Sur la convergence d'un système différentiel de premier ordre, vectoriel, ordinaire, linéaire non-homogene et non-autonome

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--~

LABORATOiRE

LYSE ET

MODÉLiSA-TiON

DE SYSTEMES POUR

AIDE À LA DÉCISION

.

CNRS

SUR LA CONVERGENCE D'UN SYSTÈME

DIFFÉREN-TIEL DE PREMIER ORDRE, VECTORIEL, ORDINAIRE,

LINÉAIRE NON-HOMOGÈNE ET NON-AUTONOME

jlTÉ DE RECHERCHE

ASSOCIÉE

ESA

7024

.

Joël Biot

CERMSEM,UniversitéParis1

106-112, boulevardde l'Hôpital

75647 ParisCedex 13, France

Joel.Blot@univ-parisl.fr

UNiVERSITE

PARIS DAUPHINE

PLACE DU \1' DE

LATTRE DE

CEDEX

16 Vangelis Paschos

LAMSADE, Université Paris-Dauphine

Place du Maréchal De Lattre de Tassigny

75775 Paris Cedex 16, France

paschos@lamsade.dauphine.fr

TASS GNY

F-75775

PARIS

.

TELEPHONE

33 1)(01)

44 05 44 34

TÉLECOPIE

\ 44 05 40 91

.

E-MAIL

_{NOTE DE RECHERCHE (RESEARCH PAPER) n° 27}

_{septembre 2000}

t!amsade

aauph nG.fr

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nefr

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ordinaire, linéaire non-homogene et non-autonome

On the onvergen e of a ve torial rst-order linear non-homogeneous

non-autonomous ordinary dierential equation

Joe lBlot

CERMSEM, UniversitéParis I

106-112boulevard de l'Hpital 75647 Paris Cedex 13,Fran e

Joel.Blotuniv-paris1.fr

VangelisTh. Pas hos

LAMSADE, Université Paris-Dauphine

Pla e duMaré hal deLattre de Tassigny 75775 Paris Cedex16,Fran e pas hoslamsade.dauphine.fr

Abstra t

We present a detailed study of the asymptoti al behaviourof the solution of a ve to-rialrst-orderlinearnon-homogeneousnon-autonomousordinarydierentialequation. This systemdes ribes,in termsof ontinuous mathemati s,the behaviour of thenatural greedy algorithmforthe minimumhittingset problemandit isdened in ourpaperAverage ase analysis ofgreedyalgorithms for optimisationproblems onsetsystems.

Résumé

Nousprésentonsuneétudedétailléedu omportementasymptotiquedelasolutiond'un système diérentiel de premier ordre, ve toriel, ordinaire, linéaire non-homogene et non-autonome.Cesystèmemodéliseentermesdemathématiques ontinuesle omportementde l'algorithme glouton pourle problème dutransversalminimum d'un hypergrapheet il est dénidansnotrearti leAverage aseanalysisofgreedyalgorithmsforoptimisationproblems onsetsystems.

In[2 ℄wehaveprovedthatthebehaviourofthenaturalgreedyalgorithmforminimumhittingset problem(or, equivalently, for minimum set overingproblem) an be des ribed bythe following

systemof equations: Em j = (j+1)m j+1 S jm j S; 1jh 1; Em h = 1 hm h S:

Then, usingstandard results on erningthe approximationofsmall stepsMarkov hains ([3℄), the solution of the above system is well asymptoti ally approximated by the solution ~y of the

following systemofdierential equations(where we useNewton'snotation for dy=d)

(S ;h ) 8 < : _ y ;h;j () = (j+1) h(k 1) R ;h hk y ;h;j+1 () j h(k 1) R ;h hk y ;h;j (); 1jh 1 _ y ;h;h () = 1 h h(k 1) R ;h hk y ;h;h ()

in the sensethat, sup

jy ;h;j

() (1=n)m

j

()j!0asn!1, inprobabilityfor ea h1j h ifonly the initial onditions y

;h;j

(0), 1j tend to xed limits (where nis the size of the instan eof theminimum hittingsetproblem; forthe rest ofthe notations, the interestedreader

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;h ; y I ;;1 (0) = e + 1 X j=+1 e j (j 1)! y I ;;j (0) = e j j! ; 1j:

Observe that for ea h j, the orresponding RHS of the above equalities is equal to the almost sure limit ofm j (0)=n asn!1. Also,R ; = P j=1 jy I ;;j . Wedenote by ; the rst time at

whi h the last oordinate be omes0:

;

=minf>0:y ;;

()=0g

and set, for1j 1,

y I ; 1;j = y ;;j ( ; ) R ; 1 = 1 X j=1 jy I ; 1;j :

Then, weturnto thesystem(S ; 1

) ofdimension( 1)( 1)whi h weintegrate with the initial onditions: =0 andy

; 1;j (0)=y I ; 1;j , 1j 1. We dene ; 1 by ; 1 =minf >0:y ; 1; 1 ()=0g

and we repeat indu tively this pro edure for ea h h 1, taking as initial onditions for the

system(S

;h

)the nal onditionsfor (S ;h+1 ). We dene = X h=1 ;h :

In [2℄ we have given the main steps and intermediate results that prove the onvergen e of

. Sin e this proof is not only usefull in des ribing the asymptoti behaviour of the hitting set

algorithm but, also, interesting by itself, in this manus ript we present in details the study of the asymptoti behaviourof the sequen e(

)

2IN .

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[1℄ N. Bourbaki,Fon tions d'une Variable Réelle(Hermann, Paris, 1976).

[2℄ J. Blot, W. Fernandez de la Vega, V. Th. Pas hos and R. Saad, Average ase analysis of greedy algorithms for optimisation problems on set systems,Theoreti al Computer

S i-en e,147 (1995), 267298.

[3℄ T. G. Kurtz, Solutions of ordinary dierential equations as limits of pure jump Markov

pro esses,J. Appl.Prob.7 (1970) 4958.

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