标题： 一种针对无约束优化问题的新型数值方法 显示英文标题

标题： A novel numerical method tailored for unconstrained optimization problems

摘要： 无约束优化问题随着人工智能的快速发展，在科学计算和工程应用中变得越来越普遍，人们也越来越关注能更快速、更高效地解决这些问题的数值方法。 此外，迫切需要一种能够最小化各种目标函数（尤其是非光滑目标函数）的有效方法。 因此，本文的重点是提出一种新颖的数值方法，用于解决无论目标函数是否平滑的无约束优化问题。 具体来说，基于变分程序来改进梯度和Hessian矩阵的近似值，建立了一个具有$2n$约束条件的有效二次模型。 此外，为了提高计算效率，还提出了一个具有2个约束条件的简化模型，在该模型中梯度和Hessian矩阵可以显式更新，并且其余$2n-2$约束条件的有界性也得到了推导。 另一方面，总结了这种新颖的数值方法，并分析展示了关于导数信息的近似结果。 测试了涉及平滑、导数爆炸和非平滑问题的数值实验，证明了其可行性和效率。 与现有方法相比，我们提出的方法首次能够有效地解决平滑和非平滑的无约束优化问题，而且代码编写非常容易，这表明我们的方法不仅具有巨大的应用前景，而且对于探索实际复杂的工程和科学问题也非常有意义。 显示英文摘要

摘要： Unconstrained optimization problems become more common in scientific computing and engineering applications with the rapid development of artificial intelligence, and numerical methods for solving them more quickly and efficiently have been getting more attention and research. Moreover, an efficient method to minimize all kinds of objective functions is urgently needed, especially the nonsmooth objective function. Therefore, in the current paper, we focus on proposing a novel numerical method tailored for unconstrained optimization problems whether the objective function is smooth or not. To be specific, based on the variational procedure to refine the gradient and Hessian matrix approximations, an efficient quadratic model with $2n$ constrained conditions is established. Moreover, to improve the computational efficiency, a simplified model with 2 constrained conditions is also proposed, where the gradient and Hessian matrix can be explicitly updated, and the corresponding boundedness of the remaining $2n-2$ constrained conditions is derived. On the other hand, the novel numerical method is summarized, and approximation results on derivative information are also analyzed and shown. Numerical experiments involving smooth, derivative blasting, and non-smooth problems are tested, demonstrating its feasibility and efficiency. Compared with existing methods, our proposed method can efficiently solve smooth and non-smooth unconstrained optimization problems for the first time, and it is very easy to program the code, indicating that our proposed method not also has great application prospects, but is also very meaningful to explore practical complex engineering and scientific problems.