Generalized Gauss inequalities via semidefinite programming

A sharp upper bound on the probability of a random vector falling outside a polytope, based solely on the first and second moments of its distribution, can be computed efficiently using semidefinite programming. However, this Chebyshev-type bound tends to be overly conservative since it is determine...

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Vydáno v:	Mathematical programming Ročník 156; číslo 1-2; s. 271 - 302
Hlavní autoři:	Van Parys, Bart P. G., Goulart, Paul J., Kuhn, Daniel
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Berlin/Heidelberg Springer Berlin Heidelberg 01.03.2016 Springer Nature B.V
Témata:	Calculus of Variations and Optimal Control; Optimization Chebyshev approximation Combinatorics Computation Computer programming Decision theory Falling Full Length Paper Inequalities Inequality Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical programming Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Optimization Probability Random variables Representations Studies Theoretical Vectors (mathematics) United States > US 90C15 Stochastic programming 90C22 Semidefinite programming
ISSN:	0025-5610, 1436-4646
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Abstract	A sharp upper bound on the probability of a random vector falling outside a polytope, based solely on the first and second moments of its distribution, can be computed efficiently using semidefinite programming. However, this Chebyshev-type bound tends to be overly conservative since it is determined by a discrete worst-case distribution. In this paper we obtain a less pessimistic Gauss-type bound by imposing the additional requirement that the random vector’s distribution must be unimodal. We prove that this generalized Gauss bound still admits an exact and tractable semidefinite representation. Moreover, we demonstrate that both the Chebyshev and Gauss bounds can be obtained within a unified framework using a generalized notion of unimodality. We also offer new perspectives on the computational solution of generalized moment problems, since we use concepts from Choquet theory instead of traditional duality arguments to derive semidefinite representations for worst-case probability bounds.
AbstractList	A sharp upper bound on the probability of a random vector falling outside a polytope, based solely on the first and second moments of its distribution, can be computed efficiently using semidefinite programming. However, this Chebyshev-type bound tends to be overly conservative since it is determined by a discrete worst-case distribution. In this paper we obtain a less pessimistic Gauss-type bound by imposing the additional requirement that the random vector's distribution must be unimodal. We prove that this generalized Gauss bound still admits an exact and tractable semidefinite representation. Moreover, we demonstrate that both the Chebyshev and Gauss bounds can be obtained within a unified framework using a generalized notion of unimodality. We also offer new perspectives on the computational solution of generalized moment problems, since we use concepts from Choquet theory instead of traditional duality arguments to derive semidefinite representations for worst-case probability bounds.
Author	Van Parys, Bart P. G. Goulart, Paul J. Kuhn, Daniel
Author_xml	– sequence: 1 givenname: Bart P. G. surname: Van Parys fullname: Van Parys, Bart P. G. email: bartvan@control.ee.ethz.ch organization: Automatic Control Laboratory, Swiss Federal Institute of Technology Zürich (ETHZ) – sequence: 2 givenname: Paul J. surname: Goulart fullname: Goulart, Paul J. organization: Department of Engineering Science, University of Oxford – sequence: 3 givenname: Daniel surname: Kuhn fullname: Kuhn, Daniel organization: Risk Analytics and Optimization Chair, École Polytechnique Fédérale de Lausanne (EPFL)
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SubjectTerms	Calculus of Variations and Optimal Control; Optimization Chebyshev approximation Combinatorics Computation Computer programming Decision theory Falling Full Length Paper Inequalities Inequality Mathematical analysis Mathematical and Computational Physics Mathematical Methods in Physics Mathematical programming Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Optimization Probability Random variables Representations Studies Theoretical Vectors (mathematics)
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