arXiv subjects

Marc Goerigk

Publications and source records attributed to Marc Goerigk.

At least 37 records · Page 2Linked to original sources

arXiv2023

A Framework for Inherently Interpretable Optimization Models

With dramatic improvements in optimization software, the solution of large-scale problems that seemed intractable decades ago are now a routine task. This puts even more real-world applications into the reach of optimizers. At the same time, solving optimization problems often turns out to be one of the smaller difficulties when putting solutions into practice. One major barrier is that the optimization software can be perceived as a black box, which may produce solutions of high quality, but can create completely different solutions when circumstances change leading to low acceptance of optimized solutions. Such issues of interpretability and explainability have seen significant attention in other areas, such as machine learning, but less so in optimization. In this paper we propose an optimization framework that inherently comes with an easily interpretable optimization rule, that explains under which circumstances certain solutions are chosen. Focusing on decision trees to represent interpretable optimization rules, we propose integer programming formulations as well as a heuristic method that ensure applicability of our approach even for large-scale problems. Computational experiments using random and real-world data indicate that the costs of inherent interpretability can be very small.

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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

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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arXiv2022

A faster exact method for solving the robust multi-mode resource-constrained project scheduling problem

This paper presents a mixed-integer linear programming formulation for the multi-mode resource-constrained project scheduling problem with uncertain activity durations. We consider a two-stage robust optimisation approach and find solutions that minimise the worst-case project makespan, whilst assuming that activity durations lie in a budgeted uncertainty set. Computational experiments show that this easy-to-implement formulation is many times faster than the current state-of-the-art solution approach for this problem, whilst solving over 40% more instances to optimality over the same benchmarking set.

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arXiv2022

Recoverable Robust Single Machine Scheduling with Polyhedral Uncertainty

This paper considers a recoverable robust single-machine scheduling problem under polyhedral uncertainty with the objective of minimising the total flow time. In this setting, a decision-maker must determine a first-stage schedule subject to the uncertain job processing times. Then following the realisation of these processing times, they have the option to swap the positions of up to Delta disjoint pairs of jobs to obtain a second-stage schedule. We first formulate this scheduling problem using a general recoverable robust framework, before we examine the incremental subproblem in further detail. We prove a general result for max-weight matching problems, showing that for edge weights of a specific form, the matching polytope can be fully characterised by polynomially many constraints. We use this result to derive a matching-based compact formulation for the full problem. Further analysis of the incremental problem leads to an additional assignment-based compact formulation. Computational results on budgeted uncertainty sets compare the relative strengths of the three compact models we propose.

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arXiv2022

Investigating the Recoverable Robust Single Machine Scheduling Problem Under Interval Uncertainty

We investigate the recoverable robust single machine scheduling problem under interval uncertainty. In this setting, jobs have first-stage processing times p and second-stage processing times q and we aim to find a first-stage and second-stage schedule with a minimum combined sum of completion times, such that at least Delta jobs share the same position in both schedules. We provide positive complexity results for some important special cases of this problem, as well as derive a 2-approximation algorithm to the full problem. Computational experiments examine the performance of an exact mixed-integer programming formulation and the approximation algorithm, and demonstrate the strength of a proposed polynomial time greedy heuristic.

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arXiv2022

Benchmarking Problems for Robust Discrete Optimization

Robust discrete optimization is a highly active field of research where a plenitude of combinations between decision criteria, uncertainty sets and underlying nominal problems are considered. Usually, a robust problem becomes harder to solve than its nominal counterpart, even if it remains in the same complexity class. For this reason, specialized solution algorithms have been developed. To further drive the development of stronger solution algorithms and to facilitate the comparison between methods, a set of benchmark instances is necessary but so far missing. In this paper we propose a further step towards this goal by proposing several instance generation procedures for combinations of min-max, min-max regret, two-stage and recoverable robustness with interval, discrete or budgeted uncertainty sets. Besides sampling methods that go beyond the simple uniform sampling method that is the de-facto standard to produce instances, also optimization models to construct hard instances are considered. Using a selection problem for the nominal ground problem, we are able to generate instances that are several orders of magnitudes harder to solve than uniformly sampled instances when solving them with a general mixed-integer programming solver. All instances and generator codes are made available online.

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arXiv2021

Two-Stage Robust Optimization Problems with Two-Stage Uncertainty

We consider two-stage robust optimization problems, which can be seen as games between a decision maker and an adversary. After the decision maker fixes part of the solution, the adversary chooses a scenario from a specified uncertainty set. Afterwards, the decision maker can react to this scenario by completing the partial first-stage solution to a full solution. We extend this classic setting by adding another adversary stage after the second decision-maker stage, which results in min-max-min-max problems, thus pushing two-stage settings further towards more general multi-stage problems. We focus on budgeted uncertainty sets and consider both the continuous and discrete case. For the former, we show that a wide range of robust combinatorial optimization problems can be decomposed into polynomially many subproblems, which can be solved in polynomial time for example in the case of (\textsc{representative}) \textsc{selection}. For the latter, we prove NP-hardness for a wide range of problems, but note that the special case where first- and second-stage adversarial costs are equal can remain solvable in polynomial time.

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arXiv2021

Combinatorial Optimization Problems with Balanced Regret

For decision making under uncertainty, min-max regret has been established as a popular methodology to find robust solutions. In this approach, we compare the performance of our solution against the best possible performance had we known the true scenario in advance. We introduce a generalization of this setting which allows us to compare against solutions that are also affected by uncertainty, which we call balanced regret. Using budgeted uncertainty sets, this allows for a wider range of possible alternatives the decision maker may choose from. We analyze this approach for general combinatorial problems, providing an iterative solution method and insights into solution properties. We then consider a type of selection problem in more detail and show that, while the classic regret setting with budgeted uncertainty sets can be solved in polynomial time, the balanced regret problem becomes NP-hard. In computational experiments using random and real-world data, we show that balanced regret solutions provide a useful trade-off for the performance in classic performance measures.

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arXiv2021

On the Recoverable Traveling Salesman Problem

In this paper we consider the Recoverable Traveling Salesman Problem (TSP). Here the task is to find two tours simultaneously, such that the intersection between the tours is at least a given minimum size, while the sum of travel distances with respect to two different distance metrics is minimized. Building upon the classic double-tree method, we derive a 4-approximation algorithm for the RecoverableTSP. We also show that if the required size of the intersection between the tours is constant, a 2-approximation guarantee can be achieved, even if more than two tours need to be constructed. We discuss consequences for approximability results in the more general area of recoverable robust optimization.

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arXiv2021

Data-Driven Robust Optimization using Unsupervised Deep Learning

Robust optimization has been established as a leading methodology to approach decision problems under uncertainty. To derive a robust optimization model, a central ingredient is to identify a suitable model for uncertainty, which is called the uncertainty set. An ongoing challenge in the recent literature is to derive uncertainty sets from given historical data that result in solutions that are robust regarding future scenarios. In this paper we use an unsupervised deep learning method to learn and extract hidden structures from data, leading to non-convex uncertainty sets and better robust solutions. We prove that most of the classical uncertainty classes are special cases of our derived sets and that optimizing over them is strongly NP-hard. Nevertheless, we show that the trained neural networks can be integrated into a robust optimization model by formulating the adversarial problem as a convex quadratic mixed-integer program. This allows us to derive robust solutions through an iterative scenario generation process. In our computational experiments, we compare this approach to a similar approach using kernel-based support vector clustering. We find that uncertainty sets derived by the unsupervised deep learning method find a better description of data and lead to robust solutions that outperform the comparison method both with respect to objective value and feasibility.

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arXiv2021

A Comparison of Discrete and Polyhedral Uncertainty Sets for Robust Network Design

We consider a network design and expansion problem, where we need to make a capacity investment now, such that uncertain future demand can be satisfied as closely as possible. To use a robust optimization approach, we need to construct an uncertainty set that contains all scenarios that we believe to be possible. In this paper we discuss how to construct two common types of uncertainty sets, which are discrete and polyhedral uncertainty, using real-world data. We employ clustering to generate a discrete uncertainty set, and place hyperplanes sequentially to generate a polyhedral uncertainty set. We then compare the performance of the resulting robust solutions for these two types of models on real-world data. Our results indicate that polyhedral models, while being popular in the recent network design literature, are less effective than discrete models both in terms of computational burden and solution quality with respect to the performance measure considered.

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arXiv2021

Recoverable Robust Representatives Selection Problems with Discrete Budgeted Uncertainty

Recoverable robust optimization is a multi-stage approach, where it is possible to adjust a first-stage solution after the uncertain cost scenario is revealed. We analyze this approach for a class of selection problems. The aim is to choose a fixed number of items from several disjoint sets, such that the worst-case costs after taking a recovery action are as small as possible. The uncertainty is modeled as a discrete budgeted set, where the adversary can increase the costs of a fixed number of items. While special cases of this problem have been studied before, its complexity has remained open. In this work we make several contributions towards closing this gap. We show that the problem is NP-hard and identify a special case that remains solvable in polynomial time. We provide a compact mixed-integer programming formulation and two additional extended formulations. Finally, computational results are provided that compare the efficiency of different exact solution approaches.

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arXiv2021

Multistage Robust Discrete Optimization via Quantified Integer Programming

Decision making needs to take an uncertain environment into account. Over the last decades, robust optimization has emerged as a preeminent method to produce solutions that are immunized against uncertainty. The main focus in robust discrete optimization has been on the analysis and solution of one- or two-stage problems, where the decision maker has limited options in reacting to additional knowledge gained after parts of the solution have been fixed. Due to its computational difficulty, multistage problems beyond two stages have received less attention. In this paper we argue that multistage robust discrete problems can be seen through the lens of quantified integer programs, where powerful tools to reduce the search tree size have been developed. By formulating both integer and quantified integer programming formulations, it is possible to compare the performance of state-of-the-art solvers from both worlds. Using selection, assignment, lot-sizing and knapsack problems as a testbed, we show that problems with up to nine stages can be solved to optimality in reasonable time.

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arXiv2020

Robust Optimization Approaches for Routing and Scheduling of Multi-Skilled Teams under Uncertain Job Skill Requirements

We consider a combined problem of teaming and scheduling of multi-skilled employees that have to perform jobs with uncertain qualification requirements. We propose two modeling approaches that generate solutions that are robust to possible data variations. Both approaches use variants of budgeted uncertainty, where deviations in qualification requirements are bounded by a constraint. In the first approach, we aggregate uncertain constraints to ensure that the total number of job qualifications present at a job is not less than a worst-case value. We show that these values can be computed beforehand, resulting in a robust model with little additional complexity compared with the nominal model. In our second approach, we bound the overall qualification deviation over all jobs. While this approach is more complex, we show that it is still possible to derive a compact problem formulation by using a linear programming formulation for the adversarial problem based on a dynamic program. The performance of both approaches is analyzed on a test bed of instances which were originally provided for a deterministic problem version. Our experiments show the effectiveness of the proposed approaches in the presence of data uncertainty and reveal the price and gain of robustness.

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arXiv2020

Robust Combinatorial Optimization with Locally Budgeted Uncertainty

Budgeted uncertainty sets have been established as a major influence on uncertainty modeling for robust optimization problems. A drawback of such sets is that the budget constraint only restricts the global amount of cost increase that can be distributed by an adversary. Local restrictions, while being important for many applications, cannot be modeled this way. We introduce new variant of budgeted uncertainty sets, called locally budgeted uncertainty. In this setting, the uncertain parameters become partitioned, such that a classic budgeted uncertainty set applies to each partition, called region. In a theoretical analysis, we show that the robust counterpart of such problems for a constant number of regions remains solvable in polynomial time, if the underlying nominal problem can be solved in polynomial time as well. If the number of regions is unbounded, we show that the robust selection problem remains solvable in polynomial time, while also providing hardness results for other combinatorial problems. In computational experiments using both random and real-world data, we show that using locally budgeted uncertainty sets can have considerable advantages over classic budgeted uncertainty sets.

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arXiv2020

Combinatorial two-stage minmax regret problems under interval uncertainty

In this paper a class of combinatorial optimization problems is discussed. It is assumed that a feasible solution can be constructed in two stages. In the first stage the objective function costs are known while in the second stage they are uncertain and belong to an interval uncertainty set. In order to choose a solution, the minmax regret criterion is used. Some general properties of the problem are established and results for two particular problems, namely the shortest path and the selection problem, are shown.

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arXiv2020

Automatic Generation of Algorithms for Black-Box Robust Optimisation Problems

We develop algorithms capable of tackling robust black-box optimisation problems, where the number of model runs is limited. When a desired solution cannot be implemented exactly the aim is to find a robust one, where the worst case in an uncertainty neighbourhood around a solution still performs well. This requires a local maximisation within a global minimisation. To investigate improved optimisation methods for robust problems, and remove the need to manually determine an effective heuristic and parameter settings, we employ an automatic generation of algorithms approach: Grammar-Guided Genetic Programming. We develop algorithmic building blocks to be implemented in a Particle Swarm Optimisation framework, define the rules for constructing heuristics from these components, and evolve populations of search algorithms. Our algorithmic building blocks combine elements of existing techniques and new features, resulting in the investigation of a novel heuristic solution space. As a result of this evolutionary process we obtain algorithms which improve upon the current state of the art. We also analyse the component level breakdowns of the populations of algorithms developed against their performance, to identify high-performing heuristic components for robust problems.

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