Analytical Shortcuts to Multiple-Objective Portfolio Optimization: Investigating the Non-Negativeness of Portfolio Weight Vectors of Equality-Constraint-Only Models and Implications for Capital Asset Pricing Models - Yue Qi; Yue Wang; Jianing Huang; Yushu Zhang (Mathematics)
Computing optimal-solution sets has long been a topic in multiple-objective optimization. Despite substantial progress, there are still research limitations in the multiple-objective portfolio optimization area. The optimal-solution sets’ structure is barely known. Public-domain software for even three objectives is absent. Alternatively, researchers scrutinize equality-constraint-only models and analytically resolve them. Within this context, this paper extends these analytical methods for nonnegative constraints and thus theoretically contributes to the literature. We prove the existence of positive elements and negative elements for the optimal-solution sets. Practically, we prove that non-negative subsets of the optimal-solution sets can exist. Consequently, the possible existence endorses these analytical methods, because researchers bypass mathematical programming, analytically resolve, and pinpoint some non-negative optima. Moreover, we elucidate these analytical methods’ alignment with capital asset pricing models (CAPMs). Furthermore, we generalize for k-objective models. In conclusion, this paper theoretically reinforces these analytical methods and hints the optimal-solution sets’ structure for multiple-objective portfolio optimization.