ED3-Mathfi - CSC215 Introduction aux mathmatiques de la...

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CSC215 : Introduction aux math´ ematiques de la finance 2006 - 2007 E.D. 3 Probl` eme d’optimisation avec contraintes Soit la fonction coˆut J ( x, y, z, t ) = 1 2 [ x 2 + 2 y 2 + z 2 - xy - yz ] - ( x + y + z ) Question 1 Ecrire J ( x, y, z ) sous la forme J ( x, y, z ) = 1 2 ( AX, X ) - ( B, X ) o`u X est le vecteur de IR 3 de composantes x, y, z , ( ., . ) le produit scalaire de IR 3 , A une matrice sym´ etrique et B un vecteur de IR 3 que l’on pr´ ecisera. Question 2 Montrer que la matrice A est d´ efinie positive. En d´ eduire que minimiser J sur IR 3 ´ equivaut `a r´ esoudre un syst` eme lin´ eaire que l’on ´ ecrira. Question 3 On consid` ere maintenant le probl` eme de minimisation de J sous les contraintes x + y = 1 et z - y = 1. Utiliser les th´ eor` emes du cours pour en d´ eduire l’existence et l’unicit´ e du minimum. Question 4 Ecrire le Lagrangien du probl` eme et les conditions d’optimalit´ e. Donner leur ex- pression matricielle. Calculer la solution de ce probl` eme de minimum sous contraintes.
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