In the presence of multiple constraints such as the risk tolerance constraint and the budget constraint, many extensively studied (Pareto-)optimal reinsurance problems based on general distortion risk measures become technically challenging and have only been solved using ad hoc methods for certain special cases. In this paper, we extend the method developed in Lo (2017a) by proposing a generalized Neyman-Pearson framework to identify the optimal forms of the solutions. We then develop a dual formulation and show that the infinite-dimensional constrained optimization problems can be reduced to finite-dimensional unconstrained ones. With the support of the Nelder-Mead algorithm, we are able to obtain optimal solutions efficiently. We illustrate the versatility of our approach by working out several detailed numerical examples, many of which in the literature were only partially resolved.

在存在风险承受力约束和预算约束等多重约束的情况下，许多基于一般扭曲风险度量的广泛研究的（帕累托）最优再保险问题在技术上变得具有挑战性，并且只能针对某些特殊情况使用临时方法来解决。在本文中，我们扩展了Lo (2017a)中开发的方法通过提出一个广义的内曼-皮尔逊框架来确定解决方案的最佳形式。然后，我们开发了一个对偶公式，并证明无限维约束优化问题可以简化为有限维无约束问题。在Nelder-Mead算法的支持下，我们能够高效地获得最优解。我们通过制定几个详细的数值示例来说明我们方法的多功能性，其中许多示例在文献中仅得到部分解决。