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The price of the robustness

01 Jan 2004-Vol. 52, Iss: 1, pp 35-53
TL;DR: An approach is proposed that flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations, and an attractive aspect of this method is that the new robust formulation is also a linear optimization problem, so it naturally extend to discrete optimization problems in a tractable way.
Abstract: A robust approach to solving linear optimization problems with uncertain data was proposed in the early 1970s and has recently been extensively studied and extended. Under this approach, we are willing to accept a suboptimal solution for the nominal values of the data in order to ensure that the solution remains feasible and near optimal when the data changes. A concern with such an approach is that it might be too conservative. In this paper, we propose an approach that attempts to make this trade-off more attractive; that is, we investigate ways to decrease what we call the price of robustness. In particular, we flexibly adjust the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations. An attractive aspect of our method is that the new robust formulation is also a linear optimization problem. Thus we naturally extend our methods to discrete optimization problems in a tractable way. We report numerical results for a portfolio optimization problem, a knapsack problem, and a problem from the Net Lib library.
Citations
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Journal ArticleDOI
TL;DR: This work proposes 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, and proposes 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.
Abstract: 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 NP-hard α-approximable 0−1 discrete optimization problem, remains α-approximable. 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.

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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Journal Article
TL;DR: In this article, the authors survey the primary research, both theoretical and applied, in the area of robust optimization and highlight applications of RO across a wide spectrum of domains, including finance, statistics, learning, and various areas of engineering.
Abstract: In this paper we survey the primary research, both theoretical and applied, in the area of Robust Optimization (RO). Our focus is on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology. In addition to surveying prominent theoretical results of RO, we also present some recent results linking RO to adaptable models for multi-stage decision-making problems. Finally, we highlight applications of RO across a wide spectrum of domains, including finance, statistics, learning, and various areas of engineering.

1,633 citations

Journal ArticleDOI
TL;DR: In this paper, a two-stage adaptive robust unit commitment model for the security constrained unit commitment problem in the presence of nodal net injection uncertainty is proposed, which only requires a deterministic uncertainty set, rather than a hard-to-obtain probability distribution on the uncertain data.
Abstract: Unit commitment, one of the most critical tasks in electric power system operations, faces new challenges as the supply and demand uncertainty increases dramatically due to the integration of variable generation resources such as wind power and price responsive demand. To meet these challenges, we propose a two-stage adaptive robust unit commitment model for the security constrained unit commitment problem in the presence of nodal net injection uncertainty. Compared to the conventional stochastic programming approach, the proposed model is more practical in that it only requires a deterministic uncertainty set, rather than a hard-to-obtain probability distribution on the uncertain data. The unit commitment solutions of the proposed model are robust against all possible realizations of the modeled uncertainty. We develop a practical solution methodology based on a combination of Benders decomposition type algorithm and the outer approximation technique. We present an extensive numerical study on the real-world large scale power system operated by the ISO New England. Computational results demonstrate the economic and operational advantages of our model over the traditional reserve adjustment approach.

1,454 citations

Journal ArticleDOI
TL;DR: The main focus will be on the different approaches to perform robust optimization in practice including the methods of mathematical programming, deterministic nonlinear optimization, and direct search methods such as stochastic approximation and evolutionary computation.

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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Journal ArticleDOI
TL;DR: A large deviation-type approximation, referred to as “Bernstein approximation,” of the chance constrained problem is built that is convex and efficiently solvable and extended 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.
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 independent-of-each-other random variables, we build a large deviation-type approximation, referred to as “Bernstein approximation,” of the chance constrained problem. This approximation is convex and efficiently solvable. We propose a simulation-based scheme for bounding the optimal value in the chance constrained problem and report numerical experiments aimed at comparing the Bernstein and well-known 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.

1,099 citations

References
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Journal ArticleDOI
TL;DR: If U is an ellipsoidal uncertainty set, then for some of the most important generic convex optimization problems (linear programming, quadratically constrained programming, semidefinite programming and others) the corresponding robust convex program is either exactly, or approximately, a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods.
Abstract: We study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set U, yet the constraints must hold for all possible values of the data from U. The ensuing optimization problem is called robust optimization. In this paper we lay the foundation of robust convex optimization. In the main part of the paper we show that if U is an ellipsoidal uncertainty set, then for some of the most important generic convex optimization problems (linear programming, quadratically constrained programming, semidefinite programming and others) the corresponding robust convex program is either exactly, or approximately, a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods.

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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Journal ArticleDOI
TL;DR: This note formulates a convex mathematical programming problem in which the usual definition of the feasible region is replaced by a significantly different strategy via set containment.
Abstract: This note formulates a convex mathematical programming problem in which the usual definition of the feasible region is replaced by a significantly different strategy. Instead of specifying the feasible region by a set of convex inequalities, fi(x) ≦ bi, i = 1, 2, …, m, the feasible region is defined via set containment. Here n convex activity sets {Kj, j = 1, 2, …, n} and a convex resource set K are specified and the feasible region is given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$X =\{x \in R^{n}\mid x_{1}K_{1} + x_{2}K_{2} + \cdots + x_{n}K_{n} \subseteq K, x_{j}\geq 0\}$$ \end{document} where the binary operation + refers to addition of sets. The problem is then to find x ∈ X that maximizes the linear function c · x. When the res...

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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Journal ArticleDOI
TL;DR: It is shown that the RC of an LP with ellipsoidal uncertainty set is computationally tractable, since it leads to a conic quadratic program, which can be solved in polynomial time.

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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Journal ArticleDOI
TL;DR: The Robust Optimization methodology is applied to produce “robust” solutions of the above LPs which are in a sense immuned against uncertainty for the NETLIB problems.
Abstract: Optimal solutions of Linear Programming problems may become severely infeasible if the nominal data is slightly perturbed. We demonstrate this phenomenon by studying 90 LPs from the well-known NETLIB collection. We then apply the Robust Optimization methodology (Ben-Tal and Nemirovski [1–3]; El Ghaoui et al. [5, 6]) to produce “robust” solutions of the above LPs which are in a sense immuned against uncertainty. Surprisingly, for the NETLIB problems these robust solutions nearly lose nothing in optimality.

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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Journal ArticleDOI
TL;DR: This work considers least-squares problems where the coefficient matrices A,b are unknown but bounded and minimize the worst-case residual error using (convex) second-order cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A.
Abstract: We consider least-squares problems where the coefficient matrices A,b are unknown but bounded. We minimize the worst-case residual error using (convex) second-order cone programming, yielding an algorithm with complexity similar to one singular value decomposition of A. The method can be interpreted as a Tikhonov regularization procedure, with the advantage that it provides an exact bound on the robustness of solution and a rigorous way to compute the regularization parameter. When the perturbation has a known (e.g., Toeplitz) structure, the same problem can be solved in polynomial-time using semidefinite programming (SDP). We also consider the case when A,b are rational functions of an unknown-but-bounded perturbation vector. We show how to minimize (via SDP) upper bounds on the optimal worst-case residual. We provide numerical examples, including one from robust identification and one from robust interpolation.

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....

    [...]

  • ...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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