Results 1  10
of
33
Robust solutions of uncertain linear programs
 Operations Research Letters
, 1999
"... 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 ..."
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Cited by 232 (14 self)
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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.
Robust discrete optimization and network flows
 Mathematical Programming Series B
, 2003
"... 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 ..."
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Cited by 125 (23 self)
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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 NPhard αapproximable 0 − 1 discrete optimization problem, remains αapproximable. 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.
Robust solutions of Linear Programming problems contaminated with uncertain data
 Mathematical Programming
, 2000
"... 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 wellknown NETLIB collection. We then apply the Robust Optimization methodology (BenTal and Nemirovski [13]; El Ghao ..."
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Cited by 100 (6 self)
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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 wellknown NETLIB collection. We then apply the Robust Optimization methodology (BenTal and Nemirovski [13]; 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
Robust optimization  methodology and applications
, 2002
"... Robust Optimization (RO) is a modeling methodology, combined with computational tools, to process optimization problems in which the data are uncertain and is only known to belong to some uncertainty set. The paper surveys the main results of RO as applied to uncertain linear, conic quadratic and s ..."
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Cited by 84 (3 self)
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Robust Optimization (RO) is a modeling methodology, combined with computational tools, to process optimization problems in which the data are uncertain and is only known to belong to some uncertainty set. The paper surveys the main results of RO as applied to uncertain linear, conic quadratic and semidefinite programming. For these cases, computationally tractable robust counterparts of uncertain problems are explicitly obtained, or good approximations of these counterparts are proposed, making RO a useful tool for realworld applications. We discuss some of these applications, specifically: antenna design, truss topology design and stability analysis/synthesis in uncertain dynamic systems. We also describe a case study of 90 LPs from the NETLIB collection. The study reveals that the feasibility properties of the usual solutions of real world LPs can be severely affected by small perturbations of the data and that the RO methodology can be successfully used to overcome this phenomenon.
Robust Solutions To Uncertain Semidefinite Programs
 SIAM J. OPTIMIZATION
, 1998
"... In this paper we consider semidefinite programs (SDPs) whose data depend on some unknown but bounded perturbation parameters. We seek "robust" solutions to such programs, that is, solutions which minimize the (worstcase) objective while satisfying the constraints for every possible value of paramet ..."
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Cited by 82 (8 self)
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In this paper we consider semidefinite programs (SDPs) whose data depend on some unknown but bounded perturbation parameters. We seek "robust" solutions to such programs, that is, solutions which minimize the (worstcase) objective while satisfying the constraints for every possible value of parameters within the given bounds. Assuming the data matrices are rational functions of the perturbation parameters, we show how to formulate sufficient conditions for a robust solution to exist as SDPs. When the perturbation is "full," our conditions are necessary and sufficient. In this case, we provide sufficient conditions which guarantee that the robust solution is unique and continuous (Hölderstable) with respect to the unperturbed problem's data. The approach can thus be used to regularize illconditioned SDPs. We illustrate our results with examples taken from linear programming, maximum norm minimization, polynomial interpolation, and integer programming.
On tractable approximations of uncertain linear matrix inequalities affected by interval uncertainty
 SIAM Journal on Optimization
, 2002
"... Abstract. We present efficiently verifiable sufficient conditions for the validity of specific NPhard semiinfinite systems of Linear Matrix Inequalities (LMI’s) arising from LMI’s with uncertain data and demonstrate that these conditions are “tight ” up to an absolute constant factor. In particular ..."
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Cited by 38 (10 self)
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Abstract. We present efficiently verifiable sufficient conditions for the validity of specific NPhard semiinfinite systems of Linear Matrix Inequalities (LMI’s) arising from LMI’s with uncertain data and demonstrate that these conditions are “tight ” up to an absolute constant factor. In particular, we prove that given an n × n interval matrix Uρ = {A  Aij − A ∗ ij  ≤ ρCij}, one can build a computable lower bound, accurate within the factor π, on the supremum of those ρ for which 2 all instances of Uρ share a common quadratic Lyapunov function. We then obtain a similar result for the problem of Quadratic Lyapunov Stability Synthesis. Finally, we apply our techniques to the problem of maximizing a homogeneous polynomial of degree 3 over the unit cube. Key words. Robust semidefinite optimization, data uncertainty, Lyapunov stability synthesis, relaxations of combinatorial problems AMS subject classifications. 90C05, 90C25, 90C30
Tractable approximations of robust conic optimization problems
"... Abstract. In earlier proposals, the robust counterpart of conic optimization problems exhibits a lateral increase in complexity, i.e., robust linear programming problems (LPs) become second order cone problems (SOCPs), robust SOCPs become semidefinite programming problems (SDPs), and robust SDPs bec ..."
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Cited by 33 (11 self)
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Abstract. In earlier proposals, the robust counterpart of conic optimization problems exhibits a lateral increase in complexity, i.e., robust linear programming problems (LPs) become second order cone problems (SOCPs), robust SOCPs become semidefinite programming problems (SDPs), and robust SDPs become NPhard. We propose a relaxed robust counterpart for general conic optimization problems that (a) preserves the computational tractability of the nominal problem; specifically the robust conic optimization problem retains its original structure, i.e., robust LPs remain LPs, robust SOCPs remain SOCPs and robust SDPs remain SDPs, and (b) allows us to provide a guarantee on the probability that the robust solution is feasible when the uncertain coefficients obey independent and identically distributed normal distributions. Key words. Robust Optimization – Conic Optimization – Stochastic Optimization 1.
An introduction to convex optimization for communications and signal processing
 IEEE J. Sel. Areas Commun
, 2006
"... Abstract—Convex optimization methods are widely used in the ..."
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Cited by 29 (2 self)
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Abstract—Convex optimization methods are widely used in the
Robust Filtering for DiscreteTime Systems with Bounded Noise and Parametric Uncertainty
 IEEE Trans. Aut. Control
, 2001
"... This paper presents a new approach to finitehorizon guaranteed state prediction for discretetime systems affected by bounded noise and unknownbutbounded parameter uncertainty. Our framework handles possibly nonlinear dependence of the statespace matrices on the uncertain parameters. The main re ..."
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Cited by 24 (2 self)
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This paper presents a new approach to finitehorizon guaranteed state prediction for discretetime systems affected by bounded noise and unknownbutbounded parameter uncertainty. Our framework handles possibly nonlinear dependence of the statespace matrices on the uncertain parameters. The main result is that a minimal confidence ellipsoid for the state, consistent with the measured output and the uncertainty description, may be recursively computed in polynomial time, using interiorpoint methods for convex optimization. With n states, l uncertain parameters appearing linearly in the statespace matrices, with rankone matrix coefficients, the worstcase complexity grows as O(l(n + l) 3:5 ). With unstructured uncertainty in all system matrices, the worstcase complexity reduces to O(n 3:5 ).