Optimizing over Pareto set of semistrictly quasiconcave vector maximization and application to stochastic portfolio selection

Abstract

<p style='text-indent:20px;'>Optimization over Pareto set of a semistrictly quasiconcave vector maximization problem has many applications in economics and technology but it is a challenging task because of the nonconvexity of objective functions and constraint sets. In this article, we propose a novel approach, which is a Branch-and-Bound algorithm for maximizing a composite function <inline-formula><tex-math id="M1">\begin{document}$ \varphi(f(x)) $\end{document}</tex-math></inline-formula> over the non-dominated solution set of the <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>-objective programming problem, where <inline-formula><tex-math id="M3">\begin{document}$ p\geq 2, p \in \mathbb{N}, $\end{document}</tex-math></inline-formula> the function <inline-formula><tex-math id="M4">\begin{document}$ \varphi $\end{document}</tex-math></inline-formula> is increasing and the objective function <inline-formula><tex-math id="M5">\begin{document}$ f $\end{document}</tex-math></inline-formula> is semistrictly quasiconcave. By utilizing the nice properties of Pareto set to define the partitions of branch and bound scheme, the proposed algorithms are promised to be more accurate and efficient than ones using the multi-objective evolutionary approach such as NSGA-III. This is validated by some computational experiments. The Stochastic Portfolio Selection Problem is chosen as an application of our algorithm, where Sharpe ratio is a semistrictly quasiconcave objective function.</p>