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97
Adaptive Robust Optimization for the Security Constrained Unit Commitment Problem
, 2011
"... 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 p ..."
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Cited by 41 (2 self)
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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 twostage 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 hardtoobtain 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 realworld 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.
Constructing uncertainty sets for robust linear optimization
, 2006
"... doi 10.1287/opre.1080.0646 ..."
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 33 (4 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 ).
A Robust Optimization Solution to the Data Hiding Problem using Distributed Source Coding Principles
 in Proc. of SPIE Vol. 3974: Image and Video Communications and Processing 2000
, 2000
"... Inspired by a recently proposed constructive framework for the distributed source coding problem, 1 we propose a powerful constructive approach to the watermarking problem, emphasizing the dual roles of "source codes" and "channel codes." In our framework, we explore various sour ..."
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Cited by 32 (1 self)
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Inspired by a recently proposed constructive framework for the distributed source coding problem, 1 we propose a powerful constructive approach to the watermarking problem, emphasizing the dual roles of "source codes" and "channel codes." In our framework, we explore various source and channel codes to achieve watermarks that are robust to attackers in terms of maximizing the distortion between the corrupted codedsource signal and the original signal while holding the distortion between the codedsource signal and the original signal constant. We solve the resulting combinatorial optimization problem using an original technique based on robust optimization and convex programming. Keywords: Data Hiding, Digital Watermarking, Multimedia, Convex Optimization, Robustness 1. INTRODUCTION Digital watermarking (data hiding) is an emerging research area that has received a considerable amount of attention in recent years. The basic idea behind digital watermarking is to embed information...
Robust Convex Quadratically Constrained Programs
 Mathematical Programming
, 2002
"... In this paper we study robust convex quadratically constrained programs, a subset of the class of robust convex programs introduced by BenTal and Nemirovski [4]. Unlike [4], our focus in this paper is to identify uncertainty structures that allow the corresponding robust quadratically constrained p ..."
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Cited by 32 (2 self)
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In this paper we study robust convex quadratically constrained programs, a subset of the class of robust convex programs introduced by BenTal and Nemirovski [4]. Unlike [4], our focus in this paper is to identify uncertainty structures that allow the corresponding robust quadratically constrained programs to be reformulated as secondorder cone programs. We propose three classes of uncertainty sets that satisfy this criterion and present examples where these classes of uncertainty sets are natural. 1 Problem formulation A generic quadratically constrained program (QCP) is defined as follows.
Cuts for mixed 01 conic programming
, 2005
"... In this we paper we study techniques for generating valid convex constraints for mixed 01 conic programs. We show that many of the techniques developed for generating linear cuts for mixed 01 linear programs, such as the Gomory cuts, the liftandproject cuts, and cuts from other hierarchies of ti ..."
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Cited by 31 (0 self)
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In this we paper we study techniques for generating valid convex constraints for mixed 01 conic programs. We show that many of the techniques developed for generating linear cuts for mixed 01 linear programs, such as the Gomory cuts, the liftandproject cuts, and cuts from other hierarchies of tighter relaxations, extend in a straightforward manner to mixed 01 conic programs. We also show that simple extensions of these techniques lead to methods for generating convex quadratic cuts. Gomory cuts for mixed 01 conic programs have interesting implications for comparing the semidefinite programming and the linear programming relaxations of combinatorial optimization problems, e.g. we show that all the subtour elimination inequalities for the traveling salesman problem are rank1 Gomory cuts with respect to a single semidefinite constraint. We also include results from our preliminary computational experiments with these cuts.
Stochastic Algorithms for Exact and Approximate Feasibility of Robust LMIs
, 2001
"... In this note, we discuss fast randomized algorithms for determining an admissible solution for robust linear matrix inequalities (LMIs) of the form ( 1) 0, where is the optimization variable and 1 is the uncertainty, which belongs to a given set 1. The proposed algorithms are based on uncertainty r ..."
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Cited by 29 (4 self)
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In this note, we discuss fast randomized algorithms for determining an admissible solution for robust linear matrix inequalities (LMIs) of the form ( 1) 0, where is the optimization variable and 1 is the uncertainty, which belongs to a given set 1. The proposed algorithms are based on uncertainty randomization: the first algorithm finds a robust solution in a finite number of iterations with probability one, if a strong feasibility condition holds. In case no robust solution exists, the second algorithm computes an approximate solution which minimizes the expected value of a suitably selected feasibility indicator function. The theory is illustrated by examples of application to uncertain linear inequalities and quadratic stability of interval matrices.
Distributionally robust optimization and its tractable approximations
 Operations Research
"... In this paper, we focus on a linear optimization problem with uncertainties, having expectations in the objective and in the set of constraints. We present a modular framework to obtain an approximate solution to the problem that is distributionally robust, and more flexible than the standard techni ..."
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Cited by 24 (4 self)
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In this paper, we focus on a linear optimization problem with uncertainties, having expectations in the objective and in the set of constraints. We present a modular framework to obtain an approximate solution to the problem that is distributionally robust, and more flexible than the standard technique of using linear rules. Our framework begins by firstly affinelyextending the set of primitive uncertainties to generate new linear decision rules of larger dimensions, and are therefore more flexible. Next, we develop new piecewiselinear decision rules which allow a more flexible reformulation of the original problem. The reformulated problem will generally contain terms with expectations on the positive parts of the recourse variables. Finally, we convert the uncertain linear program into a deterministic convex program by constructing distributionally robust bounds on these expectations. These bounds are constructed by first using different pieces of information on the distribution of the underlying uncertainties to develop separate bounds, and next integrating them into a combined bound that is better than each of the individual bounds.
Robust design of biological experiments
 In NIPS
, 2006
"... We address the problem of robust, computationallyefficient design of biological experiments. Classical optimal experiment design methods have not been widely adopted in biological practice, in part because the resulting designs can be very brittle if the nominal parameter estimates for the model ar ..."
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Cited by 20 (0 self)
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We address the problem of robust, computationallyefficient design of biological experiments. Classical optimal experiment design methods have not been widely adopted in biological practice, in part because the resulting designs can be very brittle if the nominal parameter estimates for the model are poor, and in part because of computational constraints. We present a method for robust experiment design based on a semidefinite programming relaxation. We present an application of this method to the design of experiments for a complex calcium signal transduction pathway, where we have found that the parameter estimates obtained from the robust design are better than those obtained from an “optimal ” design. 1
Constructing risk measures from uncertainty sets
, 2005
"... We propose a unified theory that links uncertainty sets in robust optimization to risk measures in portfolio optimization. We illustrate the correspondence between uncertainty sets and some popular risk measures in finance, and show how robust optimization can be used to generalize the concepts of t ..."
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Cited by 19 (1 self)
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We propose a unified theory that links uncertainty sets in robust optimization to risk measures in portfolio optimization. We illustrate the correspondence between uncertainty sets and some popular risk measures in finance, and show how robust optimization can be used to generalize the concepts of these measures. We also show that by using properly defined uncertainty sets in robust optimization models, one can in fact construct coherent risk measures. Our approach to creating coherent risk measures is easy to apply in practice, and computational experiments suggest that it may lead to superior portfolio performance. Our results have implications for efficient portfolio optimization under different measures of risk.