Optimization of Perron Eigenvectors and Applications: from Web Ranking to Chronotherapeutics Olivier Fercoq

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Optimization of Perron Eigenvectors and Applications: from Web Ranking to Chronotherapeutics Olivier Fercoq Optimization of Perron eigenvectors and applications: from web ranking to chronotherapeutics Olivier Fercoq To cite this version: Olivier Fercoq. Optimization of Perron eigenvectors and applications: from web ranking to chronother- apeutics. Optimization and Control [math.OC]. Ecole Polytechnique X, 2012. English. pastel- 00743187 HAL Id: pastel-00743187 https://pastel.archives-ouvertes.fr/pastel-00743187 Submitted on 18 Oct 2012 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Th`esepr´esent´eepour obtenir le titre de DOCTEUR DE L’ECOLE´ POLYTECHNIQUE Sp´ecialit´e: Math´ematiques appliqu´ees par Olivier Fercoq Optimization of Perron eigenvectors and applications: From web ranking to chronotherapeutics Optimisation de vecteurs propres de Perron et applications : Du r´ef´erencement de pages web `ala chronoth´erapie Soutenue le 17 septembre 2012 devant le jury compos´ede : Marianne Akian INRIA Saclay et CMAP Ecole Polytechnique co-directrice Konstantin Avratchenkov INRIA Sophia Antipolis rapporteur Mustapha Bouhtou France T´el´ecomR & D co-directeur Jean Clairambault INRIA Rocquencourt et Universit´eParis 6 pr´esident du jury Michel De Lara Ecole des Ponts ParisTech et Universit´eParis-Est examinateur St´ephane Gaubert INRIA Saclay et CMAP Ecole Polytechnique directeur Roberto Tempo Politecnico di Torino examinateur Paul Van Dooren Universit´eCatholique de Louvain rapporteur 2 Summary Search engines play a key role in the World Wide Web. They gather information on the web pages and for each query of a web surfer they give a sorted list of relevant web pages. Internet search engines use a variety of algorithms to sort web pages based on their text content or on the hyperlink structure of the web. Here, we focus on algorithms that use the latter hyperlink structure, called link-based algorithms, among them PageRank, HITS, SALSA and HOTS. The basic notion for all these algorithms is the web graph, which is a digraph with a node for each web page and an arc between nodes i and j if there is a hyperlink from page i to page j. The original problem considered in the present work, carried out as part of a collaboration between INRIA and Orange Labs, is the optimization of the ranking of the pages of a given web site. It consists in finding an optimal outlink strategy maximizing a scalar function of a given ranking subject to design constraints. PageRank, HITS and SALSA are Perron vector rankings, which means that they correspond to the principal eigenvector of a (elementwise) nonnegative matrix. When optimizing the ranking, we thus optimize a scalar utility func- tion of the Perron eigenvector over a set of nonnegative irreducible matrices. The matrix is constructed from the web graph, so controlling the hyperlinks corresponds to controlling the matrix itself. We first study general PageRank optimization problems with a “total income” utility function and design constraints. This case is of particular interest since the value of the PageRank is an acknowledged economic issue. We reduced the PageRank optimization prob- lem to Markov decision problems such that the action sets are implicitly defined as the vertices of polytopes that have a polynomial time separation oracle. We show that such Markov de- cision problems are solvable in polynomial time and we provide a scalable algorithm for the effective resolution of the PageRank optimization problem on large dataset. Then, we study the general problem of optimizing a scalar utility function of the Perron eigenvector over a set of nonnegative irreducible matrices. This covers all Perron vector rankings, including HITS and SALSA. We show that the matrix of partial derivatives of the objective has a low rank and can be computed by an algorithm with the same convergence properties as the power algorithm used to compute the ranking and so the value of the objective. We give an optimization algorithm that couples power and gradient iterations and prove its convergence to a stationary point of the optimization problem. Considering HOTS 4 as a nonlinear Perron vector, we show that the HOTS algorithm converges with a linear rate of convergence, that the objective of the HOTS optimization problem has a low rank and that the coupled power and gradient algorithm applies. Finally, we extend the domain of application of the Perron eigenvalue and eigenvector optimization methods to the optimization of chemotherapy under the McKendrick model of population dynamics. We consider here that the cells behave differently at an hour of the day or another. We want to take advantage of this feature to minimize the growth rate of cancer cell population while we maintain the growth rate of healthy cell population over a given toxicity threshold. The objective and the constraint can be written as the Floquet eigenvalues of age-structured PDE models with periodic coefficients, and they are approximated by Perron eigenvalues in the discretized problem. We search for locally optimal drug infusion strategies by a method of multipliers, where the unconstrained minimizations are performed using the coupled power and gradient algorithm that we have developed in the context of web ranking optimization. R´esum´e Les moteurs de recherche jouent un rˆoleessentiel sur le Web. Ils rassemblent des informations sur les pages web et pour chaque requˆeted’un internaute, ils donnent une liste ordonn´eede pages pertinentes. Ils utilisent divers algorithmes pour classer les pages en fonction de leur contenu textuel ou de la structure d’hyperlien du Web. Ici, nous nous concentrons sur les algorithmes qui utilisent cette structure d’hyperliens, comme le PageRank, HITS, SALSA et HOTS. La notion fondamentale pour tous ces algorithmes est le graphe du web. C’est le graphe orient´equi a un nœud pour chaque page web et un arc entre les nœuds i et j si il y a un hyperlien entre les pages i et j. Le probl`emeoriginal consid´er´edans cette th`ese,r´ealis´ee dans le cadre d’une collabora- tion entre INRIA et Orange Labs, est l’optimisation du r´ef´erencement des pages d’un site web donn´e. Il consiste `atrouver une strat´egieoptimale de liens qui maximise une fonction scalaire d’un classement donn´esous des contraintes de design. Le PageRank, HITS et SALSA classent les pages par un vecteur de Perron, c’est-`a-dire qu’ils correspondent au vecteur pro- pre principal d’une matrice `acoefficients positifs. Quand on optimise le r´ef´erencement, on optimise donc une fonction scalaire du vecteur propre de Perron sur un ensemble de matrices positives irr´eductibles. La matrice est construite `apartir du graphe du web, donc commander les hyperliens revient `acommander la matrice elle-mˆeme. Nous ´etudions d’abord un probl`emeg´en´erald’optimisation du PageRank avec une fonction d’utilit´ecorrespondant au revenu total du site et des contraintes de design. Ce cas est d’un int´erˆetparticulier car pour de nombreux sites la valeur du PageRank est corr´el´ee au chiffre d’affaires. Nous avons r´eduit le probl`emed’optimisation du PageRank `ades probl`emesde d´ecisionmarkoviens dont les ensembles d’action sont d´efinis implicitement comme ´etant les points extrˆemesde polytopes qui ont un oracle de s´eparation polynomial. Nous montrons que de tels probl`emesde d´ecisionmarkoviens sont solubles en temps polynomial et nous donnons un algorithme qui passe `al’´echelle pour la r´esolution effective du probl`emed’optimisation du PageRank sur de grandes bases de donn´ees. Ensuite, nous ´etudions le probl`emeg´en´eralde l’optimisation d’une fonction scalaire du vecteur propre de Perron sur un ensemble de matrices positives irr´eductibles. Cela couvre tous les classements par vecteur de Perron, HITS et SALSA compris. Nous montrons que la matrice des d´eriv´eespartielles de la fonction objectif a un petit rang et peut ˆetrecalcul´ee 6 par un algorithme qui a les mˆemespropri´et´esde convergence que la m´ethode de la puissance utilis´eepour calculer le classement. Nous donnons un algorithme d’optimisation qui couple les it´erations puissance et gradient et nous prouvons sa convergence vers un point station- naire du probl`emed’optimisation. En consid´erant HOTS comme un vecteur de Perron non lin´eaire,nous montrons que l’algorithme HOTS converge g´eom´etriquement et nous r´esolvons l’optimisation locale de HOTS. Finalement, nous ´etendons le domaine d’application des m´ethodes d’optimisation du vecteur propre et de la valeur propre de Perron `al’optimisation de la chimioth´erapie, sous l’hypoth`eseque les cellules se comportent diff´eremment suivant l’heure de la journ´ee. Nous voulons profiter de cette caract´eristique pour minimiser le taux de croissance des cellules canc´ereuses tout en maintenant le taux de croissance des cellules saines au dessus d’un seuil de toxicit´edonn´e. L’objectif et la contrainte peuvent s’´ecrirecomme les valeurs propres de Floquet de mod`eles d’EDP structur´esen ˆageavec des coefficients p´eriodiques, qui sont ap- proch´espar des valeurs propres de Perron dans le probl`emediscr´etis´e. Nous cherchons des strat´egiesd’injection de m´edicament localement optimales par une m´ethode des multiplica- teurs o`ules minimisations sans contrainte sont faites en utilisant l’algorithme couplant les it´erations puissance et gradient d´evelopp´edans le cadre de l’optimisation du r´ef´erencement.
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