You need JavaScript to view this

Some applications of the notion of duality in the research of support functions of the solutions of linear equations; Quelques applications de la notion de dualite a la recherche de fonctions d'appui de solutions d'equations lineaires

Abstract

Let us consider the linear equation: Ax = b where A is a linear continue application of E in F (E and F are Banach spaces), b is an element of F approximately known x the unknown: a priori x belongs to C (C is a convex set of E). The set of solutions for this problem is defined by: D =[x belongs to C| ||Ax - b|| {<=}{epsilon}] {epsilon} given positive number. D is described by the set of values that <c,x> takes when x belongs to D (c is a fixed element of the dual of E). The convex optimisation problems must be solved: (1) inf (sup) (<c,x> | x belongs to D). We find that the dual problems of (1) are very interesting intermediaries for the theoretic study (existence) as well as for the search of approximate solutions algorithms. The following example (which is by itself interesting: moment problem) is carefully studied:D= [{mu}: RADON non negative measures on [0,1] | {sigma} i = 1 to n [{integral}0 to 1 ai(t)d{mu}-bi]{sup 2} {<=} {epsilon}{sup 2}] inf (sup)[{integral}0 to 1 c(t)d{mu} | x belongs to D]. (author) [French] Soit l'equation lineaire: Ax = b ou A est une application  More>>
Authors:
Martinet, B [1] 
  1. Commissariat a l'Energie Atomique, Grenoble (France). Centre d'Etudes Nucleaires
Publication Date:
Jul 01, 1969
Product Type:
Technical Report
Report Number:
CEA-R-3871
Resource Relation:
Other Information: PBD: 1969
Subject:
99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; BANACH SPACE; DUALITY; EQUATIONS; FUNCTIONS; MATHEMATICAL SOLUTIONS; MATHEMATICS
OSTI ID:
20552704
Research Organizations:
CEA Grenoble, 38 (France)
Country of Origin:
France
Language:
French
Other Identifying Numbers:
TRN: FR04R3871007961
Availability:
Available from INIS in electronic form
Submitting Site:
FRN
Size:
29 pages
Announcement Date:
Feb 20, 2005

Citation Formats

Martinet, B. Some applications of the notion of duality in the research of support functions of the solutions of linear equations; Quelques applications de la notion de dualite a la recherche de fonctions d'appui de solutions d'equations lineaires. France: N. p., 1969. Web.
Martinet, B. Some applications of the notion of duality in the research of support functions of the solutions of linear equations; Quelques applications de la notion de dualite a la recherche de fonctions d'appui de solutions d'equations lineaires. France.
Martinet, B. 1969. "Some applications of the notion of duality in the research of support functions of the solutions of linear equations; Quelques applications de la notion de dualite a la recherche de fonctions d'appui de solutions d'equations lineaires." France.
@misc{etde_20552704,
title = {Some applications of the notion of duality in the research of support functions of the solutions of linear equations; Quelques applications de la notion de dualite a la recherche de fonctions d'appui de solutions d'equations lineaires}
author = {Martinet, B}
abstractNote = {Let us consider the linear equation: Ax = b where A is a linear continue application of E in F (E and F are Banach spaces), b is an element of F approximately known x the unknown: a priori x belongs to C (C is a convex set of E). The set of solutions for this problem is defined by: D =[x belongs to C| ||Ax - b|| {<=}{epsilon}] {epsilon} given positive number. D is described by the set of values that <c,x> takes when x belongs to D (c is a fixed element of the dual of E). The convex optimisation problems must be solved: (1) inf (sup) (<c,x> | x belongs to D). We find that the dual problems of (1) are very interesting intermediaries for the theoretic study (existence) as well as for the search of approximate solutions algorithms. The following example (which is by itself interesting: moment problem) is carefully studied:D= [{mu}: RADON non negative measures on [0,1] | {sigma} i = 1 to n [{integral}0 to 1 ai(t)d{mu}-bi]{sup 2} {<=} {epsilon}{sup 2}] inf (sup)[{integral}0 to 1 c(t)d{mu} | x belongs to D]. (author) [French] Soit l'equation lineaire: Ax = b ou A est une application lineaire continue de E dans F (E,F sont des espaces de Banach), b un element de F connu approximativement, x l'inconnue: on sait a priori que x appartient a C (C partie convexe de E). On definit l'ensemble des solutions de ce probleme par: D =[x appartient a C| ||Ax - b|| {<=}{epsilon}] {epsilon} nombre positif donne. On decrit D par l'ensemble des valeurs que peut prendre <c,x> lorsque x parcourt D (c element fixe du dual de E), On doit resoudre les problemes d'optimisation convexe: (1) inf (sup) (<c,x> | x appartient a D). Les problemes duaux de (1) apparaissent comme des intermediaires tres interessants tant pour l'etude theorique (existence) que pour la recherche d'algorithmes de resolution approchee. On etudie tout particulierement l'exemple suivant (qui presente un interet propre: probleme des moments): D[{mu}: mesures de RADON non negatives sur [0,1] | {sigma} i = 1 to n [{integral}0 to 1 ai(t)d{mu}-bi]{sup 2} {<=} {epsilon}{sup 2}] inf (sup)[{integral}0 to 1 c(t)d{mu} | x appartient a D]. (auteur)}
place = {France}
year = {1969}
month = {Jul}
}