arXiv subjects

Jannis Kurtz

Publications and source records attributed to Jannis Kurtz.

At least 19 recordsLinked to original sources

arXiv2026

The complexity landscape of robust (integer) linear programming

We study the computational complexity of the decision versions of three classic robust optimization problems: static robust optimization, two-stage (adjustable) robust optimization, and $K$-adaptability. We consider that the feasibility, uncertainty, and recourse sets are polyhedra or integer-programming-representable sets, the uncertainty is decision-independent or decision-dependent, and the feasible regions are bounded or unbounded. While these problems are well-established in the current literature, we give a systematic classification that places the resulting problems in $\mathsf{P}$, $\mathsf{NP}$, $\mathsf{coNP}$, and higher levels of the polynomial hierarchy ($\Sigma_2^p$ and $\Sigma_3^p$), and, once decision-dependent uncertainty introduces quadratic constraints, in the existential theory of the reals and its hierarchy ($\exists\mathbb{R}$, $\Sigma_2\mathbb{R}$, and $\Sigma_3\mathbb{R}$) or among the undecidable problems. Beyond hardness reductions, we pay particular attention to membership proofs, establishing polynomial-size certificates even though the sets considered generally contain vectors of exponential encoding length. As a by-product, we give an alternative proof that bilevel linear optimization lies in $\mathsf{NP}$, exploiting its connection to decision-dependent robust optimization. For $K$-adaptability, we relate the problem to its static and two-stage counterparts, showing that hardness grows monotonically with $K$ and that, for fully discrete decision-dependent instances, $K$-adaptability reduces back to the static problem.

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arXiv2026

BREAD: Baseline-Referenced Explanations for Anomaly Diagnosis

Artificial Intelligence (AI)-based prospective anomaly detection methods are increasingly deployed in high-dimensional and nonlinear settings. Among these approaches, AI-based statistical process monitoring (SPM) is widely used, providing a structured framework for prospective monitoring. Once an anomaly is detected, a diagnosis method is needed to identify the features driving the flagged observation away from normal behaviour. Traditional SPM diagnosis methods are typically designed for specific detection models and cannot be directly applied to AI-based methods. Model-agnostic explainable AI (XAI) offers a general framework for feature relevance explanation. However, existing methods suffer from scalability limitations or assign relevance to noise features, reducing diagnosis accuracy. We propose a scalable, baseline-referenced diagnosis method that uses both the anomalous observation and normal baseline information. We provide mathematical guarantees that under a mean-shift anomaly setting, the proposed method achieves higher faithfulness in detecting the features causing the anomaly compared to LIME. Simulation studies and a real-world case study validate the effectiveness of the proposed method and show that it generates more faithful and accurate diagnosis results for AI-based prospective anomaly detection methods.

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arXiv2026

Globalized Adversarial Regret Optimization: Robust Decisions with Uncalibrated Predictions

Optimization problems routinely depend on uncertain parameters that must be predicted before a decision is made. Classical robust and regret formulations are designed to handle erroneous predictions and can provide statistical error bounds in simple settings. However, when predictions lack rigorous error bounds (as is typical of modern machine learning methods) classical robust models often yield vacuous guarantees, while regret formulations can paradoxically produce decisions that are more optimistic than even a nominal solution. We introduce Globalized Adversarial Regret Optimization (GARO), a decision framework that controls adversarial regret, defined as the gap between the worst-case cost and the oracle robust cost, uniformly across all possible uncertainty set sizes. By design, GARO delivers absolute or relative performance guarantees against an oracle with full knowledge of the prediction error, without requiring any probabilistic calibration of the uncertainty set. We show that GARO equipped with a relative rate function generalizes the classical adaptation method of Lepski to downstream decision problems. We derive exact tractable reformulations for problems with affine worst-case cost functions and polyhedral norm uncertainty sets, and provide a discretization and a constraint-generation algorithm with convergence guarantees for general settings. Finally, experiments demonstrate that GARO yields solutions with a more favorable trade-off between worst-case and mean out-of-sample performance, as well as stronger global performance guarantees.

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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.

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arXiv2026

Counterfactual Explanations for Integer Optimization Problems

Counterfactual explanations (CEs) offer a human-understandable way to explain decisions by identifying specific changes to the input parameters of a base or present model that would lead to a desired change in the outcome. For optimization models, CEs have primarily been studied in limited contexts and little research has been done on CEs for general integer optimization problems. In this work, we address this gap. We first show that the general problem of constructing a CE is $Σ_2^p$-complete even for binary integer programs with just a single mutable constraint. Second, we propose solution algorithms for several of the most tractable special cases: (i) mutable objective parameters, (ii) a single mutable constraint, (iii) mutable right-hand-side, and (iv) all input parameters can be modified. We evaluate our approach using classical knapsack problem instances, focusing on cases with mutable constraint parameters. Additionally, we present experiments on the resource constrained shortest path problem.

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arXiv2025

Bounding the Optimal Number of Policies for Robust K-Adaptability

In the realm of robust optimization the k-adaptability approach is one promising method to derive approximate solutions for two-stage robust optimization problems. Instead of allowing all possible second-stage decisions, the k-adaptability approach aims at calculating a limited set of k such decisions already in the first-stage before the uncertainty is revealed. The parameter k can be adjusted to control the quality of the approximation. However, not much is known on how many solutions k are needed to achieve an optimal solution for the two-stage robust problem. In this work we derive bounds on k which guarantee optimality for general non-linear problems with integer decisions where the uncertainty appears in the objective function or in the constraints. For convex uncertainty sets we show that for objective uncertainty the bound depends linearly on the dimension of the uncertainty, while for constraint uncertainty the dependence can be exponential, still providing the first generic bound for a wide class of problems. Additionally, we provide approximation guarantees if k is smaller than the derived bounds. The results give new insights on how many solutions are needed for problems as the decision dependent information discovery problem or the capital budgeting problem with constraint uncertainty. Finally, for finite uncertainty sets we show that calculating the minimal k for which k-adaptable and two-stage problems are equivalent is NP-hard and derive a greedy method which approximates this k for the case where no first-stage decisions exist.

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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.

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arXiv2025

Coherent Local Explanations for Mathematical Optimization

The surge of explainable artificial intelligence methods seeks to enhance transparency and explainability in machine learning models. At the same time, there is a growing demand for explaining decisions taken through complex algorithms used in mathematical optimization. However, current explanation methods do not take into account the structure of the underlying optimization problem, leading to unreliable outcomes. In response to this need, we introduce Coherent Local Explanations for Mathematical Optimization (CLEMO). CLEMO provides explanations for multiple components of optimization models, the objective value and decision variables, which are coherent with the underlying model structure. Our sampling-based procedure can provide explanations for the behavior of exact and heuristic solution algorithms. The effectiveness of CLEMO is illustrated by experiments for the shortest path problem, the knapsack problem, and the vehicle routing problem.

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arXiv2024

A Frank-Wolfe Algorithm for Oracle-based Robust Optimization

We tackle robust optimization problems under objective uncertainty in the oracle model, i.e., when the deterministic problem is solved by an oracle. The oracle-based setup is favorable in many situations, e.g., when a compact formulation of the feasible region is unknown or does not exist. We propose an iterative method based on a Frank-Wolfe type algorithm applied to a smoothed version of the piecewise linear objective function. Our approach bridges several previous efforts from the literature, attains the best known oracle complexity for the problem and performs better than state-of-the-art on high-dimensional problem instances, in particular for larger uncertainty sets.

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arXiv2024

Deep Learning for Two-Stage Robust Integer Optimization

Robust optimization is an established framework for modeling optimization problems with uncertain parameters. While static robust optimization is often criticized for being too conservative, two-stage (or adjustable) robust optimization (2RO) provides a less conservative alternative by allowing some decisions to be made after the uncertain parameters have been revealed. Unfortunately, in the case of integer decision variables, existing solution methods for 2RO typically fail to solve large-scale instances, limiting the applicability of this modeling paradigm to simple cases. We propose Neur2RO, a deep-learning-augmented instantiation of the column-and-constraint-generation (CCG) algorithm, which expands the applicability of the 2RO framework to large-scale instances with integer decisions in both stages. A custom-designed neural network is trained to estimate the optimal value and feasibility of the second-stage problem. The network can be incorporated into CCG, leading to more computationally tractable subproblems in each of its iterations. The resulting algorithm enjoys approximation guarantees which depend on the neural network's prediction error. In our experiments, Neur2RO produces high-quality solutions quickly, outperforming state-of-the-art methods on two-stage knapsack, capital budgeting, and facility location problems. Compared to existing methods, which often run for hours, Neur2RO finds better solutions in a few seconds or minutes. Our code is available at https://github.com/khalil-research/Neur2RO.

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arXiv2024

Neur2BiLO: Neural Bilevel Optimization

Bilevel optimization deals with nested problems in which a 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.

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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.

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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.

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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.

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arXiv2023

Approximation Algorithms for Min-max-min Robust Optimization and K-Adaptability under Objective Uncertainty

In this work we investigate the min-max-min robust optimization problem and the k-adaptability robust optimization problem for binary problems with uncertain costs. The idea of the first approach is to calculate a set of k feasible solutions which are worst-case optimal if in each possible scenario the best of the k solutions is implemented. It is known that the min-max-min robust problem can be solved efficiently if k is at least the dimension of the problem, while it is theoretically and computationally hard if k is small. However, nothing is known about the intermediate case, i.e. k lies between one and the dimension of the problem. We approach this open question and present an approximation algorithm which achieves good problem-specific approximation guarantees for the cases where k is close to or where k is a fraction of the dimension. The derived bounds can be used to show that the min-max-min robust problem is solvable in oracle-polynomial time under certain conditions even if k is smaller than the dimension. We extend the previous results to the robust k-adaptability problem. As a consequence we can provide bounds on the number of necessary second-stage policies to approximate the exact two-stage robust problem. We derive an approximation algorithm for the k-adaptability problem which has similar guarantees as for the min-max-min problem. Finally, we test both algorithms on knapsack and shortest path problems and related two-stage variants. The experiments show that both algorithms calculate solutions with relatively small optimality gap in seconds.

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arXiv2022

Data-driven Prediction of Relevant Scenarios for Robust Combinatorial Optimization

We study iterative methods for (two-stage) robust combinatorial optimization problems with discrete uncertainty. We propose a machine-learning-based heuristic to determine starting scenarios that provide strong lower bounds. To this end, we design dimension-independent features and train a Random Forest Classifier on small-dimensional instances. Experiments show that our method improves the solution process for larger instances than contained in the training set and also provides a feature importance-score which gives insights into the role of scenario properties.

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arXiv2022

Ensemble Methods for Robust Support Vector Machines using Integer Programming

In this work we study binary classification problems where we assume that our training data is subject to uncertainty, i.e. the precise data points are not known. To tackle this issue in the field of robust machine learning the aim is to develop models which are robust against small perturbations in the training data. We study robust support vector machines (SVM) and extend the classical approach by an ensemble method which iteratively solves a non-robust SVM on different perturbations of the dataset, where the perturbations are derived by an adversarial problem. Afterwards for classification of an unknown data point we perform a majority vote of all calculated SVM solutions. We study three different variants for the adversarial problem, the exact problem, a relaxed variant and an efficient heuristic variant. While the exact and the relaxed variant can be modeled using integer programming formulations, the heuristic one can be implemented by an easy and efficient algorithm. All derived methods are tested on random and realistic datasets and the results indicate that the derived ensemble methods have a much more stable behaviour when changing the protection level compared to the classical robust SVM model.

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arXiv2021

Efficient and Robust Mixed-Integer Optimization Methods for Training Binarized Deep Neural Networks

Compared to classical deep neural networks its binarized versions can be useful for applications on resource-limited devices due to their reduction in memory consumption and computational demands. In this work we study deep neural networks with binary activation functions and continuous or integer weights (BDNN). We show that the BDNN can be reformulated as a mixed-integer linear program with bounded weight space which can be solved to global optimality by classical mixed-integer programming solvers. Additionally, a local search heuristic is presented to calculate locally optimal networks. Furthermore to improve efficiency we present an iterative data-splitting heuristic which iteratively splits the training set into smaller subsets by using the k-mean method. Afterwards all data points in a given subset are forced to follow the same activation pattern, which leads to a much smaller number of integer variables in the mixed-integer programming formulation and therefore to computational improvements. Finally for the first time a robust model is presented which enforces robustness of the BDNN during training. All methods are tested on random and real datasets and our results indicate that all models can often compete with or even outperform classical DNNs on small network architectures confirming the viability for applications having restricted memory or computing power.

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