Multi-constrained optimal reinsurance model from the duality perspectives

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.