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139
Robust discrete optimization and network flows
 Mathematical Programming Series B
, 2003
"... We propose an approach to address data uncertainty for discrete optimization and network flow problems that allows controlling the degree of conservatism of the solution, and is computationally tractable both practically and theoretically. In particular, when both the cost coefficients and the data ..."
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Cited by 125 (23 self)
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We propose an approach to address data uncertainty for discrete optimization and network flow problems that allows controlling the degree of conservatism of the solution, and is computationally tractable both practically and theoretically. In particular, when both the cost coefficients and the data in the constraints of an integer programming problem are subject to uncertainty, we propose a robust integer programming problem of moderately larger size that allows controlling the degree of conservatism of the solution in terms of probabilistic bounds on constraint violation. When only the cost coefficients are subject to uncertainty and the problem is a 0 − 1 discrete optimization problem on n variables, then we solve the robust counterpart by solving at most n+1 instances of the original problem. Thus, the robust counterpart of a polynomially solvable 0−1 discrete optimization problem remains polynomially solvable. In particular, robust matching, spanning tree, shortest path, matroid intersection, etc. are polynomially solvable. We also show that the robust counterpart of an NPhard αapproximable 0 − 1 discrete optimization problem, remains αapproximable. Finally, we propose an algorithm for robust network flows that solves the robust counterpart by solving a polynomial number of nominal minimum cost flow problems in a modified network.
Convex approximations of chance constrained programs
 SIAM Journal of Optimization
, 2006
"... Abstract. We consider a chance constrained problem, where one seeks to minimize a convex objective over solutions satisfying, with a given close to one probability, a system of randomly perturbed convex constraints. This problem may happen to be computationally intractable; our goal is to build its ..."
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Cited by 74 (8 self)
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Abstract. We consider a chance constrained problem, where one seeks to minimize a convex objective over solutions satisfying, with a given close to one probability, a system of randomly perturbed convex constraints. This problem may happen to be computationally intractable; our goal is to build its computationally tractable approximation, i.e., an efficiently solvable deterministic optimization program with the feasible set contained in the chance constrained problem. We construct a general class of such convex conservative approximations of the corresponding chance constrained problem. Moreover, under the assumptions that the constraints are affine in the perturbations and the entries in the perturbation vector are independentofeachother random variables, we build a large deviationtype approximation, referred to as “Bernstein approximation, ” of the chance constrained problem. This approximation is convex and efficiently solvable. We propose a simulationbased scheme for bounding the optimal value in the chance constrained problem and report numerical experiments aimed at comparing the Bernstein and wellknown scenario approximation approaches. Finally, we extend our construction to the case of ambiguous chance constrained problems, where the random perturbations are independent with the collection of distributions known to belong to a given convex compact set rather than to be known exactly, while the chance constraint should be satisfied for every distribution given by this set.
TWOSTAGE ROBUST NETWORK FLOW AND DESIGN UNDER DEMAND UNCERTAINTY
 FORTHCOMING IN OPERATIONS RESEARCH
, 2004
"... We describe a twostage robust optimization approach for solving network flow and design problems with uncertain demand. In twostage network optimization one defers a subset of the flow decisions until after the realization of the uncertain demand. Availability of such a recourse action allows one ..."
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Cited by 36 (3 self)
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We describe a twostage robust optimization approach for solving network flow and design problems with uncertain demand. In twostage network optimization one defers a subset of the flow decisions until after the realization of the uncertain demand. Availability of such a recourse action allows one to come up with less conservative solutions compared to singlestage optimization. However, this advantage often comes at a price: twostage optimization is, in general, significantly harder than singestage optimization. For network flow and design under demand uncertainty we give a characterization of the firststage robust decisions with an exponential number of constraints and prove that the corresponding separation problem is N Phard even for a network flow problem on a bipartite graph. We show, however, that if the secondstage network topology is totally ordered or an arborescence, then the separation problem is tractable. Unlike singlestage robust optimization under demand uncertainty, twostage robust optimization allows one to control conservatism of the solutions by means of an allowed “budget for demand uncertainty.” Using a budget of uncertainty we provide an upper
Robust game theory
, 2006
"... We present a distributionfree model of incompleteinformation games, both with and without private information, in which the players use a robust optimization approach to contend with payoff uncertainty. Our “robust game” model relaxes the assumptions of Harsanyi’s Bayesian game model, and provides ..."
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Cited by 35 (0 self)
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We present a distributionfree model of incompleteinformation games, both with and without private information, in which the players use a robust optimization approach to contend with payoff uncertainty. Our “robust game” model relaxes the assumptions of Harsanyi’s Bayesian game model, and provides an alternative distributionfree equilibrium concept, which we call “robustoptimization equilibrium, ” to that of the ex post equilibrium. We prove that the robustoptimization equilibria of an incompleteinformation game subsume the ex post equilibria of the game and are, unlike the latter, guaranteed to exist when the game is finite and has bounded payoff uncertainty set. For arbitrary robust finite games with bounded polyhedral payoff uncertainty sets, we show that we can compute a robustoptimization equilibrium by methods analogous to those for identifying a Nash equilibrium of a finite game with complete information. In addition, we present computational results.
Tractable approximations of robust conic optimization problems
, 2006
"... In earlier proposals, the robust counterpart of conic optimization problems exhibits a lateral increase in complexity, i.e., robust linear programming problems (LPs) become second order cone problems (SOCPs), robust SOCPs become semidefinite programming problems (SDPs), and robust SDPs become NPha ..."
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Cited by 34 (11 self)
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In earlier proposals, the robust counterpart of conic optimization problems exhibits a lateral increase in complexity, i.e., robust linear programming problems (LPs) become second order cone problems (SOCPs), robust SOCPs become semidefinite programming problems (SDPs), and robust SDPs become NPhard. We propose a relaxed robust counterpart for general conic optimization problems that (a) preserves the computational tractability of the nominal problem; specifically the robust conic optimization problem retains its original structure, i.e., robust LPs remain LPs, robust SOCPs remain SOCPs and robust SDPs remain SDPs, and (b) allows us to provide a guarantee on the probability that the robust solution is feasible when the uncertain coefficients obey independent and identically distributed normal distributions.
Facility Location under Uncertainty: A Review
 IIE Transactions
, 2004
"... Plants, distribution centers, and other facilities generally function for years or decades, during which time the environment in which they operate may change substantially. Costs, demands, travel times, and other inputs to classical facility location models may be highly uncertain. This has made th ..."
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Cited by 33 (6 self)
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Plants, distribution centers, and other facilities generally function for years or decades, during which time the environment in which they operate may change substantially. Costs, demands, travel times, and other inputs to classical facility location models may be highly uncertain. This has made the development of models for facility location under uncertainty a high priority for researchers in both the logistics and stochastic/robust optimization communities. Indeed, a large number of the approaches that have been proposed for optimization under uncertainty have been applied to facility location problems. This paper reviews the literature...
A robust optimization approach to supply chain management
 Operations Research
, 2003
"... Abstract. We propose a general methodology based on robust optimization to address the problem of optimally controlling a supply chain subject to stochastic demand in discrete time. The attractive features of the proposed approach are: (a) It incorporates a wide variety of phenomena, including deman ..."
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Cited by 25 (3 self)
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Abstract. We propose a general methodology based on robust optimization to address the problem of optimally controlling a supply chain subject to stochastic demand in discrete time. The attractive features of the proposed approach are: (a) It incorporates a wide variety of phenomena, including demands that are not identically distributed over time and capacity on the echelons and links; (b) it uses very little information on the demand distributions; (c) it leads to qualitatively similar optimal policies (basestock policies) as in dynamic programming; (d) it is numerically tractable for large scale supply chain problems even in networks, where dynamic programming methods face serious dimensionality problems; (e) in preliminary computational experiments, it often outperforms dynamic programming based solutions for a wide range of parameters. 1
Theory and applications of Robust Optimization
, 2007
"... In this paper we survey the primary research, both theoretical and applied, in the field of Robust Optimization (RO). Our focus will be on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying the most pr ..."
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Cited by 24 (5 self)
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In this paper we survey the primary research, both theoretical and applied, in the field of Robust Optimization (RO). Our focus will be on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying the most prominent theoretical results of RO over the past decade, we will also present some recent results linking RO to adaptable models for multistage decisionmaking problems. Finally, we will highlight successful applications of RO across a wide spectrum of domains, including, but not limited to, finance, statistics, learning, and engineering.
How to pay, come what may: Approximation algorithms for demandrobust covering problems
 In FOCS
, 2005
"... Robust optimization has traditionally focused on uncertainty in data and costs in optimization problems to formulate models whose solutions will be optimal in the worstcase among the various uncertain scenarios in the model. While these approaches may be thought of defining data or costrobust prob ..."
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Cited by 23 (6 self)
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Robust optimization has traditionally focused on uncertainty in data and costs in optimization problems to formulate models whose solutions will be optimal in the worstcase among the various uncertain scenarios in the model. While these approaches may be thought of defining data or costrobust problems, we formulate a new “demandrobust” model motivated by recent work on twostage stochastic optimization problems. We propose this in the framework of general covering problems and prove a general structural lemma about special types of firststage solutions for such problems: there exists a firststage solution that is a minimal feasible solution for the union of the demands for some subset of the scenarios and its objective function value is no more than twice the optimal. We then provide approximation algorithms for a variety of standard discrete covering problems in this setting, including minimum cut, minimum multicut, shortest paths, Steiner trees, vertex cover and uncapacitated facility location. While many of our results draw from rounding approaches recently developed for stochastic programming problems, we also show new applications of old metric rounding techniques for cut problems in this demandrobust setting.