An analytical derivation of properly efficient sets in multi-objective portfolio selection

Computing efficient sets has long been a topic in multiple-objective optimization and research has made substantial progress. However, there are still limitations in the multiple-objective portfolio selection and optimization areas. Firstly, researchers typically focus on models containing only one quadratic objective. Secondly, few researchers pursue multiple quadratic objectives, but their algorithms could be relatively elusive and it could be a pity that they do not explicitly demonstrate the efficient sets’ structure. Lastly, researchers mostly limit their scope to three objectives. Within this context, this paper makes theoretical contributions to the literature. Operating with multiple quadratic objectives, we analytically derive closed-form formulae for the computation of the properly efficient and weakly efficient sets of problems and demonstrate the efficient sets’ structure in the form of a sequence of pyramids in decision space. Although we are restricted to equality-constraint-only models, our results have implications for general-constraint models. In addition, our methods can be extended to general k-quadratic objective models.