}\right) \)</EquationSource> </InlineEquation>, which explains the numerical instability observed for excessively small finite difference steps. Extensive numerical experiments on a broad set of benchmark problems demonstrate that the Padé [2/2] method achieves competitive or superior performance compared to classical quasi–Newton and curvature scaled gradient methods in terms of iteration count, function evaluations, and CPU time. In addition, the results indicate that the simpler Padé [1/1] approximation already provides a meaningful curvature estimate, while the [2/2] model offers improved robustness and accuracy without sacrificing computational efficiency.</p>

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A curvature scaled quasi-Newton method based on Padé approximation

  • 期刊论文

  • 发表日期：

    2026年5月2日

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  • Stefan Panic

摘要

A novel quasi–Newton optimization method based on a low–order rational Padé approximation, for estimating curvature information along the descent direction, has been proposed. The method constructs a Padé [2/2] model of the objective function restricted to a one dimensional search line and extracts a scalar curvature surrogate that replaces the Hessian in a directional sense. This leads to an adaptive curvature scaled gradient update that requires neither Hessian evaluations nor matrix updates. It is shown that the proposed Padé based curvature estimate is a consistent approximation of the directional Rayleigh quotient of the Hessian. Global convergence to stationary points is established under standard smoothness assumptions when the method is combined with Armijo backtracking, and linear convergence is obtained under strong convexity. A detailed truncation/round-off analysis reveals that the Padé curvature estimate satisfies: \(\widehat{\gamma }_k = \lambda _k + \mathcal {O}(h^2) + \mathcal {O}\!\left( \frac{u}{h^p}\right) \) , which explains the numerical instability observed for excessively small finite difference steps. Extensive numerical experiments on a broad set of benchmark problems demonstrate that the Padé [2/2] method achieves competitive or superior performance compared to classical quasi–Newton and curvature scaled gradient methods in terms of iteration count, function evaluations, and CPU time. In addition, the results indicate that the simpler Padé [1/1] approximation already provides a meaningful curvature estimate, while the [2/2] model offers improved robustness and accuracy without sacrificing computational efficiency.