A new quasi-Newton method with a diagonal updating matrix is suggested, where the diagonal elements are determined by forward or by central finite differences. The search direction is a direction of sufficient descent. The algorithm is equipped with an acceleration scheme. The convergence of the algorithm is linear. The preliminary computational experiments use a set of 75 unconstrained optimization test problems classified in five groups according to the structure of their Hessian: diagonal, block-diagonal, band (tri- or penta-diagonal), sparse and dense. Subject to the CPU time metric, intensive numerical experiments show that, for problems with Hessian in a diagonal, block-diagonal or band structure, the algorithm with diagonal approximation of the Hessian by finite differences is top performer versus the well-established algorithms: the steepest descent and the Broyden–Fletcher–Goldfarb–Shanno. On the other hand, as a by-product of this numerical study, we show that the Broyden–Fletcher–Goldfarb–Shanno algorithm is faster for problems with sparse Hessian, followed by problems with dense Hessian.

中文翻译：

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无约束优化的有限差分 Hessian 对角线逼近

提出了一种新的具有对角更新矩阵的拟牛顿法，其中对角元素由前向或中心有限差分确定。搜索方向是足够下降的方向。该算法配备了加速方案。算法的收敛性是线性的。初步计算实验使用一组 75 个无约束优化测试问题，根据 Hessian 的结构分为五组：对角线、块对角线、带状（三对角线或五对角线）、稀疏和密集。根据 CPU 时间指标，密集的数值实验表明，对于 Hessian 在对角线、块对角线或带结构中的问题，通过有限差分对 Hessian 进行对角逼近的算法与完善的算法相比性能最佳：最陡峭的下坡和 Broyden-Fletcher-Goldfarb-Shanno。另一方面，作为该数值研究的副产品，我们表明 Broyden-Fletcher-Goldfarb-Shanno 算法对于稀疏 Hessian 问题更快，其次是稠密 Hessian 问题。