We consider an iterative computation of negative curvature directions, in large-scale unconstrained optimization frameworks, needed for ensuring the convergence toward stationary points which satisfy second-order necessary optimality conditions. We show that to the latter purpose, we can fruitfully couple the conjugate gradient (CG) method with a recently introduced approach involving the use of the numeral called Grossone. In particular, recalling that in principle the CG method is well posed only when solving positive definite linear systems, our proposal exploits the use of grossone to enhance the performance of the CG, allowing the computation of negative curvature directions in the indefinite case, too. Our overall method could be used to significantly generalize the theory in state-of-the-art literature. Moreover, it straightforwardly allows the solution of Newton’s equation in optimization frameworks, even in nonconvex problems. We remark that our iterative procedure to compute a negative curvature direction does not require the storage of any matrix, simply needing to store a couple of vectors. This definitely represents an advance with respect to current results in the literature.

中文翻译：

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大规模优化中负曲率方向的基于迭代格罗松计算

我们考虑在大规模无约束优化框架中对负曲率方向进行迭代计算，以确保向满足二阶必要最优性条件的平稳点收敛。我们表明，对于后一个目的，我们可以将共轭梯度 (CG) 方法与最近引入的涉及使用称为 Grossone 的数字的方法有效地结合起来。特别是，回想起原则上 CG 方法仅在求解正定线性系统时才适合，我们的提议利用了grossone 的使用来增强 CG 的性能，也允许在不定情况下计算负曲率方向。我们的整体方法可用于在最先进的文献中显着推广该理论。而且，它直接允许在优化框架中求解牛顿方程，即使在非凸问题中也是如此。我们注意到，我们计算负曲率方向的迭代过程不需要存储任何矩阵，只需要存储几个向量。这绝对代表了相对于当前文献结果的进步。