arXiv subjects

Marc Goerigk

Publications and source records attributed to Marc Goerigk.

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

Sparse Linear Surrogates for Interpretable Budget Allocation

To address the demand for inherently interpretable optimization methods, we introduce novel linear surrogates for budget allocation problems. These surrogates consist of sparse linear rules that map instances to feature-based representations of solutions. We present an exact approach based on mixed-integer programming as well as a heuristic for their computation. The performance of both approaches is analyzed through computational experiments.

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arXiv2026

A Robust Optimization Approach for Scheduling with Uncertain Start-Time Dependent Costs

In this work, we study a single-machine scheduling problem that aims at minimizing the total cost of a schedule subject to start-time dependent costs. This framework naturally captures scenarios where costs fluctuate throughout the day, such as time-varying energy or labor prices. To model more realistic scenarios, we assume that these costs lie within a budgeted uncertainty set and propose a two-stage robust optimization approach. In a first stage, the order in which activities should be executed is decided. After a cost scenario has been revealed, the starting times for each activity are established, subject to the ordering from the first stage. We demonstrate that the proposed problem is NP-hard and not approximable, implying the complexity of its robust counterpart. Furthermore, we show that already evaluating a first-stage solution is NP-hard when the uncertainty set is discrete. We develop models and solution methods for both continuous and discrete budgeted uncertainty. In computational experiments, we compare these approaches and demonstrate the advantages of including uncertainty beforehand.

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arXiv2026

The Complexity Landscape of Two-Stage Robust Selection Problems with Budgeted Uncertainty

A standard type of uncertainty set in robust optimization is budgeted uncertainty, where an interval of possible values for each parameter is given and the total deviation from their lower bounds is bounded. In the two-stage setting, discrete and continuous budgeted uncertainty have to be distinguished. The complexity of such problems is largely unexplored, in particular if the underlying nominal optimization problem is simple, such as for selection problems. In this paper, we give a comprehensive answer to long-standing open complexity questions for three types of selection problems and three types of budgeted uncertainty sets. In particular, we demonstrate that the two-stage selection problem with continuous budgeted uncertainty is NP-hard, while the corresponding two-stage representative selection problem is solvable in polynomial time. Our hardness result implies that also the two-stage assignment problem with continuous budgeted uncertainty is NP-hard.

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arXiv2025

Feature Selection for Data-driven Explainable Optimization

Mathematical optimization, although often leading to NP-hard models, is now capable of solving even large-scale instances within reasonable time. However, the primary focus is often placed solely on optimality. This implies that while obtained solutions are globally optimal, they are frequently not comprehensible to humans, in particular when obtained by black-box routines. In contrast, explainability is a standard requirement for results in Artificial Intelligence, but it is rarely considered in optimization yet. There are only a few studies that aim to find solutions that are both of high quality and explainable. In recent work, explainability for optimization was defined in a data-driven manner: A solution is considered explainable if it closely resembles solutions that have been used in the past under similar circumstances. To this end, it is crucial to identify a preferably small subset of features from a presumably large set that can be used to measure instance similarity. In this work, we formally define the feature selection problem for explainable optimization and prove that its decision version is NP-complete. We introduce mathematical models for optimized feature selection. As their global solution requires significant computation time with modern mixed-integer linear solvers, we employ local heuristics. Our computational study using data that reflect real-world scenarios demonstrates that the problem can be solved practically efficiently for instances of reasonable size.

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arXiv2025

A fast approximate column-and-constraint generation method for two-stage robust mixed-integer programs

This paper presents a new column-and-constraint generation method for two-stage robust mixed-integer programs with finite uncertainty sets. Our method combines and extends speed-up techniques used in previous column-and-constraint generation methods and introduces several new techniques. In particular, it uses dual bounds for second-stage problems in order to allow a faster identification of the next promising scenario to be added to the master problem. Moreover, adaptive time limits are imposed to avoid getting stuck on particularly hard second-stage problems, and a gap propagation between master problem and second-stage problems is used to stop solving them earlier if only a given non-zero optimality gap is to be reached overall. This makes our method particularly effective for problems where solving the second-stage problem is computationally challenging. To evaluate the method's performance, we compare it to two recent column-and-constraint generation methods from the literature on two applications: a robust capacitated location routing problem and a robust integrated berth allocation and quay crane assignment and scheduling problem. The first problem features a particularly hard second stage, and we show that our method is able to solve considerably more and larger instances in a given time limit. Using the second problem, we verify the general applicability of our method, even for problems where the second stage is relatively easy.

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arXiv2025

The robust selection problem with information discovery

We explore a multiple-stage variant of the min-max robust selection problem with budgeted uncertainty that includes queries. First, one queries a subset of items and gets the exact values of their uncertain parameters. Given this information, one can then choose the set of items to be selected, still facing uncertainty on the unobserved parameters. In this paper, we study two specific variants of this problem. The first variant considers objective uncertainty and focuses on selecting a single item. The second variant considers constraint uncertainty instead, which means that some selected items may fail. We show that both problems are NP-hard in general. We also propose polynomial-time algorithms for special cases of the sets of items that can be queried. For the problem with constraint uncertainty, we also show how the objective function can be expressed as a linear program, leading to a mixed-integer linear programming reformulation for the general case. We illustrate the performance of this formulation using numerical experiments.

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arXiv2024

Towards Robust Interpretable Surrogates for Optimization

An important factor in the practical implementation of optimization models is the acceptance by the intended users. This is influenced among other factors by the interpretability of the solution process. Decision rules that meet this requirement can be generated using the framework for inherently interpretable optimization models. In practice, there is often uncertainty about the parameters of an optimization problem. An established way to deal with this challenge is the concept of robust optimization. The goal of our work is to combine both concepts: to create decision trees as surrogates for the optimization process that are more robust to perturbations and still inherently interpretable. For this purpose we present suitable models based on different variants to model uncertainty, and solution methods. Furthermore, the applicability of heuristic methods to perform this task is evaluated. Both approaches are compared with the existing framework for inherently interpretable optimization models.

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arXiv2024

An extension of Ordered Weighted Averaging over intervals with application to optimization under risk

The Ordered Weighted Averaging (OWA) operator is a traditional and commonly used criterion for aggregating discrete values of uncertain quantities. In this paper, it is shown that the discrete OWA naturally extends to the continuous case by using the concept of a distortion risk measure. It is shown how to apply the distortion risk measure to optimization problems with a linear objective function, whose coefficients are random variables with continuous distribution functions supported on intervals. The case where these coefficients are independent, uniformly distributed random variables is explored in more detail. The computational complexity of the resulting optimization problem is analyzed, and solution methods with approximation guarantees are proposed. These methods are also verified through computational experiments.

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arXiv2024

Problem-Driven Scenario Reduction and Scenario Approximation for Robust Optimization

In robust optimization, we would like to find a solution that is immunized against all scenarios that are modeled in an uncertainty set. Which scenarios to include in such a set is therefore of central importance for the tractability of the robust model and practical usefulness of the resulting solution. We consider problems with a discrete uncertainty set affecting only the objective function. Our aim is reduce the size of the uncertainty set, while staying as true as possible to the original robust problem, measured by an approximation guarantee. Previous reduction approaches ignored the structure of the set of feasible solutions in this process. We show how to achieve better uncertainty sets by taking into account what solutions are possible, providing a theoretical framework and models to this end. In computational experiments, we note that our new framework achieves better uncertainty sets than previous methods or a simple K-means approach.

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arXiv2024

Feature-Based Interpretable Surrogates for Optimization

For optimization models to be used in practice, it is crucial that users trust the results. A key factor in this aspect is the interpretability of the solution process. A previous framework for inherently interpretable optimization models used decision trees to map instances to solutions of the underlying optimization model. Based on this work, we investigate how we can use more general optimization rules to further increase interpretability and, at the same time, give more freedom to the decision-maker. The proposed rules do not map to a concrete solution but to a set of solutions characterized by common features. To find such optimization rules, we present an exact methodology using mixed-integer programming formulations as well as heuristics. We also outline the challenges and opportunities that these methods present. In particular, we demonstrate the improvement in solution quality that our approach offers compared to existing interpretable surrogates for optimization, and we discuss the relationship between interpretability and performance. These findings are supported by experiments using both synthetic and real-world data.

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arXiv2024

A new formulation for the collection and delivery problem of biomedical specimen

We study the collection and delivery problem of biomedical specimens (CDSP) with multiple trips, time windows, a homogeneous fleet, and the objective of minimizing total completion time of delivery requests. This is a prominent problem in healthcare logistics, where specimens (blood, plasma, urin etc.) collected from patients in doctor's offices and hospitals are transported to a central laboratory for advanced analysis. To the best of our knowledge, available exact solution approaches for CDSP have been able to solve only small instances with up to 9 delivery requests. In this paper, we propose a two-index mixed-integer programming formulation that, when used with an off-the-shelf solver, results in a fast exact solution approach. Computational experiments on a benchmark data set confirm that the proposed formulation outperforms both the state-of-the-art model and the state-of-the-art metaheuristic from the literature, solving 80 out of 168 benchmark instances to optimality, including a significant number of instances with 100 delivery requests.

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arXiv2023

A Framework for Data-Driven Explainability in Mathematical Optimization

Advancements in mathematical programming have made it possible to efficiently tackle large-scale real-world problems that were deemed intractable just a few decades ago. However, provably optimal solutions may not be accepted due to the perception of optimization software as a black box. Although well understood by scientists, this lacks easy accessibility for practitioners. Hence, we advocate for introducing the explainability of a solution as another evaluation criterion, next to its objective value, which enables us to find trade-off solutions between these two criteria. Explainability is attained by comparing against (not necessarily optimal) solutions that were implemented in similar situations in the past. Thus, solutions are preferred that exhibit similar features. Although we prove that already in simple cases the explainable model is NP-hard, we characterize relevant polynomially solvable cases such as the explainable shortest path problem. Our numerical experiments on both artificial as well as real-world road networks show the resulting Pareto front. It turns out that the cost of enforcing explainability can be very small.

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arXiv2023

Robust Min-Max (Regret) Optimization using Ordered Weighted Averaging

In decision-making under uncertainty, several criteria have been studied to aggregate the performance of a solution over multiple possible scenarios. This paper introduces a novel variant of ordered weighted averaging (OWA) for optimization problems. It generalizes the classic OWA approach, which includes robust min-max optimization as a special case, as well as min-max regret optimization. We derive new complexity results for this setting, including insights into the inapproximability and approximability of this problem. In particular, we provide stronger positive approximation results that asymptotically improve the previously best-known bounds for the classic OWA approach. In computational experiments, we evaluate the quality of the proposed methods and compare the proposed setting with classic OWA and min-max regret approaches.

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arXiv2023

Robust Combinatorial Optimization Problems Under Budgeted Interdiction Uncertainty

In robust combinatorial optimization, we would like to find a solution that performs well under all realizations of an uncertainty set of possible parameter values. How we model this uncertainty set has a decisive influence on the complexity of the corresponding robust problem. For this reason, budgeted uncertainty sets are often studied, as they enable us to decompose the robust problem into easier subproblems. We propose a variant of discrete budgeted uncertainty for cardinality-based constraints or objectives, where a weight vector is applied to the budget constraint. We show that while the adversarial problem can be solved in linear time, the robust problem becomes NP-hard and not approximable. We discuss different possibilities to model the robust problem and show experimentally that despite the hardness result, some models scale relatively well in the problem size.

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arXiv2023

Robust optimization with belief functions

In this paper, an optimization problem with uncertain objective function coefficients is considered. The uncertainty is specified by providing a discrete scenario set, containing possible realizations of the objective function coefficients. The concept of belief function in the traditional and possibilistic setting is applied to define a set of admissible probability distributions over the scenario set. The generalized Hurwicz criterion is then used to compute a solution. In this paper, the complexity of the resulting problem is explored. Some exact and approximation methods of solving it are proposed.

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arXiv2022

On the Complexity of Robust Multi-Stage Problems in the Polynomial Hierarchy

We study the computational complexity of multi-stage robust optimization problems. Such problems are formulated with alternating min/max quantifiers and therefore naturally fall into a higher stage of the polynomial hierarchy. Despite this, almost no hardness results with respect to the polynomial hierarchy are known. In this work, we examine the hardness of robust two-stage adjustable and robust recoverable optimization with budgeted uncertainty sets. Our main technical contribution is the introduction of a technique tailored to prove $\Sigma^p_3$-hardness of such problems. We highlight a difference between continuous and discrete budgeted uncertainty: In the discrete case, indeed a wide range of problems becomes complete for the third stage of the polynomial hierarchy; in particular, this applies to the TSP, independent set, and vertex cover problems. However, in the continuous case this does not happen and problems remain in the first stage of the hierarchy. Finally, if we allow the uncertainty to not only affect the objective, but also multiple constraints, then this distinction disappears and even in the continuous case we encounter hardness for the third stage of the hierarchy. This shows that even robust problems which are already NP-complete can still exhibit a significant computational difference between column-wise and row-wise uncertainty.

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arXiv2022

Optimal Scenario Reduction for One- and Two-Stage Robust Optimization

Robust optimization typically follows a worst-case perspective, where a single scenario may determine the objective value of a given solution. Accordingly, it is a challenging task to reduce the size of an uncertainty set without changing the resulting objective value too much. On the other hand, robust optimization problems with many scenarios tend to be hard to solve, in particular for two-stage problems. Hence, a reduced uncertainty set may be central to find solutions in reasonable time. We propose scenario reduction methods that give guarantees on the performance of the resulting robust solution. Scenario reduction problems for one- and two-stage robust optimization are framed as optimization problems that only depend on the uncertainty set and not on the underlying decision making problem. Experimental results indicate that objective values for the reduced uncertainty sets are closely correlated to original objective values, resulting in better solutions than when using general-purpose clustering methods such as K-means.

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