A novel numerical method tailored for unconstrained optimization problems

Abstract: 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.