The abundance of data has led to the emergence of a variety of optimization techniques that attempt to leverage available side information to provide more anticipative decisions. The wide range of methods and contexts of application have motivated the design of a universal unitless measure of performance known as the coefficient of prescriptiveness. This coefficient was designed to quantify both the quality of contextual decisions compared to a reference one and the prescriptive power of side information. To identify policies that maximize the former in a data-driven context, this paper introduces a distributionally robust contextual optimization model where the coefficient of prescriptiveness substitutes for the classical empirical risk minimization objective. We present a bisection algorithm to solve this model, which relies on solving a series of linear programs when the distributional ambiguity set has an appropriate nested form and polyhedral structure. Studying a contextual shortest path problem, we evaluate the robustness of the resulting policies against alternative methods when the out-of-sample dataset is subject to varying amounts of distribution shift.

The Benjamini-Hochberg (BH) procedure is widely used to control the false detection rate (FDR) in large-scale hypothesis testing. Although this control may be distributionally robust, we investigate its adversarial robustness to perturbations of the p-values. In particular, we present a class of data-perturbation algorithms, discuss when BH does and does not exhibit adversarial robustness, and perform computational experiments. With our algorithms, we demonstrate that it is possible for BH's control to be badly broken with relatively few (even just one) p-value perturbation(s), and provide non-asymptotic guarantees on the expected adversarial-adjustment to FDR. Our technical analysis involves reframing the BH procedure as a ``balls into bins'' process and drawing a connection to generalized ballot problems that facilitates an information-theoretic approach for deriving non-asymptotic lower bounds. This study demonstrates an instance of how distributional robustness may come at the price of adversarial robustness.

This paper investigates the bounds on the expectation of combinatorial optimization given moment information for each individual random variable. A popular approach to solving this problem, known as the marginal moment model (MMM), is to reformulate it as a semidefinite program (SDP). In this paper, we investigate the structure of MMM with up to fourth-order marginal moments and reformulate them as second-order cone programs (SOCP). Additionally, we establish that this SOCP formulation is equivalent to a convex optimization problem over the convex hull of the feasible region of the original combinatorial optimization, presenting closed-form expressions for both the objective function and its derivative. These reformulations enable more efficient computation of the bounds and persistency value. In addition, we explore two types of ambiguity sets characterized by incomplete moment information. We further discuss the relationship between the aforementioned MMMs and another widely-used model, the marginal distribution model (MDM). Beyond bounding the worst-case expectations, our approaches can be modified to bound the worst-case conditional value at risk (CVaR) of the combinatorial optimization.

Building on this theoretical advancement, we explore two applications. First, we consider the project crashing problem, wherein both the means and variances of activity durations can be controlled through effort. We demonstrate that the distributionally robust project crashing problem, incorporating up to fourth-order moment information, can be reformulated as either an SOCP or a convex minimization over a simple polytope. Numerical analysis reveals that MMM with fourth moment information yields tighter bounds on expected delays and requires a significantly smaller budget than the mean-variance model for a fixed delay guarantee. Second, we apply our reformulations to solve the distributionally robust newsvendor problem with moment information, extending the well-known Scarf’s model. We derive several new closed-form solutions and explore how the optimal order quantities depend on skewness and kurtosis. Numerically, we show that incorporating additional moment information can lead to better performance, especially in the high service level regime.