arXiv subjects

Marc Goerigk

Publications and source records attributed to Marc Goerigk.

At least 55 records · Page 3Linked to original sources

arXiv2020

A compact reformulation of the two-stage robust resource-constrained project scheduling problem

This paper considers the resource-constrained project scheduling problem with uncertain activity durations. We assume that activity durations lie in a budgeted uncertainty set, and follow a robust two-stage approach, where a decision maker must resolve resource conflicts subject to the problem uncertainty, but can determine activity start times after the uncertain activity durations become known. We introduce a new reformulation of the second-stage problem, which enables us to derive a compact robust counterpart to the full two-stage adjustable robust optimisation problem. Computational experiments show that this compact robust counterpart can be solved using standard optimisation software significantly faster than the current state-of-the-art algorithm for solving this problem, reaching optimality for almost 50% more instances on the same benchmark set.

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arXiv2020

An Efficient Approach to Distributionally Robust Network Capacity Planning

In this paper, we consider a network capacity expansion problem in the context of telecommunication networks, where there is uncertainty associated with the expected traffic demand. We employ a distributionally robust stochastic optimization (DRSO) framework where the ambiguity set of the uncertain demand distribution is constructed using the moments information, the mean and variance. The resulting DRSO problem is formulated as a bilevel optimization problem. We develop an efficient solution algorithm for this problem by characterizing the resulting worst-case two-point distribution, which allows us to reformulate the original problem as a convex optimization problem. In computational experiments the performance of this approach is compared to that of the robust optimization approach with a discrete uncertainty set. The results show that solutions from the DRSO model outperform the robust optimization approach on highly risk-averse performance metrics, whereas the robust solution is better on the less risk-averse metric.

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arXiv2020

Particle Swarm Metaheuristics for Robust Optimisation with Implementation Uncertainty

We consider global non-convex optimisation problems under uncertainty. In this setting, it is not possible to implement a desired solution exactly. Instead, any other solution within some distance to the intended solution may be implemented. The aim is to find a robust solution, i.e., one where the worst possible solution nearby still performs as well as possible. Problems of this type exhibit another maximisation layer to find the worst case solution within the minimisation level of finding a robust solution, which makes them harder to solve than classic global optimisation problems. So far, only few methods have been provided that can be applied to black-box problems with implementation uncertainty. We improve upon existing techniques by introducing a novel particle swarm based framework which adapts elements of previous approaches, combining them with new features in order to generate more effective techniques. In computational experiments, we find that our new method outperforms state of the art comparator heuristics in almost 80% of cases.

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arXiv2019

The Multi-level Bottleneck Assignment Problem: Complexity and Solution Methods

We study the multi-level bottleneck assignment problem (MBA), which has important applications in scheduling and quantitative finance. Given a weight matrix, the task is to rearrange entries in each column such that the maximum sum of values in each row is as small as possible. We analyze the complexity of this problem in a generalized setting, where there are restrictions in how values in columns can be permuted. We present a lower bound on its approximability by giving a non-trivial gap reduction from three-dimensional matching to MBA. To solve MBA, a greedy method has been used in the literature. We present new solution methods based on an extension of the greedy method, an integer programming formulation, and a column generation heuristic. In computational experiments we show that it is possible to outperform the standard greedy approach by around 10% on random instances.

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arXiv2019

Min-Max-Min Robustness for Combinatorial Problems with Discrete Budgeted Uncertainty

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 Gamma 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 NPhard 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 knapsack and shortest path instances previously used in the scientific literature and find that the heuristics obtain good quality solutions in short computational times.

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arXiv2019

Flexible here-and-now decisions for two-stage multi-objective optimization: Method and application to energy system design selection

The synthesis of energy systems is a two-stage optimization problem where design decisions have to be implemented here-and-now (first stage), while for the operation of installed components, we can wait-and-see (second stage). To identify a sustainable design, we need to account for both economical and environmental criteria leading to multi-objective optimization problems. However, multi-objective optimization leads not to one optimal design but to multiple Pareto-efficient design options in general. Thus, the decision maker usually has to decide manually which design should finally be implemented. In this paper, we propose the flexible here-and-now decision (flex-hand) approach for automatic identification of one single design for multi-objective optimization. The approach minimizes the distance of the Pareto front based on one fixed design to the Pareto front allowing multiple designs. Uncertainty regarding parameters of future operations can be easily included through a robust extension of the flex-hand approach. Results of a real-world case study show that the obtained design is highly flexible to adapt operation to the considered objective functions. Thus, the design provides an energy system with the ability to adapt to a changing focus in decision criteria, e. g., due to changing political aims.

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arXiv2019

Robust two-stage combinatorial optimization problems under convex uncertainty

In this paper a class of robust two-stage combinatorial optimization problems is discussed. It is assumed that the uncertain second stage costs are specified in the form of a convex uncertainty set, in particular polyhedral or ellipsoidal ones. It is shown that the robust two-stage versions of basic network and selection problems are NP-hard, even in a very restrictive cases. Some exact and approximation algorithms for the general problem are constructed. Polynomial and approximation algorithms for the robust two-stage versions of basic problems, such as the selection and shortest path problems, are also provided.

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arXiv2019

Faster Algorithms for Min-max-min Robustness for Combinatorial Problems with Budgeted Uncertainty

We consider robust combinatorial optimization problems where the decision maker can react to a scenario by choosing from a finite set of $k$ solutions. This approach is appropriate for decision problems under uncertainty where the implementation of decisions requires preparing the ground. We focus on the case that the set of possible scenarios is described through a budgeted uncertainty set and provide three algorithms for the problem. The first algorithm solves heuristically the dualized problem, a non-convex mixed-integer non-linear program (MINLP), via an alternating optimization approach. The second algorithm solves the MINLP exactly for $k=2$ through a dedicated spatial branch-and-bound algorithm. The third approach enumerates $k$-tuples, relying on strong bounds to avoid a complete enumeration. We test our methods on shortest path instances that were used in the previous literature and on randomly generated knapsack instances, and find that our methods considerably outperform previous approaches. Many instances that were previously not solved within hours can now be solved within few minutes, often even faster.

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arXiv2019

Generating Hard Instances for Robust Combinatorial Optimization

While research in robust optimization has attracted considerable interest over the last decades, its algorithmic development has been hindered by several factors. One of them is a missing set of benchmark instances that make algorithm performance better comparable, and makes reproducing instances unnecessary. Such a benchmark set should contain hard instances in particular, but so far, the standard approach to produce instances has been to sample values randomly from a uniform distribution. In this paper we introduce a new method to produce hard instances for min-max combinatorial optimization problems, which is based on an optimization model itself. Our approach does not make any assumptions on the problem structure and can thus be applied to any combinatorial problem. Using the Selection and Traveling Salesman problems as examples, we show that it is possible to produce instances which are up to 500 times harder to solve for a mixed-integer programming solver than the current state-of-the-art instances.

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arXiv2019

Mixed Uncertainty Sets for Robust Combinatorial Optimization

In robust optimization, the uncertainty set is used to model all possible outcomes of uncertain parameters. In the classic setting, one assumes that this set is provided by the decision maker based on the data available to her. Only recently it has been recognized that the process of building useful uncertainty sets is in itself a challenging task that requires mathematical support. In this paper, we propose an approach to go beyond the classic setting, by assuming multiple uncertainty sets to be prepared, each with a weight showing the degree of belief that the set is a "true" model of uncertainty. We consider theoretical aspects of this approach and show that it is as easy to model as the classic setting. In an extensive computational study using a shortest path problem based on real-world data, we auto-tune uncertainty sets to the available data, and show that with regard to out-sample performance, the combination of multiple sets can give better results than each set on its own.

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arXiv2019

A Comparison of Models for Uncertain Network Design

To solve a real-world problem, the modeler usually needs to make a trade-off between model complexity and usefulness. This is also true for robust optimization, where a wide range of models for uncertainty, so-called uncertainty sets, have been proposed. However, while these sets have been mainly studied from a theoretical perspective, there is little research comparing different sets regarding their usefulness for a real-world problem. In this paper we consider a network design problem in a telecommunications context. We need to invest into the infrastructure, such that there is sufficient capacity for future demand which is not known with certainty. There is a penalty for an unsatisfied realized demand, which needs to be outsourced. We consider three approaches to model demand: using a discrete uncertainty set, using a polyhedral uncertainty set, and using the mean of a per-commodity fitted zero-inflated uniform distribution. While the first two models are used as part of a robust optimization setting, the last model represents a simple stochastic optimization setting. We compare these approaches on an efficiency frontier real-world data taken from the online library SNDlib and observe that, contrary to current research trends, robust optimization using the polyhedral uncertainty set may result in less efficient solutions.

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arXiv2018

Two-stage Combinatorial Optimization Problems under Risk

In this paper a class of combinatorial optimization problems is discussed. It is assumed that a solution can be constructed in two stages. The current first-stage costs are precisely known, while the future second-stage costs are only known to belong to an uncertainty set, which contains a finite number of scenarios with known probability distribution. A partial solution, chosen in the first stage, can be completed by performing an optimal recourse action, after the true second-stage scenario is revealed. A solution minimizing the Conditional Value at Risk (CVaR) measure is computed. Since expectation and maximum are boundary cases of CVaR, the model generalizes the traditional stochastic and robust two-stage approaches, previously discussed in the existing literature. In this paper some new negative and positive results are provided for basic combinatorial optimization problems such as the selection or network problems.

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arXiv2018

A Largest Empty Hypersphere Metaheuristic for Robust Optimisation with Implementation Uncertainty

We consider box-constrained robust optimisation problems with implementation uncertainty. In this setting, the solution that a decision maker wants to implement may become perturbed. The aim is to find a solution that optimises the worst possible performance over all possible perturbances. Previously, only few generic search methods have been developed for this setting. We introduce a new approach for a global search, based on placing a largest empty hypersphere. We do not assume any knowledge on the structure of the original objective function, making this approach also viable for simulation-optimisation settings. In computational experiments we demonstrate a strong performance of our approach in comparison with state-of-the-art methods, which makes it possible to solve even high-dimensional problems.

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arXiv2018

Approximating multiobjective combinatorial optimization problems with the OWA criterion

The paper deals with a multiobjective combinatorial optimization problem with $K$ linear cost functions. The popular Ordered Weighted Averaging (OWA) criterion is used to aggregate the cost functions and compute a solution. It is well known that minimizing OWA for most basic combinatorial problems is weakly NP-hard even if the number of objectives $K$ equals two, and strongly NP-hard when $K$ is a part of the input. In this paper, the problem with nonincreasing weights in the OWA criterion and a large $K$ is considered. A method of reducing the number of objectives by appropriately aggregating the objective costs before solving the problem is proposed. It is shown that an optimal solution to the reduced problem has a guaranteed worst-case approximation ratio. Some new approximation results for the Hurwicz criterion, which is a special case of OWA, are also presented.

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arXiv2018

Constructing Representative Scenarios to Approximate Robust Combinatorial Optimization Problems

In robust combinatorial optimization with discrete uncertainty, two general approximation algorithms are frequently used, which are both based on constructing a single scenario representing the whole uncertainty set. In the midpoint method, one optimizes for the average case scenario. In the element-wise worst-case approach, one constructs a scenario by taking the worst case in each component over all scenarios. Both methods are known to be $N$-approximations, where $N$ is the number of scenarios. In this paper, these results are refined by reconsidering their respective proofs as optimization problems. We present a linear program to construct a representative scenario for the uncertainty set, which guarantees an approximation guarantee that is at least as good as for the previous methods. Incidentally, we show that the element-wise worst-case approach can have an advantage over the midpoint approach if the number of scenarios is large. In numerical experiments on the selection problem we demonstrate that our approach can improve the approximation guarantee of the midpoint approach by around 20%.

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arXiv2018

Algorithms and Uncertainty Sets for Data-Driven Robust Shortest Path Problems

We consider robust shortest path problems, where the aim is to find a path that optimizes the worst-case performance over an uncertainty set containing all relevant scenarios for arc costs. The usual approach for such problems is to assume this uncertainty set given by an expert who can advise on the shape and size of the set. Following the idea of data-driven robust optimization, we instead construct a range of uncertainty sets from the current literature based on real-world traffic measurements provided by the City of Chicago. We then compare the performance of the resulting robust paths within and outside the sample, which allows us to draw conclusions what the most suited uncertainty set is. Based on our experiments, we then focus on ellipsoidal uncertainty sets, and develop a new solution algorithm that significantly outperforms a state-of-the-art solver.

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arXiv2017

An Experimental Comparison of Uncertainty Sets for Robust Shortest Path Problems

Through the development of efficient algorithms, data structures and preprocessing techniques, real-world shortest path problems in street networks are now very fast to solve. But in reality, the exact travel times along each arc in the network may not be known. This lead to the development of robust shortest path problems, where all possible arc travel times are contained in a so-called uncertainty set of possible outcomes. Research in robust shortest path problems typically assumes this set to be given, and provides complexity results as well as algorithms depending on its shape. However, what can actually be observed in real-world problems are only discrete raw data points. The shape of the uncertainty is already a modelling assumption. In this paper we test several of the most widely used assumptions on the uncertainty set using real-world traffic measurements provided by the City of Chicago. We calculate the resulting different robust solutions, and evaluate which uncertainty approach is actually reasonable for our data. This anchors theoretical research in a real-world application and allows us to point out which robust models should be the future focus of algorithmic development.

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arXiv2017

On Recoverable and Two-Stage Robust Selection Problems with Budgeted Uncertainty

In this paper the problem of selecting $p$ out of $n$ available items is discussed, such that their total cost is minimized. We assume that costs are not known exactly, but stem from a set of possible outcomes. Robust recoverable and two-stage models of this selection problem are analyzed. In the two-stage problem, up to $p$ items is chosen in the first stage, and the solution is completed once the scenario becomes revealed in the second stage. In the recoverable problem, a set of $p$ items is selected in the first stage, and can be modified by exchanging up to $k$ items in the second stage, after a scenario reveals. We assume that uncertain costs are modeled through bounded uncertainty sets, i.e., the interval uncertainty sets with an additional linear (budget) constraint, in their discrete and continuous variants. Polynomial algorithms for recoverable and two-stage selection problems with continuous bounded uncertainty, and compact mixed integer formulations in the case of discrete bounded uncertainty are constructed.

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