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Affine recourse for the robust network design problem: Between static and dynamic routing

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TLDR
It is shown that affine routing can be seen as a generalization of the widely used static routing while still being tractable and providing cheaper solutions and that for these instances the optimal solutions based on affine routings tend to be as cheap as optimal network designs for dynamic routings.
Abstract
Affinely Adjustable Robust Counterparts provide tractable alternatives to (two-stage) robust programs with arbitrary recourse. Following Ouorou and Vial, we apply them to robust network design with polyhedral demand uncertainty, introducing the notion of affine routing. We compare the new affine routing scheme to the well-studied static and dynamic routing schemes for robust network design. It is shown that affine routing can be seen as a generalization of the widely used static routing while still being tractable and providing cheaper solutions. We investigate properties of the demand polytope under which affine routings reduce to static routings and also develop conditions on the uncertainty set leading to dynamic routings being affine. We show however that affine routings suffer from the drawback that (even totally) dominated demand vectors are not necessarily supported by affine solutions. Uncertainty sets have to be designed accordingly. Finally, we present computational results on networks from SNDlib. We conclude that for these instances the optimal solutions based on affine routings tend to be as cheap as optimal network designs for dynamic routings. In this respect the affine routing principle can be used to approximate the cost for two-stage solutions with free recourse which are hard to compute

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Journal ArticleDOI

The robust vehicle routing problem with time windows

TL;DR: This paper addresses the robust vehicle routing problem with time windows by proposing two new formulations for the robust problem, each based on a different robust approach, and develops a new cutting plane technique for robust combinatorial optimization problems with complicated constraints.

A cutting plane algorithm for multicomoodity survivable network design problems

Geir Dahl, +1 more
TL;DR: A cutting plane algorithm is presented for solving the following telecommunications network design problem: given point-to-point traffic demands in a network, specified survivability requirements and a discrete cost/capacity function for each link, find minimum cost capacity expansions satisfying the given demands.
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Robust and Adaptive Network Flows

TL;DR: This paper shows that the robust maximum flow problem can be solved in polynomial time, but the robust minimum cut problem is NP-hard, and proves that the adaptive versions are NP- hard.
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Decomposition for adjustable robust linear optimization subject to uncertainty polytope

TL;DR: A general decomposition framework to solve exactly adjustable robust linear optimization problems subject to polytope uncertainty and shows that the relative performance of the algorithms depend on whether the budget is integer or fractional.
Journal ArticleDOI

A copositive approach for two-stage adjustable robust optimization with uncertain right-hand sides

TL;DR: In this article, the affine policy is reformulated as a copositive optimization problem, which leads to a class of tractable, semidefinite-based approximations.
References
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Journal ArticleDOI

On cut-based inequalities for capacitated network design polyhedra

TL;DR: This article unify and extend polyhedral results for directed, bidirected, and undirected link capacity models, and presents a new class of facet-defining inequalities, showing as well that flow-cutset inequalities alone do not suffice to give a complete description for single-commodity, single-module cutset polyhedra in the Bidirected and Undirected case.
Journal ArticleDOI

Robust network design: Formulations, valid inequalities, and computations

TL;DR: This article considers telecommunication network design under traffic uncertainty, adapting the robust optimization approach of Bertsimas and Sim, and presents two different mathematical formulations, which provide valid inequalities, study the computational implications, and evaluate the realized robustness.
Journal IssueDOI

Provisioning virtual private networks under traffic uncertainty

TL;DR: This work investigates a network design problem under traffic uncertainty that arises when provisioning Virtual Private Networks (VPNs), and presents compact linear mixed-integer programming formulations for the problem with the classical hose traffic model and for a less conservative robust variant relying on the traffic statistics that are often available.
Journal IssueDOI

Hardness of robust network design

TL;DR: The authors settle the complexity status of the robust network design problem in undirected graphs with a single-source version of the problem where the flow-cut gap is known to be one and shows that this restricted problem is coNP-Hard.

On an extension of the maximum-flow minimum-cut theorem to multicommo))ity flows

Masao Iri
TL;DR: In this article, a simple proof to Onaga's condition is given based on the duality theorem in linear programming, where some improvement will be made also on the statement of the condition itself.
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