Higher-Order Stochastic Dominance Constraints in Optimization

TL;DR

This paper tackles optimization under higher-order stochastic dominance constraints, which are inherently infinite in number. It develops two equivalent formulations—one using expectation-based dominance $X \preccurlyeq^{(p)} Y$ and another using risk measures $\mathcal{R}_\beta$—and proves that the uncountable constraint set can be reduced to a finite set of test points. A Newton-based optimization framework is derived, employing a single aggregate constraint $\overline{g}_p(x) \le 0$ and gradient calculations via the implicit function theorem, enabling efficient handling of non-linear dominance constraints. Numerical experiments on portfolio optimization with real data (MSCI benchmark and standardized datasets) demonstrate that higher-order dominance yields higher returns and more concentrated allocations to top-performing assets, while significantly reducing constraint complexity relative to traditional SSD/TSD methods. The work provides practical tools for robust, higher-order decision-making under uncertainty and lays groundwork for broader applications and future extensions in multi-objective and multi-dominance settings.

Abstract

This contribution examines optimization problems that involve stochastic dominance constraints. These problems have uncountably many constraints. We develop methods to solve the optimization problem by reducing the constraints to a finite set of test points needed to verify stochastic dominance. This improves both theoretical understanding and computational efficiency. Our approach introduces two formulations of stochastic dominance$\unicode{x2013}$one employs expectation operators and another based on risk measures$\unicode{x2013}$allowing for efficient verification processes. Additionally, we develop an optimization framework incorporating these stochastic dominance constraints. Numerical results validate the robustness of our method, showcasing its effectiveness for solving higher-order stochastic dominance problems, with applications to fields such as portfolio optimization.

Figures (7)

• Figure 1: Illustration of the stochastic dominance verification condition from Theorem \ref{['thm:156']}. The solid curves $\|(t - X)_+\|$ (blue) and $\|(t - Y)_+\|$ (red) represent the norms under comparison, while the dashed curves show their respective derivatives, $\frac{d}{dt} \|(t - X)_+\|$ (blue, dashed) and $\frac{d}{dt} \|(t - Y)_+\|$ (red, dashed). The critical points $t_1$, $t_2$, and $t_3$ are identified at intersections of the derivatives, and they provide sufficient confirmation of the dominance relationship $X \preccurlyeq^{\|\cdot\|} Y$.
• Figure 2: Verification of stochastic dominance relations based on Theorem \ref{['thm:140']}. The plot illustrates $t_{-X}(\beta)$ and $t_{-Y}(\beta)$ (blue dashed and red dashed), respectively, for the risk levels. These functions provide a visual representation of the conditions in Equations \ref{['eq:25']} and \ref{['eq:26']}. The shaded regions represent dominance intervals, while the solid curves $\mathcal{R}_{\beta_i}(-X)$ (blue) and $\mathcal{R}_{\beta_i}(-Y)$ (red) show the relative dominance of $X$ and $Y$ at the critical risk levels $\beta_i$. The critical points, denoted by $\beta_i$ and $\gamma_i$, satisfy the dominance conditions required to verify $X \preccurlyeq^{\|\cdot\|} Y$.
• Figure 3: Function configurations of $\overline{g}(x)$, cf. \ref{['eq:51']}, illustrating various values of $t$, where $g_p(\tilde{t}, x^\top \xi) \leq 0$ and $g_{p-1}(\tilde{t}, x^\top \xi) = 0$ for $\tilde{t} \in \{t_1, t_2, t_3\}$, including cases where $g_p(t, x^\top\xi)$ satisfies or does not satisfy the condition $\forall\,t \in \mathbb{R}$.
• Figure 4: MSCI World index and 5 big tech stock prices: Jan 2020 – Oct 2024
• Figure 5: Maximized portfolio (annualized) return (top) and cumulative portfolio allocation by stochastic order (bottom)

Theorems & Definitions (28)

• Definition 2.1: Stochastic dominance
• Remark 2.2
• Remark 2.3: Partial order
• Definition 2.4
• Remark 2.5: Norm formulation and infinity order
• Remark 2.6
• Remark 2.7: Cf. Ogryczak99Ogryczak01
• Theorem 3.1
• Remark 3.2
• proof