Optimizing 3-objective portfolio selection with equality constraints and analyzing the effect of varying constraints on the efficient sets

Abstract

Markowitz proposes portfolio selection as a 2-objective model and emphasizes computing (whole) efficient sets and nondominated sets. Computing the sets has long been a topic in multiple-objective optimization. Researchers have gradually recognized other criteria in addition to variance and expected return. To formulate the additional criteria, researchers propose multiple-objective portfolio selection. However, computing the corresponding efficient set and nondominated set is not fully achieved. Moreover, discovering the sets' properties and utilizing the properties remain typically unanswered.In this paper, we extend Sharpe's and Merton's model by adding a general linear objective and imposing equality constraints. To optimize the model, we analytically derive the minimum-variance surface (defined later), prove it as a nondegenerate paraboloid, and prove the nondominated set as a paraboloidal segment. We also analytically derive the efficient set and prove it as a 2dimensional translated cone. We then prove that the set subsumes the efficient set of the corresponding traditional model, so the efficient set expands as the general linear objective is added. Furthermore, constraints can be changed or added. We utilize the translated-cone properties and readily compute the changing constraints' effect on the efficient sets by formulae or linear-equation systems.