scispace - formally typeset
Search or ask a question
Journal ArticleDOI

The Price of Robustness

01 Jan 2004-Operations Research (INFORMS)-Vol. 52, Iss: 1, pp 35-53
TL;DR: In this paper, the authors propose an approach that attempts to make this trade-off more attractive by flexibly adjusting the level of conservatism of the robust solutions in terms of probabilistic bounds of constraint violations.
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.

Content maybe subject to copyright    Report

The price of robustness
IA meeting 14/12/2020
Bertsimas, Dimitris, and Melvyn Sim. "The price of
robustness." Operations research 52.1 (2004): 35-53.

The price of robustness
Context
Quote from the case study by Ben-Tal and Nemirovski (2000):!
«!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.!»!
This observation raises the natural question of designing solution
approaches that are immune to data uncertainty; that is, they
are «"robust"».#
#
This paper designs a new robust approach that adresses the
issue of over-conservatism.

The price of robustness
Data uncertainty in linear optimization
Data uncertainty is in the matrix A. !
The coecients a_ij that are subjected to parameter uncertainty takes
values according to a symmetric distribution with a mean equal to the
nominal value a_ij in the interval [a_ij- â_ij, a_ij + â_ij].!
Row i -> J_i coecients subject to uncertainty !
Gamma_i = parameter to adjust the robustness of the proposed method
against the level of conservatism of the solution.!
0 <= Gamma_i <= J_i -> only a subset of the coecients will change in
order to adversely aect the solution.#
The higher Gamma_i, the more robust the solution is. With Gamma_i
= J_i -> maximum protection.
Linear optimization problem:

The price of robustness
Zero-one knap sack problem (MILP)
MILP:
An application of this problem is to maximize the
total value of goods to be loaded on a cargo that
has strict weight restrictions. The weight of the
individual item is assumed to be uncertain,
independent of other weights, and follows a
symmetric distribution.

The price of robustness
Zero-one knap sack problem (MILP)
The zero-one knapsack problem is the following discrete
optimization problem:
Let J the set of uncertain parameters ωj, with 0 |J| N. The weights ωj
with j J are subjected to parameter uncertainty takes values according to
a symmetric distribution with a mean equal to the nominal value ωj in the
interval [ωj ωˆj, ωj + ωˆj]. The parameter to adjust the robustness of the
approach is Γ, with 0 Γ |J| N.

Citations
More filters
Journal ArticleDOI
TL;DR: This paper surveys the primary research, both theoretical and applied, in the area of robust optimization (RO), focusing on the computational attractiveness of RO approaches, as well as the modeling power and broad applicability of the methodology.
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 multistage decision-making problems. Finally, we highlight applications of RO across a wide spectrum of domains, including finance, statistics, learning, and various areas of engineering.

1,863 citations

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

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

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
More filters
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

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

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

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

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