Titre :	EXTENSION OF INTERIOR POINT METHOD FOR LINEAR PROGRAMMING
Type de document :	texte imprimé
Auteurs :	Sarra Benfriha, Auteur ; Manal Benlamri ; Leulmi ,Assma, Directeur de thèse
Editeur :	Sétif:UFS
Année de publication :	2024
Importance :	1 vol (50 f.)
Format :	29 cm
Langues :	Anglais (eng)
Catégories :	Thèses & Mémoires:Mathématique
Mots-clés :	Linear programming Interior point methods logarithmic barrier methods Approximate functions
Index. décimale :	510-Mathématique
Résumé :	We have highlighted in our study of linear programming problem that are of great importance in problems that mimic real life. The aim of our study is to present a theoretical, algorithmic and numerical study of the logarithmic Barrier method for resolving Linear Programming problem (LP). Firstly, we determine the direction by Newton’s method. Then, we establish an efficient algorithm to compute the displacement step according to the direction with the technic of approximate functions. We present the conceptual results that ensure the existence and uniqueness of the optimal solution to the approximate problem of LP as well as its convergence to the optimal solution of the initial problem. Finally, their convergence results well confirmed. The results obtained through the numerical study allowed us to give some conclusions regarding the behavior of the proposed approximate function to find the displacement step then, the optimal solution to the problem.
Note de contenu :	Sommaire Introductionv 1 PreliminariesandFundamentals1 1.1Convexanalysis.............................1 1.1.1A¢nesetsandapplications..................1 1.1.2Convexsets...........................2 1.1.3Convexcones..........................3 1.1.4Convexfunctions........................4 1.1.5Semi-continuity.........................5 1.2Penalty(Barrier)function.......................6 2 Linearprogramming8 2.1Notionofmathematicalprogramming.................8 2.2QualiÂ…cationofconstraints.......................10 2.2.1Optimalityconditions.....................10 2.2.2Mainresultsofexistenceanduniquenessof (MP) . .....11 2.2.3Lagrangianduality........................12 2.3Generaltheoryoflinearprogramming.................12 2.3.1Usualformsofalinearprogram................13 2.3.2DualityinLinearProgramming................15 2.4Methodofsolvingalinearprogramming...............17 2.4.1Simplexmethod.........................17 2.4.2Gradientmethods........................18 2.4.3Interiorpointmethod......................18 2.4.4Linesearchmethods......................20 3 Alogarithmicbarriermethodforlinearprogrammingbasedona newapproximatefunction 25 3.1Introduction...............................25 3.2Theoreticalaspectoftheproblem (Dr) . ...............27 3.2.1Existenceanduniquenessofthesolutionofthedisturbed problem.............................27 3.2.2Convergenceofthedisturbedproblemtowardstheproblem (D) when r tendstowardszero................29 3.3Numericalaspectoftheproblem (Dr) . ................29 3.3.1Calculationofthedisplacementstep tk . ...........30 3.3.2Calculationofthedi¤erentvaluesofbt . ............31 3.4approximatefunctionsof  . ......................32 3.4.1Primaryapproximatefunction.................32 3.4.2Secondapproximatefunction(Anewapproximatefunction)33 3.4.3Calculationofthedisplacementstep.............34 3.4.4Algorithm............................35 4 Numericalexperimentsandcommentaries36 4.1NumericalSimulations.........................36 4.1.1Comparativetable.......................44 4.2Commentary..............................46 4.3Generalconclusionandperspection..................46
Côte titre :	MAM/0725

EXTENSION OF INTERIOR POINT METHOD FOR LINEAR PROGRAMMING [texte imprimé] / Sarra Benfriha, Auteur ; Manal Benlamri ; Leulmi ,Assma, Directeur de thèse . - [S.l.] : Sétif:UFS, 2024 . - 1 vol (50 f.) ; 29 cm.
Langues : Anglais (eng)

Catégories :	Thèses & Mémoires:Mathématique
Mots-clés :	Linear programming Interior point methods logarithmic barrier methods Approximate functions
Index. décimale :	510-Mathématique
Résumé :	We have highlighted in our study of linear programming problem that are of great importance in problems that mimic real life. The aim of our study is to present a theoretical, algorithmic and numerical study of the logarithmic Barrier method for resolving Linear Programming problem (LP). Firstly, we determine the direction by Newton’s method. Then, we establish an efficient algorithm to compute the displacement step according to the direction with the technic of approximate functions. We present the conceptual results that ensure the existence and uniqueness of the optimal solution to the approximate problem of LP as well as its convergence to the optimal solution of the initial problem. Finally, their convergence results well confirmed. The results obtained through the numerical study allowed us to give some conclusions regarding the behavior of the proposed approximate function to find the displacement step then, the optimal solution to the problem.
Note de contenu :	Sommaire Introductionv 1 PreliminariesandFundamentals1 1.1Convexanalysis.............................1 1.1.1A¢nesetsandapplications..................1 1.1.2Convexsets...........................2 1.1.3Convexcones..........................3 1.1.4Convexfunctions........................4 1.1.5Semi-continuity.........................5 1.2Penalty(Barrier)function.......................6 2 Linearprogramming8 2.1Notionofmathematicalprogramming.................8 2.2QualiÂ…cationofconstraints.......................10 2.2.1Optimalityconditions.....................10 2.2.2Mainresultsofexistenceanduniquenessof (MP) . .....11 2.2.3Lagrangianduality........................12 2.3Generaltheoryoflinearprogramming.................12 2.3.1Usualformsofalinearprogram................13 2.3.2DualityinLinearProgramming................15 2.4Methodofsolvingalinearprogramming...............17 2.4.1Simplexmethod.........................17 2.4.2Gradientmethods........................18 2.4.3Interiorpointmethod......................18 2.4.4Linesearchmethods......................20 3 Alogarithmicbarriermethodforlinearprogrammingbasedona newapproximatefunction 25 3.1Introduction...............................25 3.2Theoreticalaspectoftheproblem (Dr) . ...............27 3.2.1Existenceanduniquenessofthesolutionofthedisturbed problem.............................27 3.2.2Convergenceofthedisturbedproblemtowardstheproblem (D) when r tendstowardszero................29 3.3Numericalaspectoftheproblem (Dr) . ................29 3.3.1Calculationofthedisplacementstep tk . ...........30 3.3.2Calculationofthedi¤erentvaluesofbt . ............31 3.4approximatefunctionsof  . ......................32 3.4.1Primaryapproximatefunction.................32 3.4.2Secondapproximatefunction(Anewapproximatefunction)33 3.4.3Calculationofthedisplacementstep.............34 3.4.4Algorithm............................35 4 Numericalexperimentsandcommentaries36 4.1NumericalSimulations.........................36 4.1.1Comparativetable.......................44 4.2Commentary..............................46 4.3Generalconclusionandperspection..................46
Côte titre :	MAM/0725