Many critical business and societal decisions in areas such as supply chain and healthcare involve numerous potential actions, complex constraints, and goals that can be modeled as objective functions. Mathematical optimization, a core area in Operations Research (OR), provides robust, mathematically grounded methodologies to address such decisions and has shown tremendous benefits in many applications. However, its application requires the creation of accurate and efficient optimization models, necessitating rare expertise and considerable time, creating a barrier to widespread adoption in decision-making. Thus, it is a long-standing goal to make these capabilities widely accessible. The advent of Large Language Models (LLMs) has made advanced Artificial Intelligence (AI) capabilities widely accessible through natural language. LLMs can accelerate expert work in creating formal models like computer programs, and emerging research indicates they can also speed up the development of optimization models by OR experts. We, therefore, propose integrating and advancing LLM and optimization modeling to empower organizational decision-makers to model and solve such complex problems without requiring deep expertise in optimization. In this work, we present our vision for democratizing optimization modeling for organizational decision-making by such a combination of LLMs and optimization modeling. We identify a set of fundamental requirements for the vision's implementation and describe the state of the art through a literature survey and some experimentation. We show that a) LLMs already provide substantial novel capabilities relevant to realizing this vision, but that b) major research challenges remain to be addressed. We also propose possible research directions to overcome these gaps. We would like this work to serve as a call to action to bring together the LLM and OR optimization modeling communities to pursue this vision, thereby enabling much more widespread improved decision-making and increasing by orders of magnitude the benefits AI and OR can bring to enterprises and society.

Robust optimization provides a mathematical framework for modeling and solving decision-making problems under worst-case uncertainty. This work addresses two-stage robust optimization (2RO) problems (also called *adjustable robust optimization*), wherein first-stage and second-stage decisions are made before and after uncertainty is realized, respectively. This results in a nested min-max-min optimization problem which is extremely challenging computationally, especially when the decisions are discrete. We propose Neur2RO, an efficient machine learning-driven instantiation of column-and-constraint generation (CCG), a classical iterative algorithm for 2RO. Specifically, we learn to estimate the value function of the second-stage problem via a novel neural network architecture that is easy to optimize over by design. Embedding our neural network into CCG yields high-quality solutions quickly as evidenced by experiments on two 2RO benchmarks, knapsack and capital budgeting. For knapsack, Neur2RO finds solutions that are within roughly $2$% of the best-known values in a few seconds compared to the three hours of the state-of-the-art exact branch-and-price algorithm; for larger and more complex instances, Neur2RO finds even better solutions. For capital budgeting, Neur2RO outperforms three variants of the $k$-adaptability algorithm, particularly on the largest instances, with a $10$ to $100$-fold reduction in solution time. Our code and data are available at https://github.com/khalil-research/Neur2RO.

Bilevel optimization deals with nested problems in which *leader* takes the first decision to minimize their objective function while accounting for a *follower*'s best-response reaction. Constrained bilevel problems with integer variables are particularly notorious for their hardness. While exact solvers have been proposed for mixed-integer *linear* bilevel optimization, they tend to scale poorly with problem size and are hard to generalize to the non-linear case. On the other hand, problem-specific algorithms (exact and heuristic) are limited in scope. Under a data-driven setting in which similar instances of a bilevel problem are solved routinely, our proposed framework, Neur2BiLO, embeds a neural network approximation of the leader's or follower's value function, trained via supervised regression, into an easy-to-solve mixed-integer program. Neur2BiLO serves as a heuristic that produces high-quality solutions extremely fast for four applications with linear and non-linear objectives and pure and mixed-integer variables.

Counterfactual explanations (CEs) play an important role in detecting bias and improving the explainability of data-driven classification models. A CE is a minimal perturbed data point for which the decision of the model changes. Most of the existing methods can only provide one CE, which may not be achievable for the user. In this work, we derive an iterative method to calculate robust CEs (i.e., CEs that remain valid even after the features are slightly perturbed). To this end, our method provides a whole region of CEs, allowing the user to choose a suitable recourse to obtain a desired outcome. We use algorithmic ideas from robust optimization and prove convergence results for the most common machine learning methods, including decision trees, tree ensembles, and neural networks. Our experiments show that our method can efficiently generate globally optimal robust CEs for a variety of common data sets and classification models. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: This work was supported by the Nederlandse Organisatie voor Wetenschappelijk Onderzoek [Grant OCENW.GROOT.2019.015, Optimization for and with Machine Learning (OPTIMAL)]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2023.0153 .

We consider robust combinatorial optimization problems with cost uncertainty where the decision maker can prepare K solutions beforehand and chooses the best of them once the true cost is revealed. Also known as min–max–min robustness (a special case of K-adaptability), it is a viable alternative to otherwise intractable two-stage problems. The uncertainty set assumed in this paper considers that in any scenario, at most Γ of the components of the cost vectors will be higher than expected, which corresponds to the extreme points of the budgeted uncertainty set. While the classical min–max problem with budgeted uncertainty is essentially as easy as the underlying deterministic problem, it turns out that the min–max–min problem is NP-hard for many easy combinatorial optimization problems, and not approximable in general. We thus present an integer programming formulation for solving the problem through a row-and-column generation algorithm. While exact, this algorithm can only cope with small problems, so we present two additional heuristics leveraging the structure of budgeted uncertainty. We compare our row-and-column generation algorithm and our heuristics on min-knapsack and shortest path instances previously used in the scientific literature and find that the heuristics obtain good quality solutions in short computational times.