In this paper, we focus on developing an efficient method for solving nonsmooth unconstrained optimization problems arising from mathematical models, in particular those derived from differential equations with singular nonlinearities. Due to the nonsmoothness of the objective function, we construct a sequence of iteratively updated quadratic models to approximate the nonsmooth objective function. The unique solvability and key properties of these quadratic models are thoroughly discussed and rigorously proved. A classical trust-region method is then employed to minimize these quadratic models within the trust region, yielding an effective minimization method. Furthermore, a convergence analysis of the minimization method is also established. To address the challenges posed by the nonconvexity of the nonsmooth objective function, we first introduce an auxiliary function approach designed to escape from computed local minima until a global minimizer is found. Since differential equations with singular nonlinearities admit multiple solutions, the minimization method is further extended to compute multiple solutions by incorporating a deflation technique. In this extended framework, a generalization of the deflation function is developed. Numerical experiments are provided to demonstrate the efficiency and robustness of the  proposed algorithms. Furthermore, based on our algorithms, several previously unreported multiple solutions are computed and presented here for the first time. In summary, this study not only opens a new and efficient avenue for solving differential equations with singular nonlinearities from a numerical optimization perspective, but also contributes significantly to the exploration of more complex real-world problems.

李林，南华大学数理学院硕士生导师，其主要研究方向为计算数学与应用数学的交叉领域，涵盖谱方法、多解计算、数值优化及机器学习等方向。近年来，在谱方法的数值分析、多解算法及非光滑优化计算等方面取得了一系列研究成果，并注重推动计算方法在科学与工程实际问题中的有效落地。曾多次赴台湾国立中央大学数学系、美国芝加哥伊利诺伊理工大学、新加坡南洋理工大学开展合作研究、中国科学院数学与系统科学研究院进行访问研究。截至目前，已在计算数学领域期刊Journal of Scientific Computing、Journal of Computational and Applied Mathematics、Journal of Computational Mathematics等知名期刊上发表SCI论文10余篇。