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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 34 (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
Selected topics in robust convex optimization
 Math. Prog. B, this issue
, 2007
"... Abstract Robust Optimization is a rapidly developing methodology for handling optimization problems affected by nonstochastic “uncertainbutbounded” data perturbations. In this paper, we overview several selected topics in this popular area, specifically, (1) recent extensions of the basic concept ..."
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Cited by 14 (2 self)
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Abstract Robust Optimization is a rapidly developing methodology for handling optimization problems affected by nonstochastic “uncertainbutbounded” data perturbations. In this paper, we overview several selected topics in this popular area, specifically, (1) recent extensions of the basic concept of robust counterpart of an optimization problem with uncertain data, (2) tractability of robust counterparts, (3) links between RO and traditional chance constrained settings of problems with stochastic data, and (4) a novel generic application of the RO methodology in Robust Linear Control. Keywords optimization under uncertainty · robust optimization · convex programming · chance constraints · robust linear control
Relative entropy, exponential utility, and robust dynamic pricing, Operations Research. Forthcoming
, 2004
"... informs ® doi 10.1287/opre.1070.0385 © 2007 INFORMS In the area of dynamic revenue management, optimal pricing policies are typically computed on the basis of an underlying demand rate model. From the perspective of applications, this approach implicitly assumes that the model is an accurate represe ..."
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Cited by 14 (0 self)
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informs ® doi 10.1287/opre.1070.0385 © 2007 INFORMS In the area of dynamic revenue management, optimal pricing policies are typically computed on the basis of an underlying demand rate model. From the perspective of applications, this approach implicitly assumes that the model is an accurate representation of the realworld demand process and that the parameters characterizing this model can be accurately calibrated using data. In many situations, neither of these conditions are satisfied. Indeed, models are usually simplified for the purpose of tractability and may be difficult to calibrate because of a lack of data. Moreover, pricing policies that are computed under the assumption that the model is correct may perform badly when this is not the case. This paper presents an approach to singleproduct dynamic revenue management that accounts for errors in the underlying model at the optimization stage. Uncertainty in the demand rate model is represented using the notion of relative entropy, and a tractable reformulation of the “robust pricing problem ” is obtained using results concerning the change of probability measure for point processes. The optimal pricing policy is obtained through a version of the socalled Isaacs ’ equation for stochastic differential games, and the structural properties of the optimal solution are obtained through an analysis of this equation. In particular, (i) closedform solutions for the special case of an exponential nominal demand rate model,
Distributionally robust optimization and its tractable approximations
 Operations Research
"... In this paper, we focus on a linear optimization problem with uncertainties, having expectations in the objective and in the set of constraints. We present a modular framework to obtain an approximate solution to the problem that is distributionally robust, and more flexible than the standard techni ..."
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Cited by 12 (3 self)
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In this paper, we focus on a linear optimization problem with uncertainties, having expectations in the objective and in the set of constraints. We present a modular framework to obtain an approximate solution to the problem that is distributionally robust, and more flexible than the standard technique of using linear rules. Our framework begins by firstly affinelyextending the set of primitive uncertainties to generate new linear decision rules of larger dimensions, and are therefore more flexible. Next, we develop new piecewiselinear decision rules which allow a more flexible reformulation of the original problem. The reformulated problem will generally contain terms with expectations on the positive parts of the recourse variables. Finally, we convert the uncertain linear program into a deterministic convex program by constructing distributionally robust bounds on these expectations. These bounds are constructed by first using different pieces of information on the distribution of the underlying uncertainties to develop separate bounds, and next integrating them into a combined bound that is better than each of the individual bounds.
Computing robust basestock levels
 Discrete Optimization
"... This paper considers how to optimally set the basestock level for a single buffer when demand is uncertain, in a robust framework. We present a family of algorithms based on decomposition that scale well to problems with hundreds of time periods, and theoretical results on more general models. 1 ..."
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Cited by 10 (0 self)
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This paper considers how to optimally set the basestock level for a single buffer when demand is uncertain, in a robust framework. We present a family of algorithms based on decomposition that scale well to problems with hundreds of time periods, and theoretical results on more general models. 1
Robust and DataDriven Optimization: Modern DecisionMaking Under Uncertainty
, 2006
"... Traditional models of decisionmaking under uncertainty assume perfect information, i.e., accurate values for the system parameters and specific probability distributions for the random variables. However, such precise knowledge is rarely available in practice, and a strategy based on erroneous inp ..."
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Cited by 6 (0 self)
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Traditional models of decisionmaking under uncertainty assume perfect information, i.e., accurate values for the system parameters and specific probability distributions for the random variables. However, such precise knowledge is rarely available in practice, and a strategy based on erroneous inputs might be infeasible or exhibit poor performance when implemented. The purpose of this tutorial is to present a mathematical framework that is wellsuited to the limited information available in reallife problems and captures the decisionmaker’s attitude towards uncertainty; the proposed approach builds upon recent developments in robust and datadriven optimization. In robust optimization, random variables are modeled as uncertain parameters belonging to a convex uncertainty set and the decisionmaker protects the system against the worst case within that set. Datadriven optimization uses observations of the random variables as direct inputs to the mathematical programming problems. The first part of the tutorial describes the robust optimization paradigm in detail in singlestage and multistage problems. In the second part, we address the issue of constructing uncertainty sets using historical realizations of the random variables and investigate the connection between convex sets, in particular polyhedra, and a specific class of risk measures.
Provably nearoptimal samplingbased policies for stochastic inventory control models
 Proceedings, 38th Annual ACM Symposium on Theory of Computing
, 2006
"... In this paper, we consider two fundamental inventory models, the singleperiod newsvendor problem and its multiperiod extension, but under the assumption that the explicit demand distributions are not known and that the only information available is a set of independent samples drawn from the true ..."
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Cited by 5 (3 self)
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In this paper, we consider two fundamental inventory models, the singleperiod newsvendor problem and its multiperiod extension, but under the assumption that the explicit demand distributions are not known and that the only information available is a set of independent samples drawn from the true distributions. Under the assumption that the demand distributions are given explicitly, these models are wellstudied and relatively straightforward to solve. However, in most reallife scenarios, the true demand distributions are not available or they are too complex to work with. Thus, a samplingdriven algorithmic framework is very attractive, both in practice and in theory. We shall describe how to compute samplingbased policies, that is, policies that are computed based only on observed samples of the demands without any access to, or assumptions on, the true demand distributions. Moreover, we establish bounds on the number of samples required to guarantee that with high probability, the expected cost of the samplingbased policies is arbitrarily close (i.e., with arbitrarily small relative error) compared to the expected cost of the optimal policies which have full access to the demand distributions. The bounds that we develop are general, easy to compute and do not depend at all on the specific demand distributions.
Wardrop equilibria with riskaverse users
 Transportation Science
, 2010
"... Network games can be used to model competitive situations in which agents select routes to minimize their cost. Common applications include traffic, telecommunication, and distribution networks. Although traditional network models have assumed that realized costs only depend on congestion, in most a ..."
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Cited by 5 (0 self)
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Network games can be used to model competitive situations in which agents select routes to minimize their cost. Common applications include traffic, telecommunication, and distribution networks. Although traditional network models have assumed that realized costs only depend on congestion, in most applications they also have an uncertain component. We extend the traffic assignment problem first proposed by Wardrop in 1952 by adding random deviations, which are independent of the flow, to the cost functions that model congestion in each arc. We map these uncertainties into a Wardrop equilibrium model with nonadditive path costs. The cost on a path is given by the sum of the congestion on its arcs plus a constant safety margin that represents the agents ’ risk aversion. First, we prove that an equilibrium for this game always exists and is essentially unique. Then, we introduce three specific equilibrium models that fall within this framework: the percentile equilibrium where agents select paths that minimize a specified percentile of the uncertain cost; the addedvariability equilibrium where agents add a multiple of the variability of the cost of each arc to the expected cost; and the robust equilibrium where agents select paths by solving a robust optimization problem that imposes a limit on the number of arcs that can deviate from the mean. The percentile equilibrium is difficult to compute because minimizing a percentile among all paths is computationally hard. Instead, the addedvariability and robust Wardrop equilibria can be computed efficiently in practice: The former reduces to a standard Wardrop
Twostage robust power grid optimization problem
 JNL Operations Research, submitted, 2010. [Online]. Available: http://www.optimizationonline.org/DB FILE/2010/10/2769.pdf
"... For both regulated and deregulated electric power markets, due to the integration of renewable energy generation and uncertain demands, both supply and demand sides of an electric power grid are volatile and under uncertainty. Accordingly, a large amount of spinning reserve is required to maintain t ..."
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Cited by 5 (0 self)
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For both regulated and deregulated electric power markets, due to the integration of renewable energy generation and uncertain demands, both supply and demand sides of an electric power grid are volatile and under uncertainty. Accordingly, a large amount of spinning reserve is required to maintain the reliability of the power grid in traditional approaches. In this paper, we propose a novel twostage robust integer programming model to address the power grid optimization problem under supply and demand uncertainty. In our approach, uncertain problem parameters are assumed to be within a given cardinality or polyhedral uncertainty set. We study cases with and without transmission capacity and ramprate limits. We also analyze solution schemes to solve each problem that include an exact solution approach, and an efficient heuristic approach that provides a tight lower bound for the general robust power grid optimization problem. The final computational experiments on a modified IEEE 118bus system verify the effectiveness of our approaches, as compared to the worstcase scenario generated by the nominal model without considering the uncertainty. Key words: unit commitment, transmission capacity limits, mixed integer programming, separation, robust optimization 1 1
ROBUST WARDROP EQUILIBRIUM
"... Abstract. Network games can be used to model competitive situations in which players select routes to maximize their utility. Common applications include traffic, telecommunication and distribution networks. Although traditional network models have assumed that utilities only depend on congestion, i ..."
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Cited by 4 (0 self)
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Abstract. Network games can be used to model competitive situations in which players select routes to maximize their utility. Common applications include traffic, telecommunication and distribution networks. Although traditional network models have assumed that utilities only depend on congestion, in most applications they also have an uncertain component. In this work, we extend Wardrop’s network game (1952) by explicitly incorporating uncertainty in the utility functions. Players are utility maximizers and select their route by solving a robust optimization problem, which takes the uncertainty into account. We define a robust Wardrop equilibrium as a solution under which all players are assigned to an optimal solution to their robust problems. Such a solution always exists and can be computed through efficient column generation methods. We show through a computational study that a robust Wardrop equilibrium tends to be more fair than the classic Wardrop equilibrium which ignores the uncertainty. Hence, a robust Wardrop equilibrium is more stable than the nominal counterpart as it reduces the regret that players experience after the uncertainty is revealed. 1.