## Robust solutions of uncertain linear programs (1999)

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Venue: | Operations Research Letters |

Citations: | 255 - 15 self |

### BibTeX

@ARTICLE{Ben-tal99robustsolutions,

author = {A. Ben-tal and A. Nemirovski},

title = {Robust solutions of uncertain linear programs},

journal = {Operations Research Letters},

year = {1999},

volume = {25},

pages = {1--13}

}

### Years of Citing Articles

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### Abstract

We treat in this paper Linear Programming (LP) problems with uncertain data. The focus is on uncertainty associated with hard constraints: those which must be satisfied, whatever is the actual realization of the data (within a prescribed uncertainty set). We suggest a modeling methodology whereas an uncertain LP is replaced by its Robust Counterpart (RC). We then develop the analytical and computational optimization tools to obtain robust solutions of an uncertain LP problem via solving the corresponding explicitly stated convex RC program. In particular, 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.

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Citation Context ...ntative nominal value of the data is used (e.g., expected values). The classical approach in Operations Research/Management Science to deal with uncertainty is Stochastic Programming (SP) (see, e.g., =-=[6, 13]-=- and references therein). But even in this approach constraints may be violated, with certain penalty (this is the case for SP with recourse [6, 9], Scenario optimization [14], Entropic Penalty method... |

386 | Interior-point polynomial algorithms in convex programming - Nesterov, Nemirovskii - 1994 |

301 | Robust convex optimization
- Ben-Tal, Nemirovski
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(Show Context)
Citation Context ...uncertain parameters (”non-adjustable variables”), while the other part are variables that can be chosen after the realization (”adjustable variables”). We extend the Robust Optimization methodology (=-=[1, 4, 5, 6, 7, 9, 13, 14]-=-) to this situation by introducing the Adjustable Robust Counterpart (ARC) associated with an LP of the above structure. Often the ARC is significantly less conservative than the usual Robust Counterp... |

252 |
Stochastic programming
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(Show Context)
Citation Context ...ntative nominal value of the data is used (e.g., expected values). The classical approach in Operations Research/Management Science to deal with uncertainty is Stochastic Programming (SP) (see, e.g., =-=[6, 13]-=- and references therein). But even in this approach constraints may be violated, with certain penalty (this is the case for SP with recourse [6, 9], Scenario optimization [14], Entropic Penalty method... |

223 |
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Citation Context ...≤ 1, we conclude that αi(x) + 2 ξ T βi(x)t + ξ T Γi(x)ξ ≥ 0 as well, i.e., (ξ, t) satisfy the conclusion in A(i), so that A(i) is valid. □ Now let us recall the following fundamental fact [see, e.g., =-=[2, 8]-=-]: 14Lemma 4.3 (S - Lemma) Let A, B be symmetric matrices of the same size, and let the homogeneous quadratic inequality y T Ay ≥ 0 (28) be strictly feasible (i.e., ¯y T A¯y > 0 for some ¯y). A homog... |

146 |
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Citation Context ...ming (SP) (see, e.g., [6, 13] and references therein). But even in this approach constraints may be violated, with certain penalty (this is the case for SP with recourse [6, 9], Scenario optimization =-=[14]-=-, Entropic Penalty methods [1]) or with certain probability (chance constraints). In the dominating penalty approach, even when the random variables are degenerate (deterministic), the corresponding S... |

109 |
Convex programming with set-inclusive constraints and applications to inexact linear programming
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- 1973
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Citation Context ...n 4). Dealing with uncertain hard constraints is perhaps a novelty in Mathematical Programming; to the best of our knowledge, the only previous example related to this question is due to A.L. Soyster =-=[16]-=- (see below). The issue of hard uncertain constraints, however, is not a novelty for Control Theory, where it is a well-studied subject forming the area of Robust Control (see, e.g., [17] and referenc... |

95 |
Some NP-complete problems in quadratic and nonlinear programming
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Citation Context ... would be the problem of maximizing a quadratic form over the standard simplex. But the latter problem is equivalent to the problem of checking copositivity, a problem known to be NP-hard (see, e.g., =-=[15, 16]-=-). We have just seen that when the recourse matrix V is not fixed, the AARC of LPZ can become computationally intractable. The goal of this section is to utilize recent results of [6] (obtained there ... |

86 |
Robust optimization of large-scale systems
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Citation Context ... although this was not stated explicitly in the past, SP treats in fact mainly soft constraints. These remarks apply also to the recent scenario-based penalty approach of Mulvey, Vanderbei and Zenios =-=[11]-=-. In this paper we study uncertainty associated with hard constraints, i.e., those which must be satisfied whatever is the realization of the data (A, b) within, of course, a reasonable prescribed “un... |

63 | Robust solutions to uncertain semidefinite programs
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(Show Context)
Citation Context ...uncertain parameters (”non-adjustable variables”), while the other part are variables that can be chosen after the realization (”adjustable variables”). We extend the Robust Optimization methodology (=-=[1, 3, 4, 5, 6, 9, 13, 14]-=-) to this situation by introducing the Adjustable Robust Counterpart (ARC) associated with an LP of the above structure. Often the ARC is significantly less conservative than the usual Robust Counterp... |

58 | Robust truss topology design via semidefinite programming
- Ben-Tal, Nemirovski
- 1997
(Show Context)
Citation Context ...ons of the data, and even a small violation of the constraints cannot be tolerated. We have discussed such a situation elsewhere, for problems of designing engineering structures (e.g., bridges), see =-=[2]-=-. In these problems, ignoring even small changes in the forces acting on the structure may cause “violent” displacements and result in a severely unstable structure. This example is not from the world... |

46 | Parameter estimation in the presence of bounded data uncertainties
- Chandrasekaran, Golub, et al.
- 1998
(Show Context)
Citation Context ...uncertain parameters (”non-adjustable variables”), while the other part are variables that can be chosen after the realization (”adjustable variables”). We extend the Robust Optimization methodology (=-=[1, 3, 4, 5, 6, 9, 13, 14]-=-) to this situation by introducing the Adjustable Robust Counterpart (ARC) associated with an LP of the above structure. Often the ARC is significantly less conservative than the usual Robust Counterp... |

39 | Robust Semidefinite Programming
- Ben-Tal, Ghaoui, et al.
- 2000
(Show Context)
Citation Context ...uncertain parameters (”non-adjustable variables”), while the other part are variables that can be chosen after the realization (”adjustable variables”). We extend the Robust Optimization methodology (=-=[1, 3, 4, 5, 6, 9, 13, 14]-=-) to this situation by introducing the Adjustable Robust Counterpart (ARC) associated with an LP of the above structure. Often the ARC is significantly less conservative than the usual Robust Counterp... |

37 | Penalty/barrier multiplier methods for convex programming problems
- Ben-Tal, Zibulevsky
- 1997
(Show Context)
Citation Context ...where αi are fixed reals, ai and bi are fixed vectors, and Bi are fixed matrices of proper dimensions; � · � stands for the usual Euclidean norm. Recent progress in interior point methods (see, e.g., =-=[4]-=-) makes it possible to solve truly large-scale CQP’s, so that “ellipsoidal uncertainty” leads to “practically solvable” robust counterparts (PU). 3 Problem (PU) in the case of ellipsoidal uncertainty ... |

33 | Robust solutions of uncertain quadratic and conic-quadratic problems
- Ben-Tal, Nemirovski, et al.
(Show Context)
Citation Context |

27 |
A.: On the solution of two-stage linear programs under uncertainty
- Dantzig, Madansky
- 1961
(Show Context)
Citation Context ...sideration is normalized and thus write this problem as � LPZ = min u,v cT � u : Uu + V v ≤ b . (3) ζ=[U,V,b]∈Z Borrowing from the terminology of “Two-stage stochastic programming under uncertainty” (=-=[10]-=-,[17]), the matrix V is called recourse matrix. When V is not uncertain, we call the corresponding uncertain LP � min u,v cT � u : Uu + V v ≤ b . ζ=[U,b]∈Z (4) a fixed recourse one. Definition. We def... |

23 | Robust Modeling of Multi-Stage Portfolio Problems - Ben-Tal, Margalit, et al. - 2000 |

23 |
Portfolio Selection : Efficient Diversification of Investments
- Markovitz
- 1959
(Show Context)
Citation Context ...thing new in the phenomenon in question; OR financial models take care not only of the expected yields, but also of the variances and other characteristics of risk since the seminal work of Markovitz =-=[10]-=-. Note that in the particular example we treat here the robust counterpart solution resembles a lot the one given by the Markovitz approach. 7) For different data, the RC policy gives different portio... |

23 |
Robust solutions to least-squares problem with uncertain data
- ElGhaoui, Lebret
- 1997
(Show Context)
Citation Context |

21 |
A.: Stable Truss Topology Design via Semidefinite Programming
- Ben-Tal, Nemirovski
- 1997
(Show Context)
Citation Context ...uncertain parameters (”non-adjustable variables”), while the other part are variables that can be chosen after the realization (”adjustable variables”). We extend the Robust Optimization methodology (=-=[1, 4, 5, 6, 7, 9, 13, 14]-=-) to this situation by introducing the Adjustable Robust Counterpart (ARC) associated with an LP of the above structure. Often the ARC is significantly less conservative than the usual Robust Counterp... |

19 |
Uncertainty-immunized solutions in linear programming
- Guslitser
- 2002
(Show Context)
Citation Context ...lved efficiently (see [4]). Usually this is not the case with the Adjustable Robust Counterpart of LPZ (5). One simple case when ARC is tractable (in fact is an LP) is the following: Theorem 2.2 (see =-=[12]-=-) Assume that the uncertainty set Z is given as the convex hull of a finite set : Z = Conv {[U1, V1, b1], . . . , [UN, VN, bN]} . (A) Then in the case V1 = ... = VN of fixed recourse the ARC is given ... |

11 |
Geometric methods in combinatorial optimization
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- 1984
(Show Context)
Citation Context ...(PU) can be equivalently rewritten as min{c T x | x ∈ GU}, GU = {x | Ax ≥ 0 ∀A ∈ U; f T x = 1}. (12) 5sIt is clearly seen that GU is a closed convex set, so that (PU) is a convex program. It is known =-=[8]-=- that in order to minimize in a theoretically efficient manner a linear objective over a closed convex set G ⊂ R n it suffices to equip the set G with an efficient separation oracle. The latter is a r... |

10 |
The entropic penalty approach to stochastic programming
- Ben-Tal
- 1985
(Show Context)
Citation Context ...nd references therein). But even in this approach constraints may be violated, with certain penalty (this is the case for SP with recourse [6, 9], Scenario optimization [14], Entropic Penalty methods =-=[1]-=-) or with certain probability (chance constraints). In the dominating penalty approach, even when the random variables are degenerate (deterministic), the corresponding SP model does not recover neces... |

8 |
Exact Solutions of Inexact Linear Programs
- Falk
- 1976
(Show Context)
Citation Context ...ain data in hard constraints. As it was already mentioned, uncertain hard constraints in LP models were discussed (in a very specific setting) by A.L. Soyster [16] (for further developments, see also =-=[15, 7]-=-). The case considered in these papers is the one of “column-wise” uncertainty, i.e., the columns ai of the constraint matrix in the constraints Ax ≤ b, x ≥ 0 are known to belong to a given convex set... |

8 |
Convex Programming with Set-Inclusive Constraints and its Applications to Generalized Linear and Fractional Programming
- Singh
- 1982
(Show Context)
Citation Context ...ain data in hard constraints. As it was already mentioned, uncertain hard constraints in LP models were discussed (in a very specific setting) by A.L. Soyster [16] (for further developments, see also =-=[15, 7]-=-). The case considered in these papers is the one of “column-wise” uncertainty, i.e., the columns ai of the constraint matrix in the constraints Ax ≤ b, x ≥ 0 are known to belong to a given convex set... |

2 |
L.S.: “Games with continuous pay-offs
- Bonhenblust, Karlin, et al.
- 1950
(Show Context)
Citation Context ...44) As a result of (44) the system of inequalities [U(ζk)u + V (ζk)v − b(ζk)]i − ɛ < 0, i = 1, . . . , m, k = 1, . . . , N (45) in variables v has no solution in Vu. By the Karlin-Bohnenblust Theorem =-=[7]-=- it follows that there exists collection of weights {λi,k ≥ 0}, � i,k λi,k = 1, such that the corresponding combination of the left hand sides of the inequalities (45) is nonnegative everywhere on Vu,... |

2 |
Signs of Minors
- Motskin
- 1967
(Show Context)
Citation Context ... would be the problem of maximizing a quadratic form over the standard simplex. But the latter problem is equivalent to the problem of checking copositivity, a problem known to be NP-hard (see, e.g., =-=[15, 16]-=-). We have just seen that when the recourse matrix V is not fixed, the AARC of LPZ can become computationally intractable. The goal of this section is to utilize recent results of [6] (obtained there ... |

2 |
Stochastic Programming”, Klumer
- Prekopa
- 1995
(Show Context)
Citation Context ...ation is normalized and thus write this problem as � LPZ = min u,v cT � u : Uu + V v ≤ b . (3) ζ=[U,V,b]∈Z Borrowing from the terminology of “Two-stage stochastic programming under uncertainty” ([10],=-=[17]-=-), the matrix V is called recourse matrix. When V is not uncertain, we call the corresponding uncertain LP � min u,v cT � u : Uu + V v ≤ b . ζ=[U,b]∈Z (4) a fixed recourse one. Definition. We define t... |

2 | Optimization I: Convex Analysis, Nonlinear Programming Theory, Nonlinear Programming Algorithms – Lecture - Ben-Tal, Nemirovski |