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30
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
A Robust Optimization Perspective Of Stochastic Programming
, 2005
"... In this paper, we introduce an approach for constructing uncertainty sets for robust optimization using new deviation measures for bounded random variables known as the forward and backward deviations. These deviation measures capture distributional asymmetry and lead to better approximations of c ..."
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Cited by 26 (9 self)
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In this paper, we introduce an approach for constructing uncertainty sets for robust optimization using new deviation measures for bounded random variables known as the forward and backward deviations. These deviation measures capture distributional asymmetry and lead to better approximations of chance constraints. We also propose a tractable robust optimization approach for obtaining robust solutions to a class of stochastic linear optimization problems where the risk of infeasibility can be tolerated as a tradeoff to improve upon the objective value. An attractive feature of the framework is the computational scalability to multiperiod models. We show an application of the framework for solving a project management problem with uncertain activity completion time.
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 23 (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.
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
On twostage convex chance constrained problems
, 2005
"... In this paper we develop approximation algorithms for twostage convex chance constrained problems. Nemirovski and Shapiro [18] formulated this class of problems and proposed an ellipsoidlike iterative algorithm for the special case where the impact function f(x,h) is biaffine. We show that this a ..."
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Cited by 10 (0 self)
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In this paper we develop approximation algorithms for twostage convex chance constrained problems. Nemirovski and Shapiro [18] formulated this class of problems and proposed an ellipsoidlike iterative algorithm for the special case where the impact function f(x,h) is biaffine. We show that this algorithm extends to biconvex f(x,h) in a fairly straightforward fashion. The complexity of the solution algorithm as well as the quality of its output are functions of the radius r of the largest Euclidean ball that can be inscribed in the polytope defined by a random set of linear inequalities generated by the algorithm [18]. Since the polytope determining r is random, computing r is difficult. Yet, the solution algorithm requires r as an input. In this paper we provide some guidance for selecting r. We show that the largest value of r is determined by the degree of robust feasibility of the twostage chance constrained problem – the more robust the problem, the higher one can set the parameter r. Next, we formulate ambiguous twostage chance constrained problems. In this formulation, the random variables defining the chance constraint are known to have a fixed distribution; however, the decision maker is only able to estimate this distribution to within some error. We construct an algorithm that solves the ambiguous twostage chance constrained problem when the impact function f(x,h) is biaffine and the extreme points of a certain “dual ” polytope are known explicitly. 1
Constructing risk measures from uncertainty sets
, 2005
"... We propose a unified theory that links uncertainty sets in robust optimization to risk measures in portfolio optimization. We illustrate the correspondence between uncertainty sets and some popular risk measures in finance, and show how robust optimization can be used to generalize the concepts of t ..."
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Cited by 7 (1 self)
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We propose a unified theory that links uncertainty sets in robust optimization to risk measures in portfolio optimization. We illustrate the correspondence between uncertainty sets and some popular risk measures in finance, and show how robust optimization can be used to generalize the concepts of these measures. We also show that by using properly defined uncertainty sets in robust optimization models, one can in fact construct coherent risk measures. Our approach to creating coherent risk measures is easy to apply in practice, and computational experiments suggest that it may lead to superior portfolio performance. Our results have implications for efficient portfolio optimization under different measures of risk.
Tractable robust expected utility and risk models for portfolio optimization
 Mathematical Finance
"... Expected utility models in portfolio optimization is based on the assumption of complete knowledge of the distribution of random returns. In this paper, we relax this assumption to the knowledge of only the mean, covariance and support information. No additional assumption on the type of distributio ..."
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Cited by 7 (4 self)
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Expected utility models in portfolio optimization is based on the assumption of complete knowledge of the distribution of random returns. In this paper, we relax this assumption to the knowledge of only the mean, covariance and support information. No additional assumption on the type of distribution such as normality is made. The investor’s utility is modeled as a piecewiselinear concave function. We derive exact and approximate optimal trading strategies for a robust or maximin expected utility model, where the investor maximizes his worst case expected utility over a set of ambiguous distributions. The optimal portfolios are identified using a tractable conic programming approach. Using the optimized certainty equivalent (OCE) framework of BenTal and Teboulle [6], we provide connections of our results with robust or ambiguous convex risk measures, in which the investor minimizes his worst case risk under distributional ambiguity. New closed form expressions for the OCE risk measures and optimal portfolios are provided for two and three piece utility functions. Computational experiments indicate that such robust approaches can provide good trading strategies in financial markets. 1
A soft robust model for optimization under ambiguity
, 2008
"... In this paper, we propose a framework for robust optimization that relaxes the standard notion of robustness by allowing the decisionmaker to vary the protection level in a smooth way across the uncertainty set. We apply our approach to the problem of maximizing the expected value of a payoff funct ..."
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Cited by 6 (3 self)
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In this paper, we propose a framework for robust optimization that relaxes the standard notion of robustness by allowing the decisionmaker to vary the protection level in a smooth way across the uncertainty set. We apply our approach to the problem of maximizing the expected value of a payoff function when the underlying distribution is ambiguous and therefore robustness is relevant. Our primary objective is to develop this framework and relate it to the standard notion of robustness, which deals with only a single guarantee across one uncertainty set. First, we show that our approach connects closely to the theory of convex risk measures. We show that the complexity of the this approach is equivalent to that of solving a small number of standard robust problems. We then investigate the conservatism benefits and downside probability guarantees implied by this approach and compare to the standard robust approach. Finally, we illustrate the methodology on an asset allocation example consisting of historical market data over a 25year investment horizon and find in every case we explore that relaxing standard robustness with soft robustness yields a seemingly favorable riskreturn tradeoff: each case results in a higher outofsample expected return for a relatively minor degradation of outofsample downside performance.
A lineardecision based approximation approach to stochastic programming
 Oper. Res
"... Stochastic optimization, especially multistage models, is well known to be computationally excruciating. Moreover, such models require exact specifications of the probability distributions of the underlying uncertainties, which are often unavailable. In this paper, we propose tractable methods of ad ..."
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Cited by 5 (1 self)
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Stochastic optimization, especially multistage models, is well known to be computationally excruciating. Moreover, such models require exact specifications of the probability distributions of the underlying uncertainties, which are often unavailable. In this paper, we propose tractable methods of addressing a general class of multistage stochastic optimization problems, which assume only limited information of the distributions of the underlying uncertainties, such as known mean, support and covariance. One basic idea of our methods is to approximate the recourse decisions via decision rules. We first examine linear decision rules in detail and show that even for problems with complete recourse, linear decision rules can be inadequate and even lead to infeasible instances. Hence, we propose several new decision rules that improve upon linear decision rules, while keeping the approximate models computationally tractable. Specifically, our approximate models are in the forms of the socalled second order cone (SOC) programs, which could be solved efficiently both in theory and in practice. We also present computational evidence indicating that our approach is a viable alternative, and possibly advantageous, to existing stochastic optimization solution techniques in solving a twostage stochastic optimization problem with complete recourse.