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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 93 (4 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.
Selected topics in robust convex optimization
 Math. Prog. B, this issue
, 2007
"... Abstract Robust Optimization is a rapidly developing methodology for handling optimization problems affected by nonstochastic “uncertainbutbounded” data perturbations. In this paper, we overview several selected topics in this popular area, specifically, (1) recent extensions of the basic concept ..."
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Cited by 15 (2 self)
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Abstract Robust Optimization is a rapidly developing methodology for handling optimization problems affected by nonstochastic “uncertainbutbounded” data perturbations. In this paper, we overview several selected topics in this popular area, specifically, (1) recent extensions of the basic concept of robust counterpart of an optimization problem with uncertain data, (2) tractability of robust counterparts, (3) links between RO and traditional chance constrained settings of problems with stochastic data, and (4) a novel generic application of the RO methodology in Robust Linear Control. Keywords optimization under uncertainty · robust optimization · convex programming · chance constraints · robust linear control
Ellipsoidal bounds for uncertain linear equations and dynamical systems
, 2003
"... In this paper, we discuss semidefinite relaxation techniques for computing minimal size ellipsoids that bound the solution set of a system of uncertain linear equations. The proposed technique is based on the combination of a quadratic embedding of the uncertainty, and the Sprocedure. This formulat ..."
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Cited by 11 (0 self)
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In this paper, we discuss semidefinite relaxation techniques for computing minimal size ellipsoids that bound the solution set of a system of uncertain linear equations. The proposed technique is based on the combination of a quadratic embedding of the uncertainty, and the Sprocedure. This formulation leads to convex optimization problems that can be essentially solved in O(n 3)—n being the size of unknown vector — by means of suitable interior point barrier methods, as well as to closed form results in some particular cases. We further show that the uncertain linear equations paradigm can be directly applied to various statebounding problems for dynamical systems subject to setvalued noise and model uncertainty.
Bounding the Solution Set of Uncertain Linear Equations: a Convex Relaxation Approach
"... In this paper, we discuss semidefinite relaxation techniques for computing minimal size ellipsoids that bound the solution set of a system of uncertain linear equations (ULE). The proposed technique is based on the combination of a quadratic embedding of the uncertainty, and the S procedure. The ..."
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In this paper, we discuss semidefinite relaxation techniques for computing minimal size ellipsoids that bound the solution set of a system of uncertain linear equations (ULE). The proposed technique is based on the combination of a quadratic embedding of the uncertainty, and the S procedure. The resulting bounding condition is expressed as a Linear Matrix Inequality (LMI) constraint on the ellipsoid parameters and the additional scaling variables. This formulation leads to a convex optimization problem that can be e#ciently solved by means of interior point barrier methods. 1
Learning FiniteState Machines Statistical and Algorithmic Aspects
"... The present thesis addresses several machine learning problems on generative and predictive models on sequential data. All the models considered have in common that they can be defined in terms of finitestate machines. On one line of work we study algorithms for learning the probabilistic analog of ..."
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The present thesis addresses several machine learning problems on generative and predictive models on sequential data. All the models considered have in common that they can be defined in terms of finitestate machines. On one line of work we study algorithms for learning the probabilistic analog of Deterministic Finite Automata (DFA). This provides a fairly expressive generative model for sequences with very interesting algorithmic properties. Statemerging algorithms for learning these models can be interpreted as a divisive clustering scheme where the “dependency graph ” between clusters is not necessarily a tree. We characterize these algorithms in terms of statistical queries and a use this characterization for proving a lower bound with an explicit dependency on the distinguishability of the target machine. In a more realistic setting, we give an adaptive statemerging algorithm satisfying the stringent algorithmic constraints of the data streams computing paradigm. Our algorithms come with strict PAC learning guarantees. At the heart of statemerging algorithms lies a statistical test for distribution similarity. In the streaming version this is replaced with a bootstrapbased test which yields faster convergence in many situations. We also studied a wider class of models for which the statemerging paradigm also yield PAC learning algorithms. Applications of this method are given to continuoustime Markovian models and stochastic transducers on pairs of aligned sequences. The main tools used for
Robust Optimization — Methodology and Applications 1
"... 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 se ..."
Abstract
 Add to MetaCart
(Show Context)
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. AMS 1991 subject classification. Primary: 90C05, 90C25, 90C30. OR/MS subject classification. Primary: Programming/Convex. Key words. Convex optimization, data uncertainty, robustness, linear programming, quadratic programming,
Learning FiniteState Machines  Statistical and Algorithmic Aspects
, 2013
"... The present thesis addresses several machine learning problems on generative and predictive models on sequential data. All the models considered have in common that they can be defined in terms of finitestate machines. On one line of work we study algorithms for learning the probabilistic analog of ..."
Abstract
 Add to MetaCart
The present thesis addresses several machine learning problems on generative and predictive models on sequential data. All the models considered have in common that they can be defined in terms of finitestate machines. On one line of work we study algorithms for learning the probabilistic analog of Deterministic Finite Automata (DFA). This provides a fairly expressive generative model for sequences with very interesting algorithmic properties. Statemerging algorithms for learning these models can be interpreted as a divisive clustering scheme where the “dependency graph ” between clusters is not necessarily a tree. We characterize these algorithms in terms of statistical queries and a use this characterization for proving a lower bound with an explicit dependency on the distinguishability of the target machine. In a more realistic setting, we give an adaptive statemerging algorithm satisfying the stringent algorithmic constraints of the data streams computing paradigm. Our algorithms come with strict PAC learning guarantees. At the heart of statemerging algorithms lies a statistical test for distribution similarity. In the streaming version this is replaced with a bootstrapbased test which yields faster convergence in many situations. We also studied a wider class of models for which the statemerging paradigm also yield PAC learning algorithms. Applications of this method are given to continuoustime Markovian models and stochastic transducers on pairs of aligned sequences. The main tools used for