Journal ArticleDOI
Brief Probabilistic solutions to some NP-hard matrix problems
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TLDR
This paper uses standard bounds on empirical probabilities as well as recent results from statistical learning theory on the VC-dimension of families of sets defined by a finite number of polynomial inequalities to show that for each of the above problems, there exists aPolynomial-time randomized algorithm that can provide a yes or no answer to arbitrarily small levels of accuracy and confidence.About:
This article is published in Automatica.The article was published on 2001-09-01. It has received 72 citations till now. The article focuses on the topics: Bounded function & VC dimension.read more
Citations
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Journal ArticleDOI
The scenario approach to robust control design
TL;DR: A rich family of control problems which are in general hard to solve in a deterministically robust sense is therefore amenable to polynomial-time solution, if robustness is intended in the proposed risk-adjusted sense.
Journal ArticleDOI
Survey A survey of computational complexity results in systems and control
TL;DR: This paper considers problems related to stability or stabilizability of linear systems with parametric uncertainty, robust control, time-varying linear systems, nonlinear and hybrid systems, and stochastic optimal control.
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Randomized algorithms for robust controller synthesis using statistical learning theory
TL;DR: In this article, it is shown that the uniform convergence of empirical means (UCEM) property holds in any problem in which the satisfaction of a performance constraint can be expressed in terms of a finite number of polynomial inequalities.
Journal ArticleDOI
Randomized Strategies for Probabilistic Solutions of Uncertain Feasibility and Optimization Problems
TL;DR: A randomized algorithm is proposed that provides a probabilistic solution circumventing the potential conservatism of the bounds previously derived, and it is proved that the required sample size is inversely proportional to the accuracy for fixed confidence.
Journal ArticleDOI
Randomized methods for design of uncertain systems
TL;DR: The sample complexity of various constrained control problems is derived, showing the key role played by the binomial distribution and related tail inequalities, and providing the sample complexity which guarantees that the solutions obtained with SPV algorithms meet some pre-specified probabilistic accuracy and confidence.
References
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Book
The Nature of Statistical Learning Theory
TL;DR: Setting of the learning problem consistency of learning processes bounds on the rate of convergence ofLearning processes controlling the generalization ability of learning process constructing learning algorithms what is important in learning theory?
Book
Computers and Intractability: A Guide to the Theory of NP-Completeness
TL;DR: The second edition of a quarterly column as discussed by the authors provides a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NP-Completeness,” W. H. Freeman & Co., San Francisco, 1979.
Book ChapterDOI
On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities
TL;DR: This chapter reproduces the English translation by B. Seckler of the paper by Vapnik and Chervonenkis in which they gave proofs for the innovative results they had obtained in a draft form in July 1966 and announced in 1968 in their note in Soviet Mathematics Doklady.