The price of the robustness
Citations
1,747 citations
Cites background or methods from "The price of the robustness"
...Specifically, our contributions include: (a) When both the cost coefficients and the data in the constraints of an integer programming problem are subject to uncertainty, we propose, following the approach in Bertsimas and Sim [7], a robust integer programming problem of moderately larger size that allows to control the degree of conservatism of the solution in terms of probabilistic bounds on constraint violation....
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...[Bertsimas and Sim [7]] Let x∗ be an optimal solution of Problem (3)....
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...Bertsimas and Sim [7] propose a different approach to control the level of conservatism in the solution that has the advantage that leads to a linear optimization model and thus, as we examine in more detail in this paper, can be directly applied to discrete optimization models....
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...We next show that the approach in Bertsimas and Sim [7] for linear optimization extends to discrete optimization....
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...For this reason, we have calculated Bound (9), which is simple to compute and, as Bertsimas and Sim [7] show, very tight....
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1,633 citations
1,454 citations
1,435 citations
Cites background from "The price of the robustness"
...The field of robust mathematical programming has received increasing interest during the last five years [42,43,44,45,46,47]....
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1,099 citations
References
2,501 citations
"The price of the robustness" refers background or methods in this paper
...” To illustrate the importance of robustness in practical applications, we quote from the case study by Ben-Tal and Nemirovski (2000) on linear optimization problems from the Net Lib library: In real-world applications of Linear Programming, one cannot ignore the possibility that a small uncertainty in the data can make the usual optimal solution completely meaningless from a practical viewpoint....
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...35–53, © 2004 INFORMS 37 address this conservatism, Ben-Tal and Nemirovski (2000) propose the following robust problem: maximize c′x subject to ∑...
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...However, a practical drawback of such an approach is that it leads to nonlinear, although convex, models, which are more demanding computationally than the earlier linear models by Soyster (1973) (see also the discussion in Ben-Tal and Nemirovski 2000). In this research, we propose a new approach for robust linear optimization that retains the advantages of the linear frameworkofSoyster (1973).More importantly, our approach offers full control on thedegreeof conservatism for every constraint.Weprotect against violationof constraint i deterministically, when only a prespecified number %i of the coefficients changes; that is, we guarantee that the solution is feasible if less than %i uncertain coefficients change. Moreover, we provide aprobabilistic guarantee that even ifmore than%i change, then the robust solutionwill be feasible with high probability. In the process we prove a new, to the best of our knowledge, tight bound on sums of symmetrically distributed random variables. In this way, the proposed framework is at least as flexible as the one proposed by Ben-Tal and Nemirovski (1998, 1999, 2000), El-Ghaoui and Lebret (1997), El-Ghaoui et al. (1998), and possibly more. Unlike these approaches, the robust counterparts we propose are linear optimization problems, and thus our approach readily generalizes to discrete optimization problems. To the best of our knowledge, there was no similar work done in the robust discrete optimization domain that involves deterministic and probabilistic guarantees of constraints against violation. Structure of the Paper In §2, we present the different approaches for robust linear optimization from the literature and discuss their merits. In §3, we propose the new approach and show that it can be solved as a linear optimization problem. In §4, we show that the proposed robust LP has attractive probabilistic and deterministic guarantees. Moreover, we perform sensitivity analysis of the degree of protection the proposed method offers. We provide extensions to our basic framework dealing with correlated uncertain data in §5. In §6, we apply the proposed approach to a portfolio problem, a knapsack problem, and a problem from the Net Lib library. Finally, §7 contains some concluding remarks. 2. Robust Formulation of Linear Programming Problems 2.1. Data Uncertainty in Linear Optimization We consider the following nominal linear optimization problem: maximize c′x subject to Ax! b l! x! u" In the above formulation, we assume that data uncertainty only affects the elements in matrix A. We assume without loss of generality that the objective function c is not subject to uncertainty, since we can use the objective maximize z, add the constraint z−c′x! 0, and thus include this constraint into Ax! b. Model of Data Uncertainty U. Consider a particular row i of the matrix A and let Ji represent the set of coefficients in row i that are subject to uncertainty. Each entry aij , j ∈ Ji is modeled as a symmetric and bounded random variable ãij , j ∈ Ji (see Ben-Tal and Nemirovski 2000) that takes values in &aij − âij!aij + âij '. Associated with the uncertain data ãij , we define the random variable (ij = #ãij −aij$/âij , which obeys an unknown but symmetric distribution, and takes values in &−1!1'. 2.2. The Robust Formulation of Soyster As we have mentioned in the introduction, Soyster (1973) considers columnwise uncertainty....
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...A significant step forward for developing a theory for robust optimization was taken independently by Ben-Tal and Nemirovski (1998, 1999, 2000), El-Ghaoui and Lebret (1997), and El-Ghaoui et al....
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...However, a practical drawback of such an approach is that it leads to nonlinear, although convex, models, which are more demanding computationally than the earlier linear models by Soyster (1973) (see also the discussion in Ben-Tal and Nemirovski 2000). In this research, we propose a new approach for robust linear optimization that retains the advantages of the linear frameworkofSoyster (1973).More importantly, our approach offers full control on thedegreeof conservatism for every constraint.Weprotect against violationof constraint i deterministically, when only a prespecified number %i of the coefficients changes; that is, we guarantee that the solution is feasible if less than %i uncertain coefficients change. Moreover, we provide aprobabilistic guarantee that even ifmore than%i change, then the robust solutionwill be feasible with high probability. In the process we prove a new, to the best of our knowledge, tight bound on sums of symmetrically distributed random variables. In this way, the proposed framework is at least as flexible as the one proposed by Ben-Tal and Nemirovski (1998, 1999, 2000), El-Ghaoui and Lebret (1997), El-Ghaoui et al....
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1,813 citations
"The price of the robustness" refers background in this paper
...The first step in this direction was taken by Soyster (1973), who proposes a linear optimization model to construct a solution that is feasible for all data that belong in a convex set....
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...As we have mentioned in the introduction, Soyster (1973) considers columnwise uncertainty....
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...Soyster (1973) shows that the problem is equivalent to maximize c′x subject to n ∑ j=1 %Ajxj ! b! x" 0!...
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...However, a practical drawback of such an approach is that it leads to nonlinear, although convex, models, which are more demanding computationally than the earlier linear models by Soyster (1973) (see also the discussion in Ben-Tal and Nemirovski 2000)....
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1,809 citations
"The price of the robustness" refers background or methods in this paper
...Ben-Tal and Nemirovski (1999) consider the same portfolio problem using n= 150, pi = 1"15+ i 0"05 150 !...
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...In this way, the proposed framework is at least as flexible as the one proposed by Ben-Tal and Nemirovski (1998, 1999, 2000), El-Ghaoui and Lebret (1997), El-Ghaoui et al. (1998), and possibly more....
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...Given a constraint a′x ! b, with a ∈ &ā − â! ā + â', the robust counterpart of Ben-Tal and Nemirovski (1998, 1999, 2000), El-Ghaoui et al. (1998), and El-Ghaoui and Lebret (1998) in its simplest form of ellipsoidal uncertainty (Formulation (3) includes combined interval and ellipsoidal…...
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...A significant step forward for developing a theory for robust optimization was taken independently by Ben-Tal and Nemirovski (1998, 1999, 2000), El-Ghaoui and Lebret (1997), and El-Ghaoui et al. (1998)....
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...…the issue of overconservatism, these papers proposed less conservative models by considering uncertain linear problems with ellipsoidal uncertainties, which involve solving the robust counterparts of the nominal problem in the form of conic quadratic problems (see Ben-Tal and Nemirovski 1999)....
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1,674 citations
"The price of the robustness" refers background or methods in this paper
...However, a practical drawback of such an approach, is that it leads to nonlinear, although convex, models, which are more demanding computationally than the earlier linear models by Soyster [9] (see also the discussion in [1])....
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...We have argued so far that the linear optimization framework of our approach has some computational advantages over the conic quadratic framework of Ben-Tal and Nemirovski [1, 2, 3] and El-Ghaoui et al....
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...Given a constraint a′x ≤ b, with a ∈ [a−â, a+â], the robust counterpart of Ben-Tal and Nemirovski [1, 2, 3] and El-Ghaoui et al....
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...The resulting model produces solutions that are too conservative in the sense that we give up too much of optimality for the nominal problem in order to ensure robustness (see the comments of Ben-Tal and Nemirovski [1])....
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...A significant step forward for developing a theory for robust optimization was taken independently by Ben-Tal and Nemirovski [1, 2, 3] and El-Ghaoui et al....
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1,164 citations
"The price of the robustness" refers methods in this paper
...In this way, the proposed framework is at least as flexible as the one proposed by Ben-Tal and Nemirovski (1998, 1999, 2000), El-Ghaoui and Lebret (1997), El-Ghaoui et al. (1998), and possibly more....
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...A significant step forward for developing a theory for robust optimization was taken independently by Ben-Tal and Nemirovski (1998, 1999, 2000), El-Ghaoui and Lebret (1997), and El-Ghaoui et al. (1998)....
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...…so far that the linear optimization framework of our approach has some computational advantages over the conic quadratic framework of Ben-Tal and Nemirovski (1998, 1999, 2000), El-Ghaoui et al. (1998), and El-Ghaoui and Lebret (1997), especially with respect to discrete optimization problems....
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...Given a constraint a′x ! b, with a ∈ &ā − â! ā + â', the robust counterpart of Ben-Tal and Nemirovski (1998, 1999, 2000), El-Ghaoui et al. (1998), and El-Ghaoui and Lebret (1998) in its simplest form of ellipsoidal uncertainty (Formulation (3) includes combined interval and ellipsoidal uncertainty) is: ā′x+)2 3Ax2! b! where 3A is a diagonal matrix with elements âi in the diagonal....
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...We have argued so far that the linear optimization framework of our approach has some computational advantages over the conic quadratic framework of Ben-Tal and Nemirovski (1998, 1999, 2000), El-Ghaoui et al. (1998), and El-Ghaoui and Lebret (1997), especially with respect to discrete optimization problems....
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