We study inverse optimization (IO), where the goal is to use a parametric optimization program as the hypothesis class to infer relationships between input-decision pairs. Most of the literature focuses on learning only the objective function, as learning the constraint function (i.e., feasible regions) leads to nonconvex training programs. Motivated by this, we focus on learning feasible regions for known linear objectives, and introduce two training losses along with a hypothesis class to parameterize the constraint function. Our hypothesis class surpasses the previous objective-only method by naturally capturing discontinuous behaviors in input-decision pairs. We introduce a customized block coordinate descent algorithm with a smoothing technique to solve the training problems, while for further restricted hypothesis classes, we reformulate the training optimization as a tractable convex program or mixed integer linear program. Synthetic experiments and two power system applications including comparisons with state-of-the-art approaches showcase and validate the proposed approach.

Uncertain optimization problems with decision-dependent information discovery allow the decision maker to control the timing of information discovery, in contrast to the classic multistage setting where uncertain parameters are revealed sequentially based on a prescribed filtration. This problem class is useful in a wide range of applications; however, its assimilation is partly limited by the lack of efficient solution schemes. In this paper, we study two-stage robust optimization problems with decision-dependent information discovery where uncertainty appears in the objective function. The contributions of the paper are twofold: (i) we develop the first exact algorithm for this class of problems, and (ii) we improve upon the existing K-adaptability approximation by strengthening its formulation using techniques from the integer programming literature. We benchmark our approaches using the decision-dependent information discovery orienteering and shortest path problems. We demonstrate that the exact solution method outperforms at times the K-adaptability approximation; however, the strengthened K-adaptability formulation can provide good-quality solutions in larger instances while significantly outperforming existing approximation schemes even in the decision-independent information discovery setting. We leverage the effectiveness of the proposed solution schemes and the orienteering problem in a case study from Alrijne Hospital in the Netherlands, where we try to improve the collection process of empty medicine delivery crates by cooptimizing sensor placement and routing decisions. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: The authors gratefully acknowledge the financial support by the Dutch Institute of Advanced Logistics (Dinalog) [grant number 2017-2-131TKI]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0290 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0290 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

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.

We consider the parameter estimation problem of a probabilistic generative model prescribed using a natural exponential family of distributions. For this problem, the typical maximum likelihood estimator usually overfits under limited training sample size, is sensitive to noise and may perform poorly on downstream predictive tasks. To mitigate these issues, we propose a distributionally robust maximum likelihood estimator that minimizes the worst-case expected log-loss uniformly over a parametric Kullback-Leibler ball around a parametric nominal distribution. Leveraging the analytical expression of the Kullback-Leibler divergence between two distributions in the same natural exponential family, we show that the min-max estimation problem is tractable in a broad setting, including the robust training of generalized linear models. Our novel robust estimator also enjoys statistical consistency and delivers promising empirical results in both regression and classification tasks.