# Variational Analysis of Functionals of a Poisson Process

Abstract : Let $F$ be a functional of a Poisson process whose distribution is determined by the intensity measure $\mu$. Considering the expectation $\E$ as a function on the cone $\MM$ of positive finite measures $\mu$, we derive closed form expressions for the Fréchet derivatives of all orders that generalise the perturbation analysis formulae for Poisson processes. Variational methods developed for the space $\MM$ allow us to obtain first and second-order sufficient conditions for different types of constrained optimisation problems for $\E$. We study in detail optimisation in the class of measures with a fixed total mass $a$ and develop technique that often allows us to obtain the asymptotic behaviour of the optimal intensity measure in the high intensity settings when $a$ grows to infinity. We give applications of our methods to design of experiments, spline approximation of convex functions, optimal placement of stations in telecommunication studies and others. We scetch possible numerical algorithms of the steepest descend type based on the obtained explicit form of the gradient.
Keywords :
Type de document :
Rapport
RR-3302, INRIA. 1997
Domaine :
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https://hal.inria.fr/inria-00073386
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Soumis le : mercredi 24 mai 2006 - 12:40:49
Dernière modification le : samedi 27 janvier 2018 - 01:31:03
Document(s) archivé(s) le : dimanche 4 avril 2010 - 21:12:18

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• HAL Id : inria-00073386, version 1

### Citation

Ilya Molchanov, Sergei Zuyev. Variational Analysis of Functionals of a Poisson Process. RR-3302, INRIA. 1997. 〈inria-00073386〉

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