Robust network optimization under polyhedral demand uncertainty is NP-hard
01 Mar 2010-Discrete Applied Mathematics (Elsevier Science Publishers B. V.)-Vol. 158, Iss: 5, pp 597-603
TL;DR: This pending complexity issue is settled for all robust network optimization problems featuring polyhedral demand uncertainty, both for the single-commodity and multicommodity case, even if the corresponding deterministic versions are polynomially solvable as regular (continuous) linear programs.
About: This article is published in Discrete Applied Mathematics.The article was published on 2010-03-01 and is currently open access. It has received 66 citations till now. The article focuses on the topics: Robust optimization & Optimization problem.
Citations
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TL;DR: An overview of developments in robust optimization since 2007 is provided to give a representative picture of the research topics most explored in recent years, highlight common themes in the investigations of independent research teams and highlight the contributions of rising as well as established researchers both to the theory of robust optimization and its practice.
742 citations
Cites background from "Robust network optimization under p..."
...Minoux (2010) proves that the robust network design problem with uncertain demand is NP-hard....
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TL;DR: The main purpose of this paper is to solve the maximum flow in an uncertain network by under the framework of uncertainty theory.
83 citations
Cites methods from "Robust network optimization under p..."
...In other words, they mainly used stochastic optimization, chance constrained programming and robust optimization to solve the maximum flow problem in an uncertain network [1,31,32,34]....
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TL;DR: A branch-and-cut algorithm based on the proposed capacity formulation of the Robust Network Loading problem with splittable flows and dynamic routing under polyhedral uncertainty for the demands is developed.
Abstract: In this paper the Robust Network Loading problem with splittable flows and dynamic routing under polyhedral uncertainty for the demands is considered. Polyhedral results for the capacity formulation of the problem are given. The first exact approach for solving the problem is presented. A branch-and-cut algorithm based on the proposed capacity formulation is developed. Computational results using the hose polyhedron to model the demand uncertainty are discussed.
68 citations
TL;DR: The class—denoted R-LP-RHSU—of two-stage robust linear programming problems with right-hand-side uncertainty with formal proof of strong NP-hardness for the general case is investigated, and polynomially solvable subclasses are exhibited.
Abstract: We investigate here the class--denoted R-LP-RHSU--of two-stage robust linear programming problems with right-hand-side uncertainty. Such problems arise in many applications e.g: robust PERT scheduling (with uncertain task durations); robust maximum flow (with uncertain arc capacities); robust network capacity expansion problems; robust inventory management; some robust production planning problems in the context of power production/distribution systems. It is shown that such problems can be formulated as large scale linear programs with associated nonconvex separation subproblem. A formal proof of strong NP-hardness for the general case is then provided, and polynomially solvable subclasses are exhibited. Differences with other previously described robust LP problems (featuring row-wise uncertainty instead of column wise uncertainty) are highlighted.
48 citations
Cites background or result from "Robust network optimization under p..."
...It has been shown in [ 27 ] that the class R-CEP-PU of robust capacity expansion problems with polyhedral uncertainty sets (i.e....
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...As compared with the latter reference, the NP-hardness proof given in [27] is more direct in the sense that it does not require the use of the result on equivalence between separation and optimization by Grotschel et al. [15].Various polynomially solvable special cases are also discussed in [ 27 ]....
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...As compared with the latter reference, the NP-hardness proof given in [ 27 ] is more direct in the sense that it does not require the use of the result on equivalence between separation and optimization by Grotschel et al. [15].Various polynomially solvable special cases are also discussed in [27]....
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TL;DR: This paper studies the following line balancing problem with uncertain operation execution times: operations on the same product have to be assigned to the stations of a transfer line, and operations assign to the same station are executed sequentially.
43 citations
References
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01 Jan 1979
42,654 citations
01 Jan 1979
3,381 citations
"Robust network optimization under p..." refers background in this paper
...The question is then: is the maximum s− t flow value≥ W for all c ∈ C ? Problem R-MAXFLOW-KCU has been shown to be strongly NP-hard in [22] based on a reduction of MIN-CUT-INTOBOUNDED-SETS (see [10], page 210)....
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TL;DR: In this paper, the authors propose an approach that attempts to make this trade-off more attractive by flexibly adjusting the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations.
Abstract: A robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended. Under this approach, we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes. A concern with such an approach is that it might be too conservative. In this paper, we propose an approach that attempts to make this trade-off more attractive; that is, we investigate ways to decrease what we call the price of robustness. In particular, we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations. An attractive aspect of our method is that the new robust formulation is also a linear optimization problem. Thus we naturally extend our methods to discrete optimization problems in a tractable way. We report numerical results for a portfolio optimization problem, a knapsack problem, and a problem from the Net Lib library.
3,364 citations
01 Jan 2004
TL;DR: An approach is proposed that flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations, and an attractive aspect of this method is that the new robust formulation is also a linear optimization problem, so it naturally extend to discrete optimization problems in a tractable way.
Abstract: A robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended. Under this approach, we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes. A concern with such an approach is that it might be too conservative. In this paper, we propose an approach that attempts to make this trade-off more attractive; that is, we investigate ways to decrease what we call the price of robustness. In particular, we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations. An attractive aspect of our method is that the new robust formulation is also a linear optimization problem. Thus we naturally extend our methods to discrete optimization problems in a tractable way. We report numerical results for a portfolio optimization problem, a knapsack problem, and a problem from the Net Lib library.
3,359 citations
"Robust network optimization under p..." refers methods in this paper
...Also we note that knapsack-constrained uncertainty sets typically correspond to the uncertainty model investigated by Bertsimas and Sim [7,8], though in a quite different context: indeed their approach concerns robust LP (or MILP) problems with row-wise uncertainty, whereas we address here LP problems with right-hand side uncertainty (a special case of columnwise uncertainty)....
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Book•
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03 Sep 2011
TL;DR: The question the authors are trying to ask is: how many units of water can they send from the source to the sink per unit of time?
Abstract: 1 Defining Network Flow A flow network is a directed graph G = (V,E) in which each edge (u, v) ∈ E has non-negative capacity c(u, v) ≥ 0. We require that if (u, v) ∈ E, then (v, u) / ∈ E. That is, if an edge exists, then the edge between the same vertices going the reverse direction does not exist. Every flow network has a source s and a sink t, and we assume that for every v ∈ V , there is some path s→ · · · → v → · · · → t. Note that this implies that flow networks are connected. Informally, the intuition behind network flow is to think of the edges as pipes and the weights on the edges as the capacity its corresponding pipe per unit of time. The question we are trying to ask is: how many units of water can we send from the source to the sink per unit of time? Formally, a flow in G is a function f : V × V → R that satisfies the following: • Capacity constraint. For all u, v ∈ V , we require 0 ≤ f(u, v) ≤ c(u, v). Our pipe cannot hold more than is allowed as dictated by its capacity. • Flow conservation. For u ∈ V − {s, t}, we require ∑
2,426 citations