Robust combinatorial optimization under convex and discrete cost uncertainty

Buchheim, Christoph; Kurtz, Jannis

doi:10.1007/s13675-018-0103-0

Cited by 60 publications

(55 citation statements)

References 85 publications

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“…Let b 0 1 and b j 0 for j ∈ J; then we have a linear objective with a polyhedral uncertainty set. By theorem 4 of Buchheim and Kurtz (2018), the resulting problem is NP-hard. □…”

Section: Rfp[8] For General Polyhedral Uncertainty Is Np-hardmentioning

confidence: 99%

On Robust Fractional 0-1 Programming

Mehmanchi

Gillen

Gómez

et al. 2020

INFORMS Journal on Optimization

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We study single- and multiple-ratio robust fractional 0-1 programming problems (RFPs). In particular, this work considers RFPs under a wide range of disjoint and joint uncertainty sets, where the former implies separate uncertainty sets for each numerator and denominator and the latter accounts for different forms of interrelatedness between them. We first demonstrate that unlike the deterministic case, a single-ratio RFP is nondeterministic polynomial-time hard under general polyhedral uncertainty sets. However, if the uncertainty sets are imbued with a certain structure, variants of the well-known budgeted uncertainty, the disjoint and joint single-ratio RFPs are polynomially solvable when the deterministic counterpart is. We also propose mixed-integer linear programming (MILP) formulations for multiple-ratio RFPs. We conduct extensive computational experiments using test instances based on real and synthetic data sets to evaluate the performance of our MILP reformulations as well as to compare the disjoint and joint uncertainty sets. Finally, we demonstrate the value of the robust approach by examining the robust solution in a deterministic setting and vice versa.

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“…Let b 0 1 and b j 0 for j ∈ J; then we have a linear objective with a polyhedral uncertainty set. By theorem 4 of Buchheim and Kurtz (2018), the resulting problem is NP-hard. □…”

Section: Rfp[8] For General Polyhedral Uncertainty Is Np-hardmentioning

confidence: 99%

On Robust Fractional 0-1 Programming

Mehmanchi

Gillen

Gómez

et al. 2020

INFORMS Journal on Optimization

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“…Besides the NP-hardness for the general case the authors in [4] show that the group model projection problem can be solved in polynomial time by dynamic programming for the special case of loopless groups. Furthermore the authors show that if the intersection graph is bipartite the projection problem can be solved in polynomial time by relaxing problem (12). Similar results are obtained for the more general problem, where additional to the group-sparsity the classical K -sparsity is assumed, i.e.…”

Section: Group Modelsmentioning

confidence: 53%

“…The latter problem is a special case of the robust knapsack problem with discrete uncertainty (sometimes called robust selection problem) with additional uncertain constant. In [12] the authors mention that the problem with an uncertain constant is equivalent to the problem without such a constant. Furthermore using the result in [41] Problem (36) can be solved in O(MGC t ) where…”

Section: Lemma 4 For a Given Solutionmentioning

confidence: 99%

Discrete optimization methods for group model selection in compressed sensing

2020

Self Cite

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In this article we study the problem of signal recovery for group models. More precisely for a given set of groups, each containing a small subset of indices, and for given linear sketches of the true signal vector which is known to be group-sparse in the sense that its support is contained in the union of a small number of these groups, we study algorithms which successfully recover the true signal just by the knowledge of its linear sketches. We derive model projection complexity results and algorithms for more general group models than the state-of-the-art. We consider two versions of the classical iterative hard thresholding algorithm (IHT). The classical version iteratively calculates the exact projection of a vector onto the group model, while the approximate version (AM-IHT) uses a head-and a tail-approximation iteratively. We apply both variants to group models and analyse the two cases where the sensing matrix is a Gaussian matrix and a model expander matrix. To solve the exact projection problem on the group model, which is known to be equivalent to the maximum weight coverage problem, we use discrete optimization methods based on dynamic programming and Benders' decomposition. The head-and tail-approximations are derived by a classical greedy-method and LP-rounding, respectively.

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“…Single-stage robust combinatorial optimization problems, under various uncertainty sets, have been extensively discussed over the last decade. Survey of the results in this area can be found in [2,24,20,10]. For these problems a complete solution must be determined before the true scenario is revealed.…”

Section: Introductionmentioning

confidence: 99%

Two-stage combinatorial optimization problems under risk

Goerigk

Kasperski

Zieliński

2020

Theoretical Computer Science

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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.Typically, the first stage costs are known while the second stage costs can only be predicted to belong to an uncertainty set U . First such models were discussed in [16,18,26,23], where the robust two-stage spanning tree and perfect matching problems were considered. In these papers, the uncertainty set U contains K explicitly listed scenarios. Several negative and positive complexity results for this uncertainty representation were established. Some of them have been recently extended in [19], where also the robust two-stage shortest path problem has been investigated. In [25] and [13] the robust two-stage selection problem has been explored. The problem is NP-hard for discrete uncertainty representation but it is polynomially solvable under a special case of polyhedral uncertainty set, called continuous budgeted uncertainty (see [13]).Robust two-stage problems belong to the class of three-level, min-max-min optimization problems. In mathematical programming, this approach is also called adjustable robustness (see, e.g. [5,31]). Namely, some variables must be determined before the realization of the uncertain parameters, while the other part are variables that can be chosen after the realization. Several such models have been recently considered in combinatorial optimization, which can be represented as a 0-1 programming problem. Among them there is the robust two-stage problem discussed in this paper, but also the robust recoverable models [11,12] and the k-adaptability approach [9]. In general, problems of this type can be hard to solve exactly. A standard approach is to apply row and column generation techniques, which consists in solving a sequence of MIP formulations (see, e.g., [32]). However, this method can be inefficient for larger problems, especially when the underlying deterministic problem is already NP-hard. Therefore, some faster approximation algorithms can be useful in this case.In this paper we consider the class of robust two-stage combinatorial problems under convex uncertainty, i.e. when the uncertainty set U is convex. Important special cases are polyhedral and ellipsoidal uncertainty, which are widely used in single-stage robust optimization. Notice that in the problems discussed in [16,18,26,23], U contains a fixed number of scenarios, so it is not a convex set. The problem formulation and description of the uncertainty sets are provided in Section 2. The complexity status of ba...

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Robust combinatorial optimization under convex and discrete cost uncertainty

Cited by 60 publications

References 85 publications

On Robust Fractional 0-1 Programming

On Robust Fractional 0-1 Programming

Discrete optimization methods for group model selection in compressed sensing

Two-stage combinatorial optimization problems under risk

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