Min–max–min robust combinatorial optimization

Buchheim, Christoph; Kurtz, Jannis

doi:10.1007/s10107-016-1053-z

Cited by 45 publications

(87 citation statements)

References 46 publications

(109 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let us also recall that RTSt Selection is polynomially solvable under U HP 0 [13]. The MIP formulations (9) and (11)…”

Section: Robust Two-stage Selection Problemmentioning

confidence: 99%

“…In the former case we always choose the cheapest tool. Hence the problem can be simplified and the MIP formulations (9) and (11) for the problem under U HP and U E , respectively, take the following form…”

Section: Robust Two-stage Rs Problemmentioning

confidence: 99%

“…Proof. The MIP formulation for the problem under U HP 0 takes the following form (see (9)): 1] j ∈ [n] π, ρ ρ ρ ≥ 0 which can be represented, equivalently, as follows min C C C T x x x + c c c T y y y + Γπ + j∈ [n] d j max{0, y j − π}…”

mentioning

confidence: 99%

See 2 more Smart Citations

Two-stage combinatorial optimization problems under risk

Goerigk

Kasperski

Zieliński

2020

Theoretical Computer Science

View full text Add to dashboard Cite

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...

show abstract

“…Let us also recall that RTSt Selection is polynomially solvable under U HP 0 [13]. The MIP formulations (9) and (11)…”

Section: Robust Two-stage Selection Problemmentioning

confidence: 99%

Section: Robust Two-stage Rs Problemmentioning

confidence: 99%

See 1 more Smart Citation

Two-stage combinatorial optimization problems under risk

Goerigk

Kasperski

Zieliński

2020

Theoretical Computer Science

View full text Add to dashboard Cite

show abstract

“…The idea of recoverable robustness, first proposed in [18], is similar to the so-called adjustable robustness in mathematical programming [5], in which the values of some decision variables can be fixed after the true parameter realization has been revealed. It is also similar to the min-max-min approach, recently discussed in [7], where a subset of solutions is computed in the first stage and one of them is adopted, when the second-stage scenario becomes known. In the recoverable approach a modification of the first-stage solution is allowed, while in the problems studied in [7] a complete solution can be chosen from a small subset of solutions chosen in the first stage.…”

Section: Introductionmentioning

confidence: 99%

“…It is also similar to the min-max-min approach, recently discussed in [7], where a subset of solutions is computed in the first stage and one of them is adopted, when the second-stage scenario becomes known. In the recoverable approach a modification of the first-stage solution is allowed, while in the problems studied in [7] a complete solution can be chosen from a small subset of solutions chosen in the first stage.…”

Section: Introductionmentioning

confidence: 99%

Robust recoverable 0–1 optimization problems under polyhedral uncertainty

Hradovich

Kasperski

Zieliński

2019

European Journal of Operational Research

View full text Add to dashboard Cite

This paper deals with a robust recoverable approach to 0-1 programming problems. It is assumed that a solution constructed in the first stage can be modified to some extent in the second stage. This modification consists in choosing a solution in some prescribed neighborhood of the current solution. The second stage solution cost can be uncertain and a polyhedral structure of uncertainty is used. The resulting robust recoverable problem is a min-max-min problem, which can be hard to solve when the number of variables is large. In this paper we provide a framework for solving robust recoverable 0-1 programming problems with a specified polyhedral uncertainty and propose several lower bounds and approximate solutions, which can be used for a wide class of 0-1 optimization problems. The results of computational tests for two problems, namely the assignment and the knapsack ones, are also presented.

show abstract

Shortest Path Problems Under Uncertainty

Ketkov¹

2022

Encyclopedia of Optimization

View full text Add to dashboard Cite

Min–max–min robust combinatorial optimization

Cited by 45 publications

References 46 publications

Two-stage combinatorial optimization problems under risk

Two-stage combinatorial optimization problems under risk

Robust recoverable 0–1 optimization problems under polyhedral uncertainty

Shortest Path Problems Under Uncertainty

Contact Info

Product

Resources

About