arXiv subjects

Dick den Hertog

Publications and source records attributed to Dick den Hertog.

At least 19 recordsLinked to original sources

arXiv2026

Optimization-based strategic planning for geographical healthcare accessibility in developing countries: a literature review

Access to healthcare facilities is a critical issue in developing countries, where limited resources and significant challenges hinder progress toward universal health coverage, one of the targets pursued by the United Nations. As a result, the Operations Research (OR) community has become increasingly active in addressing this issue, employing various techniques, particularly optimization. However, making long-term strategic decisions remains challenging, as these often require substantial investments and directly affect large populations. Moreover, much of the existing literature is predominantly theoretical. This paper reviews studies published over the past two decades that use optimization techniques to inform strategic planning decisions aimed at improving health accessibility and specifically contain a case study conducted in a developing country. The review explores the nature of the problems tackled, considering factors such as the level of care, service delivery channels, objectives, criteria, or potential hierarchy, uncertainty, and multi-period settings. Additionally, we discuss the modeling approaches, solution methodologies, and the extent of practical implementation. The goal of this survey is not only to summarize existing work but also to provide a roadmap for future research, offering valuable insights for OR practitioners in the field.

math.OC↗

arXiv2026

Linear Model Extraction via Factual and Counterfactual Queries

In model extraction attacks, the goal is to reveal the parameters of a black-box machine learning model by querying the model for a selected set of data points. Due to an increasing demand for explanations, this may involve counterfactual queries besides the typically considered factual queries. In this work, we consider linear models and three types of queries: factual, counterfactual, and robust counterfactual. First, for an arbitrary set of queries, we derive novel mathematical formulations for the classification regions for which the decision of the unknown model is known, without recovering any of the model parameters. Second, we derive bounds on the number of queries needed to extract the model's parameters for (robust) counterfactual queries under arbitrary norm-based distances. We show that the full model can be recovered using just a single counterfactual query when differentiable distance measures are employed. In contrast, when using polyhedral distances for instance, the number of required queries grows linearly with the dimension of the data space. For robust counterfactuals, the latter number of queries doubles. Consequently, the applied distance function and robustness of counterfactuals have a significant impact on the model's security.

math.OC↗

arXiv2025

A K-adaptability Approach to Proton Radiation Therapy Robust Treatment Planning

Uncertainties such as setup and range errors can significantly compromise proton therapy. A discrete uncertainty set is often constructed to represent different uncertainty scenarios. A min-max robust optimization approach is then utilized to optimize the worst-case performance of a radiation therapy plan against the uncertainty set. However, the min-max approach can be too conservative as a single plan has to account for the entire uncertainty set. K-adaptability is a novel approach to robust optimization which covers the uncertainty set with multiple (K) solutions, reducing the conservativeness. Solving K-adaptability to optimality is known to be computationally intractable. To that end, we developed a novel and efficient K-adaptability heuristic that iteratively clusters the scenarios based on plan-scenario performance for the proton radiation therapy planning problem. Compared to the conventional robust solution, the developed K-adaptability heuristic increased the worst-case CTV Dmin dose up to 4.52 Gy on average across five head and neck patients. The developed heuristic also demonstrated its superiority in objective value and time-efficiency compared to the competing methods we tested.

math.OC↗

arXiv2025

A novel exact approach to polynomial optimization

Polynomial optimization problems represent a wide class of optimization problems, with a large number of real-world applications. Current approaches for polynomial optimization, such as the sum of squares (SOS) method, rely on large-scale semidefinite programs, and therefore the scale of problems to which they can be applied is limited and an optimality guarantee is not always provided. Moreover, the problem can have other convex nonlinear parts, that cannot be handled by these approaches. In this paper, we propose an alternative approach for polynomial optimization. We obtain a convex relaxation of the original polynomial optimization problem, by deriving a sum of linear times convex (SLC) functions decomposition for the polynomial. We prove that such SLC decompositions always exist for arbitrary degree polynomials. Moreover, we derive the SLC decomposition that results in the tightest lower bound, thus improving significantly the quality of the obtained bounds in each node of the spatial Branch and Bound method. In the numerical experiments, we show that our approach outperforms state-of-the-art methods for polynomial optimization, such as BARON and SOS. We show that with our method, we can solve polynomial optimization problems to optimality with 40 variables and degree 3, as well as 20 variables and degree 4, in less than an hour.

math.OC↗

arXiv2025

Robust Optimization of Rank-Dependent Models with Uncertain Probabilities

This paper studies distributionally robust optimization for a rich class of risk measures with ambiguity sets defined by $ϕ$-divergences. The risk measures are allowed to be non-linear in probabilities, are represented by Choquet integrals possibly induced by a probability weighting function, and encompass many well-known examples. Optimization for this class of risk measures is challenging due to their rank-dependent nature. We show that for various shapes of probability weighting functions, including concave, convex and inverse $S$-shaped, the robust optimization problem can be reformulated into a rank-independent problem. In the case of a concave probability weighting function, the problem can be reformulated further into a convex optimization problem that admits explicit conic representability for a collection of canonical examples. While the number of constraints in general scales exponentially with the dimension of the state space, we circumvent this dimensionality curse and develop two types of algorithms. They yield tight upper and lower bounds on the exact optimal value and are formally shown to converge asymptotically. This is illustrated numerically in a robust newsvendor problem and a robust portfolio choice problem.

math.OC↗

arXiv2024

Constructing Uncertainty Sets for Robust Risk Measures: A Composition of $ϕ$-Divergences Approach to Combat Tail Uncertainty

Risk measures, which typically evaluate the impact of extreme losses, are highly sensitive to misspecification in the tails. This paper studies a robust optimization approach to combat tail uncertainty by proposing a unifying framework to construct uncertainty sets for a broad class of risk measures, given a specified nominal model. Our framework is based on a parametrization of robust risk measures using two (or multiple) $ϕ$-divergence functions, which enables us to provide uncertainty sets that are tailored to both the sensitivity of each risk measure to tail losses and the tail behavior of the nominal distribution. In addition, our formulation allows for a tractable computation of robust risk measures, and elicitation of $ϕ$-divergences that describe a decision maker's risk and ambiguity preferences.

math.OC↗

arXiv2024

Counterfactual Explanations for Linear Optimization

The concept of counterfactual explanations (CE) has emerged as one of the important concepts to understand the inner workings of complex AI systems. In this paper, we translate the idea of CEs to linear optimization and propose, motivate, and analyze three different types of CEs: strong, weak, and relative. While deriving strong and weak CEs appears to be computationally intractable, we show that calculating relative CEs can be done efficiently. By detecting and exploiting the hidden convex structure of the optimization problem that arises in the latter case, we show that obtaining relative CEs can be done in the same magnitude of time as solving the original linear optimization problem. This is confirmed by an extensive numerical experiment study on the NETLIB library.

math.OC↗

arXiv2024

From Large Language Models and Optimization to Decision Optimization CoPilot: A Research Manifesto

Significantly simplifying the creation of optimization models for real-world business problems has long been a major goal in applying mathematical optimization more widely to important business and societal decisions. The recent capabilities of Large Language Models (LLMs) present a timely opportunity to achieve this goal. Therefore, we propose research at the intersection of LLMs and optimization to create a Decision Optimization CoPilot (DOCP) - an AI tool designed to assist any decision maker, interacting in natural language to grasp the business problem, subsequently formulating and solving the corresponding optimization model. This paper outlines our DOCP vision and identifies several fundamental requirements for its implementation. We describe the state of the art through a literature survey and experiments using ChatGPT. We show that a) LLMs already provide substantial novel capabilities relevant to a DOCP, and b) major research challenges remain to be addressed. We also propose possible research directions to overcome these gaps. We also see this work as a call to action to bring together the LLM and optimization communities to pursue our vision, thereby enabling much more widespread improved decision-making.

cs.AI↗

arXiv2023

Mixed-Integer Optimization with Constraint Learning

We establish a broad methodological foundation for mixed-integer optimization with learned constraints. We propose an end-to-end pipeline for data-driven decision making in which constraints and objectives are directly learned from data using machine learning, and the trained models are embedded in an optimization formulation. We exploit the mixed-integer optimization-representability of many machine learning methods, including linear models, decision trees, ensembles, and multi-layer perceptrons, which allows us to capture various underlying relationships between decisions, contextual variables, and outcomes. We also introduce two approaches for handling the inherent uncertainty of learning from data. First, we characterize a decision trust region using the convex hull of the observations, to ensure credible recommendations and avoid extrapolation. We efficiently incorporate this representation using column generation and propose a more flexible formulation to deal with low-density regions and high-dimensional datasets. Then, we propose an ensemble learning approach that enforces constraint satisfaction over multiple bootstrapped estimators or multiple algorithms. In combination with domain-driven components, the embedded models and trust region define a mixed-integer optimization problem for prescription generation. We implement this framework as a Python package (OptiCL) for practitioners. We demonstrate the method in both World Food Programme planning and chemotherapy optimization. The case studies illustrate the framework's ability to generate high-quality prescriptions as well as the value added by the trust region, the use of ensembles to control model robustness, the consideration of multiple machine learning methods, and the inclusion of multiple learned constraints.

math.OC↗

arXiv2023

Finding Regions of Counterfactual Explanations via Robust Optimization

Counterfactual explanations play an important role in detecting bias and improving the explainability of data-driven classification models. A counterfactual explanation (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 logistic regression, decision trees, random forests, 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.

cs.LG↗

arXiv2023

Robust Upper Bounds for Adversarial Training

Many state-of-the-art adversarial training methods for deep learning leverage upper bounds of the adversarial loss to provide security guarantees against adversarial attacks. Yet, these methods rely on convex relaxations to propagate lower and upper bounds for intermediate layers, which affect the tightness of the bound at the output layer. We introduce a new approach to adversarial training by minimizing an upper bound of the adversarial loss that is based on a holistic expansion of the network instead of separate bounds for each layer. This bound is facilitated by state-of-the-art tools from Robust Optimization; it has closed-form and can be effectively trained using backpropagation. We derive two new methods with the proposed approach. The first method (Approximated Robust Upper Bound or aRUB) uses the first order approximation of the network as well as basic tools from Linear Robust Optimization to obtain an empirical upper bound of the adversarial loss that can be easily implemented. The second method (Robust Upper Bound or RUB), computes a provable upper bound of the adversarial loss. Across a variety of tabular and vision data sets we demonstrate the effectiveness of our approach -- RUB is substantially more robust than state-of-the-art methods for larger perturbations, while aRUB matches the performance of state-of-the-art methods for small perturbations.

cs.LG↗

arXiv2022

Optimization with Constraint Learning: A Framework and Survey

Many real-life optimization problems frequently contain one or more constraints or objectives for which there are no explicit formulas. If data is however available, these data can be used to learn the constraints. The benefits of this approach are clearly seen, however there is a need for this process to be carried out in a structured manner. This paper therefore provides a framework for Optimization with Constraint Learning (OCL) which we believe will help to formalize and direct the process of learning constraints from data. This framework includes the following steps: (i) setup of the conceptual optimization model, (ii) data gathering and preprocessing, (iii) selection and training of predictive models, (iv) resolution of the optimization model, and (v) verification and improvement of the optimization model. We then review the recent OCL literature in light of this framework, and highlight current trends, as well as areas for future research.

cs.LG↗

arXiv2022

Pareto Adaptive Robust Optimality via a Fourier-Motzkin Elimination Lens

We formalize the concept of Pareto Adaptive Robust Optimality (PARO) for linear Adaptive Robust Optimization (ARO) problems. A worst-case optimal solution pair of here-and-now decisions and wait-and-see decisions is PARO if it cannot be Pareto dominated by another solution, i.e., there does not exist another such pair that performs at least as good in all scenarios in the uncertainty set and strictly better in at least one scenario. We argue that, unlike PARO, extant solution approaches -- including those that adopt Pareto Robust Optimality from static robust optimization -- could fail in ARO and yield solutions that can be Pareto dominated. The latter could lead to inefficiencies and suboptimal performance in practice. We prove the existence of PARO solutions, and present particular approaches for finding and approximating such solutions. We present numerical results for a facility location problem that demonstrate the practical value of PARO solutions. Our analysis of PARO relies on an application of Fourier-Motzkin Elimination as a proof technique. We demonstrate how this technique can be valuable in the analysis of ARO problems, besides PARO. In particular, we employ it to devise more concise and more insightful proofs of known results on (worst-case) optimality of decision rule structures.

math.OC↗

arXiv2021

Adjustable robust treatment-length optimization in radiation therapy

Traditionally, optimization of radiation therapy (RT) treatment plans has been done before the initiation of RT course, using population-wide estimates for patients' response to therapy. However, recent technological advancements have enabled monitoring individual patient response during the RT course, in the form of biomarkers. Although biomarker data remains subject to substantial uncertainties, information extracted from this data may allow the RT plan to be adapted in a biologically informative way. We present a mathematical framework that optimally adapts the treatment-length of an RT plan based on the acquired mid-treatment biomarker information, while accounting for the inexact nature of this information. We formulate the adaptive treatment-length optimization problem as a 2-stage problem, wherein the information about the model parameters gathered during the first stage influences the decisions in the second stage. Using Adjustable Robust Optimization (ARO) techniques we derive explicit optimal decision rules for the stage-2 decisions and solve the optimization problem. The problem allows for multiple worst-case optimal solutions. To discriminate between these, we introduce the concept of Pareto Adjustable Robustly Optimal (PARO) solutions. In numerical experiments using lung cancer patient data, the ARO method is benchmarked against several other static and adaptive methods. In case of exact biomarker information, there is sufficient space to adapt, and numerical results show that taking into account both robustness and adaptability is not necessary. In case of inexact biomarker information, accounting for adaptability and inexactness of biomarker information is particularly beneficial when robustness (w.r.t. organ-at-risk (OAR) constraint violations) is of high importance. If minor OAR violations are allowed, a nominal folding horizon approach is a good performing alternative.

physics.med-ph↗

arXiv2020

MAD dispersion measure makes extremal queue analysis simple

A notorious problem in queueing theory is to compute the worst possible performance of the GI/G/1 queue under mean-dispersion constraints for the interarrival and service time distributions. We address this extremal queue problem by measuring dispersion in terms of Mean Absolute Deviation (MAD) instead of variance, making available recently developed techniques from Distributionally Robust Optimization (DRO). Combined with classical random walk theory, we obtain explicit expressions for the extremal interarrival time and service time distributions, and hence the best possible upper bounds, for all moments of the waiting time. {We also apply the DRO techniques to obtain tight lower bounds that together with the upper bounds provide robust performance intervals. We show that all bounds are computationally tractable and remain sharp, also when the mean and MAD are not known precisely, but estimated based on available data instead.

math.PR↗

arXiv2020

Improving Tractability of Real-Time Control Schemes via Simplified $\mathcal{S}$-Lemma

Various control schemes rely on a solution of a convex optimization problem involving a particular robust quadratic constraint, which can be reformulated as a linear matrix inequality using the well-known $\mathcal{S}$-lemma. However, the computational effort required to solve the resulting semidefinite program may be prohibitively large for real-time applications requiring a repeated solution of such a problem. We use some recent advances in robust optimization that allow us to reformulate such a robust constraint as a set of linear and second-order cone constraints, which are computationally better suited to real-time applications. A numerical example demonstrates a huge speedup that can be obtained using the proposed reformulation.

math.OC↗

arXiv2020

Tight tail probability bounds for distribution-free decision making

Chebyshev's inequality provides an upper bound on the tail probability of a random variable based on its mean and variance. While tight, the inequality has been criticized for only being attained by pathological distributions that abuse the unboundedness of the underlying support and are not considered realistic in many applications. We provide alternative tight lower and upper bounds on the tail probability given a bounded support, mean and mean absolute deviation of the random variable. We obtain these bounds as exact solutions to semi-infinite linear programs. We leverage the bounds for distribution-free analysis of the newsvendor model, monopolistic pricing, and stop-loss reinsurance. We also exploit the bounds for safe approximations of sums of correlated random variables, and to find convex reformulations of single and joint ambiguous chance constraints that are ubiquitous in distributionally robust optimization.

math.OC↗

arXiv2019

Extending the Scope of Robust Quadratic Optimization

We derive computationally tractable formulations of the robust counterparts of convex quadratic and conic quadratic constraints that are concave in matrix-valued uncertain parameters. We do this for a broad range of uncertainty sets. In particular, we show how to reformulate the support functions of uncertainty sets represented in terms of matrix norms and cones. Our results provide extensions to known results from the literature. We also consider hard quadratic constraints; those that are convex in uncertain matrix-valued parameters. For the robust counterpart of such constraints we derive inner and outer tractable approximations. As application, we show how to construct a natural uncertainty set based on a statistical confidence set around a sample mean vector and covariance matrix and use this to provide a tractable reformulation of the robust counterpart of an uncertain portfolio optimization problem. We also apply the results of this paper to a norm approximation and a regression line problem.

math.OC↗