Robust convex quadratically constrained programs

Goldfarb, Donald; Iyengar, Garud

doi:10.1007/s10107-003-0425-3

Cited by 98 publications

(75 citation statements)

References 25 publications

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“…11], [19][20][21]. These were followed by many results for specific problem classes or applications; see, e.g., the survey [22]; examples include robust linear programs [2,23,24], robust least-squares [25,26], robust quadratically constrained programs [27], robust semidefinite programs [28], robust conic programming [29], robust discrete optimization [30]. Work focused on specific applications includes robust control [31,32], robust portfolio optimization [33][34][35][36], robust beamforming [37][38][39], robust machine learning [40], and many others.…”

Section: Worst-case Robust Optimizationmentioning

confidence: 99%

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Cutting-set methods for robust convex optimization with pessimizing oracles

Mutapcic

Boyd

2009

Optimization Methods and Software

132

170

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We consider a general worst-case robust convex optimization problem, with arbitrary dependence on the uncertain parameters, which are assumed to lie in some given set of possible values. We describe a general method for solving such a problem, which alternates between optimization and worst-case analysis. With exact worst-case analysis, the method is shown to converge to a robust optimal point. With approximate worst-case analysis, which is the best we can do in many practical cases, the method seems to work very well in practice, subject to the errors in our worst-case analysis. We give variations on the basic method that can give enhanced convergence, reduce data storage, or improve other algorithm properties. Numerical simulations suggest that the method finds a quite robust solution within a few tens of steps; using warm-start techniques in the optimization steps reduces the overall effort to a modest multiple of solving a nominal problem, ignoring the parameter variation. The method is illustrated with several application examples.

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Section: Worst-case Robust Optimizationmentioning

confidence: 99%

“…We now show how to compute sup |x j −x j |≤ρ, |y j −y j |≤ρ ℜ (qw j exp(2πi(x j c + y j s)) , (27) for any given q. As x j and y j range over the box, x j c + y j s varies over the interval (x j c + y j s) ± ρ(|c| + |s|).…”

Section: Exact Pessimization Oraclementioning

confidence: 99%

Cutting-set methods for robust convex optimization with pessimizing oracles

Mutapcic

Boyd

2009

Optimization Methods and Software

132

170

View full text Add to dashboard Cite

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“…This robust technique has obtained prodigious success since the late 1990s, especially in the field of optimization and control with uncertainty parameters Nemirovski 1998, 1999;El Ghaoui and Lebret 1997;Goldfarb and Iyengar 2003a). With respect to portfolio selection, the major contributions have come in the 21st century (see, for example, Rustem et al 2000;Costa and Paiva 2002;Ben-Tal et al 2002;Goldfarb and Iyengar 2003b;El Ghaoui et al 2003;Tütüncü and Koenig 2004;Pinar and Tütüncü 2005;Lutgens and Schotman 2006;Natarajan et al 2009;Garlappi et al 2007;Pinar 2007;Calafiore 2007;Huang et al 2008;Natarajan et al 2008a;Brown and Sim 2008;Natarajan et al 2008b;Shen and Zhang 2008;Elliott and Siu 2008;Zhu and Fukushima 2008).…”

Section: Introductionmentioning

confidence: 99%

Robust portfolios: contributions from operations research and finance

2009

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In this paper we provide a survey of recent contributions to robust portfolio strategies from operations research and finance to the theory of portfolio selection. Our survey covers results derived not only in terms of the standard mean-variance objective, but also in terms of two of the most popular risk measures, mean-VaR and mean-CVaR developed recently. In addition, we review optimal estimation methods and Bayesian robust approaches.

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“…The RGP (6) is a special type of robust convex optimization problem; see, e.g., Ben-Tal and Nemirovski (1998) for more on robust convex optimization. Unlike the various types of robust convex optimization problems that have been studied in the literature (e.g., Ben-Tal and Nemirovski 1999;Ben-Tal et al 2002;Lebret 1997, 1998;Goldfarb and Iyengar 2003;Boni et al 2007), the computational tractability of the RGP (6) is not clear; it is not yet known whether one can reformulate a general RGP as a tractable optimization problem that interior-point or other algorithms can efficiently solve.…”

Section: Robust Geometric Programmingmentioning

confidence: 99%

Tractable approximate robust geometric programming

2007

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The optimal solution of a geometric program (GP) can be sensitive to variations in the problem data. Robust geometric programming can systematically alleviate the sensitivity problem by explicitly incorporating a model of data uncertainty in a GP and optimizing for the worst-case scenario under this model. However, it is not known whether a general robust GP can be reformulated as a tractable optimization problem that interior-point or other algorithms can efficiently solve. In this paper we propose an approximation method that seeks a compromise between solution accuracy and computational efficiency.The method is based on approximating the robust GP as a robust linear program (LP), by replacing each nonlinear constraint function with a piecewise-linear (PWL) convex approximation. With a polyhedral or ellipsoidal description of the uncertain data, the resulting robust LP can be formulated as a standard convex optimization problem that interior-point methods can solve. The drawback of this basic method is that the number of terms in the PWL approximations required to obtain an acceptable approximation error can be very large. To overcome the "curse of dimensionality" that arises in directly approximating the nonlinear constraint functions in the original robust GP, we form a conservative approximation of the original robust GP, which contains only bivariate constraint functions. We show how to find globally optimal PWL approximations of these bivariate constraint functions.

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Robust convex quadratically constrained programs

Cited by 98 publications

References 25 publications

Cutting-set methods for robust convex optimization with pessimizing oracles

Cutting-set methods for robust convex optimization with pessimizing oracles

Robust portfolios: contributions from operations research and finance

Tractable approximate robust geometric programming

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