Beyond Mean–Variance: The Mean–Gini Approach to Optimization Under Uncertainty

Abstract

In probabilistic approaches to engineering design, including robust design, mean and variance are commonly used as the optimization objectives. This method, however, has significant limitations. For one, some mean–variance Pareto efficient designs may be stochastically dominated and should not be considered. Stochastic dominance is a mathematically rigorous concept commonly used in risk and decision analysis, based on the cumulative distribution function (CDFs), which establishes that one uncertain prospect is superior to another, while requiring minimal assumptions about the utility function of the outcome. This property makes it applicable to a wide range of engineering problems that ordinarily do not utilize techniques from normative decision analysis. In this work, we present a method to perform optimizations consistent with stochastic dominance: the Mean–Gini method. In macroeconomics, the Gini Index is the de facto metric for economic inequality, but statisticians have also proven a variant of it can be used to establish two conditions that are necessary and sufficient for both first and second-order stochastic dominance . These conditions can be used to reduce the Pareto frontier, eliminating stochastically dominated options. Remarkably, one of the conditions combines both mean and Gini, allowing for both expected outcome and uncertainty to be expressed in a single objective which, when maximized, produces a result that is not stochastically dominated given the Pareto front meets a convexity condition. We also find that, in a multi-objective optimization, the Mean–Gini optimization converges slightly faster than the mean–variance optimization.