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Titre : APPLICATION OF INTERIOR POINT METHODS FOR NON-LINEAR PROGRAMMING Type de document : document électronique Auteurs : Choubeila Souli, Auteur ; Assma Leulmi, Directeur de thèse Editeur : Setif:UFA Année de publication : 2025 Importance : 1 vol (86 f.) Format : 29 cm Langues : Anglais (eng) Catégories : Thèses & Mémoires:Mathématique Mots-clés : Nonlinear optimization problem
Interior point method
Logarithmic barrier
Minorant function
Conjugate gradient method
Global convergence
Descent directionIndex. décimale : 510 - Mathématique Résumé :
In this thesis, we focus on the theoretical analysis and numerical investigation of specific interior point
and conjugate gradient methods for solving nonlinear optimization problems.
First, we propose logarithmic barrier approaches for constrained convex nonlinear optimization
problems, where the barrier parameter is treated as a vector. This is followed by analytical studies in
which the step-length is determined using the minorant function technique. The numerical findings reveal
that the proposed methods exhibit both effectiveness and robustness.
In addition, we develop new conjugate gradient methods for solving unconstrained optimization problems.
These methods generate descent directions without requiring line search techniques. Moreover, they
exhibit global convergence under mild assumptions. Numerical results indicate that the proposed methods
are both effective and robust in addressing various unconstrained optimization and image restoration
problems.Note de contenu : Sommaire
Introduction 1
1 Preliminaries and Fundamental Concepts 5
1.1 Convex Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.1 Affine sets and functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.2 Convexity concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.3 Differentiability and convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.4 Lower and upper semicontinuous functions . . . . . . . . . . . . . . . . . . . . 10
1.2 Mathematical Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.1 Unconstrained optimization problems . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.2 Constrained optimization problems . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Conjugate GradientMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.1 Descent direction methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.2 Line search methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.3 The linear conjugate gradient methods . . . . . . . . . . . . . . . . . . . . . . . . 25
1.3.4 The nonlinear conjugate gradient methods . . . . . . . . . . . . . . . . . . . . . 26
1.4 Interior PointMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.4.1 Potential reduction methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.4.2 Central path methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.4.3 Barrier methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2 Theoretical and Numerical Results for Nonlinear Optimization Problems 31
2.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Original Problem Formulation and its Perturbed Version . . . . . . . . . . . . . . . . 32
2.2.1 The perturbed associated problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 Theoretical Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Solution of the Perturbed Problem (P) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.1 The descent direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.2 Step-length determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4.3 The new approximating minorant function . . . . . . . . . . . . . . . . . . . . . 39
2.4.4 The auxiliary function 1 of G1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.6 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.6.1 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3 A Hybrid CG Algorithm for Nonlinear Unconstrained Optimization with Application
in Image Restoration 47
3.1 The ProposedMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.1.1 The sufficient descent condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.2 The global convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2.1 Image restoration problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4 An Efficient Hybrid Conjugate Gradient Method for Unconstrained Optimization
and Image Restoration Problems 62
4.1 The Proposed Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1.1 The sufficient descent condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.1.2 The convergence analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2.1 Image restoration problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Côte titre : DM/0210 APPLICATION OF INTERIOR POINT METHODS FOR NON-LINEAR PROGRAMMING [document électronique] / Choubeila Souli, Auteur ; Assma Leulmi, Directeur de thèse . - [S.l.] : Setif:UFA, 2025 . - 1 vol (86 f.) ; 29 cm.
Langues : Anglais (eng)
Catégories : Thèses & Mémoires:Mathématique Mots-clés : Nonlinear optimization problem
Interior point method
Logarithmic barrier
Minorant function
Conjugate gradient method
Global convergence
Descent directionIndex. décimale : 510 - Mathématique Résumé :
In this thesis, we focus on the theoretical analysis and numerical investigation of specific interior point
and conjugate gradient methods for solving nonlinear optimization problems.
First, we propose logarithmic barrier approaches for constrained convex nonlinear optimization
problems, where the barrier parameter is treated as a vector. This is followed by analytical studies in
which the step-length is determined using the minorant function technique. The numerical findings reveal
that the proposed methods exhibit both effectiveness and robustness.
In addition, we develop new conjugate gradient methods for solving unconstrained optimization problems.
These methods generate descent directions without requiring line search techniques. Moreover, they
exhibit global convergence under mild assumptions. Numerical results indicate that the proposed methods
are both effective and robust in addressing various unconstrained optimization and image restoration
problems.Note de contenu : Sommaire
Introduction 1
1 Preliminaries and Fundamental Concepts 5
1.1 Convex Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.1 Affine sets and functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.2 Convexity concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.3 Differentiability and convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.4 Lower and upper semicontinuous functions . . . . . . . . . . . . . . . . . . . . 10
1.2 Mathematical Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.1 Unconstrained optimization problems . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.2 Constrained optimization problems . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Conjugate GradientMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.1 Descent direction methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.2 Line search methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3.3 The linear conjugate gradient methods . . . . . . . . . . . . . . . . . . . . . . . . 25
1.3.4 The nonlinear conjugate gradient methods . . . . . . . . . . . . . . . . . . . . . 26
1.4 Interior PointMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.4.1 Potential reduction methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.4.2 Central path methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.4.3 Barrier methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2 Theoretical and Numerical Results for Nonlinear Optimization Problems 31
2.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Original Problem Formulation and its Perturbed Version . . . . . . . . . . . . . . . . 32
2.2.1 The perturbed associated problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3 Theoretical Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Solution of the Perturbed Problem (P) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.1 The descent direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4.2 Step-length determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4.3 The new approximating minorant function . . . . . . . . . . . . . . . . . . . . . 39
2.4.4 The auxiliary function 1 of G1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5 The Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.6 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.6.1 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3 A Hybrid CG Algorithm for Nonlinear Unconstrained Optimization with Application
in Image Restoration 47
3.1 The ProposedMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.1.1 The sufficient descent condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.2 The global convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.2 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2.1 Image restoration problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4 An Efficient Hybrid Conjugate Gradient Method for Unconstrained Optimization
and Image Restoration Problems 62
4.1 The Proposed Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1.1 The sufficient descent condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.1.2 The convergence analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2.1 Image restoration problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Côte titre : DM/0210 Exemplaires (1)
Code-barres Cote Support Localisation Section Disponibilité DM/0210 DM/0210 Thèse Bibliothèque des sciences Anglais Disponible
DisponibleEtude comparative entre la technique des fonctions minorantes et majorantes pour la programmation linéaire / Oumaima Boudra
Titre : Etude comparative entre la technique des fonctions minorantes et majorantes pour la programmation linéaire Type de document : texte imprimé Auteurs : Oumaima Boudra, Auteur ; Chaima Maouche, Auteur ; Assma Leulmi, Directeur de thèse Année de publication : 2022 Importance : 1 vol (49 f.) Format : 29 cm Langues : Français (fre) Catégories : Mathématique Mots-clés : Programmation linéaire
Méthode de KarmakarIndex. décimale : 510-Mathématique Résumé :
Ce mémoire est consacré à l’étude de la méthode projectif de
Karmarkar pour la résolution d’un programme linéaire, par laquelle on
a proposé une alternative de calcul du pas de déplacement d’une façon
explicite basée sur l’idée des fonctions minorantes et majorantes, cela
a entraîné une diminution significative du nombre d’itérations, et cela
a été fait en faisant une comparaison numérique entre eux. Les tests
numériques sont encourageants, et favorisent notre approche.Côte titre : MAM/0584 En ligne : https://drive.google.com/file/d/1DKkwfispvHOg51fNLfA848AXl0s3Wmzp/view?usp=share [...] Format de la ressource électronique : Etude comparative entre la technique des fonctions minorantes et majorantes pour la programmation linéaire [texte imprimé] / Oumaima Boudra, Auteur ; Chaima Maouche, Auteur ; Assma Leulmi, Directeur de thèse . - 2022 . - 1 vol (49 f.) ; 29 cm.
Langues : Français (fre)
Catégories : Mathématique Mots-clés : Programmation linéaire
Méthode de KarmakarIndex. décimale : 510-Mathématique Résumé :
Ce mémoire est consacré à l’étude de la méthode projectif de
Karmarkar pour la résolution d’un programme linéaire, par laquelle on
a proposé une alternative de calcul du pas de déplacement d’une façon
explicite basée sur l’idée des fonctions minorantes et majorantes, cela
a entraîné une diminution significative du nombre d’itérations, et cela
a été fait en faisant une comparaison numérique entre eux. Les tests
numériques sont encourageants, et favorisent notre approche.Côte titre : MAM/0584 En ligne : https://drive.google.com/file/d/1DKkwfispvHOg51fNLfA848AXl0s3Wmzp/view?usp=share [...] Format de la ressource électronique : Exemplaires (1)
Code-barres Cote Support Localisation Section Disponibilité MAM/0584 MAM/0584 Mémoire Bibliothèque des sciences Français Disponible
Disponible
Titre : Programming Tools PT 2 : (MATLAB) Type de document : texte imprimé Auteurs : Assma Leulmi, Auteur Année de publication : 2025 Importance : 1 vol (117 p.) Format : 29 cm Langues : Français (fre) Catégories : Publications pédagogiques:Mathématiaue P/P Mots-clés : MATLAB Vectors Matrices Graphs Data visualization : Matlab Index. décimale : 510-Mathématique Note de contenu :
Table of Contents
Introduction ....................................................................... 1
Chapter I: Introduction to the Matlab environment ................................... 3
1. Introduction and Opening a Matlab Session: ....................................... 3
1. General: .......................................................................... 3
2. MATLAB environnement:.............................................................. 4
A. Editor/Debugger: .................................................................. 6
B. M-Files:............................................................................ 6
D. Help: ............................................................................. 8
2.1 First interaction with MATLAB:.................................................... 9
2.2 The numbers in MATLAB:............................................................ 11
2.3 The main constants, functions and commands:........................................14
2.4 The priority of operations in an expression: ...................................... 15
TP1........................................................................................ 17
Exercice 01: ............................................................................ 17
Exercice 02: ............................................................................. 17
Exercice 03: ............................................................................. 18
Exercice 04: ............................................................................. 18
Solution TP 1 ............................................................................ 21
Chapter II: Vectors and Matrices ........................................................ 21
1. The vectors: .......................................................................... 21
1.1 Referencing and access to vector elements: ........................................... 23
1.2 Element-by-element operations for vectors: ............................................24
1.3 The linspace function: ............................................................... 25
2. The matrices: ......................................................................... 26
2.1 Referencing and access to matrix elements: ........................................... 28
2.2 Automatic generation of matrices:..................................................... 30
Example 1: ............................................................................... 32
Example 2: ............................................................................... 32
2.3 Basic operations on the matrices: .................................................... 32
2.4 Useful functions for matrix processing:............................................... 33
TP2......................................................................................... 36
Exercice 01: ............................................................................36
Exercice 02: ............................................................................. 36
Exercice 03: .............................................................................. 36
Exercice 04: .............................................................................. 37
Exercice 05: .............................................................................37
Exercice 06: ............................................................................37
Exercice 07: .......................................................................... 38
Solution TP 2 .......................................................................... 39
Chapter III: Introduction to programming with Matlab................................. 45
1. General: ......................................................................... 45
1.1 Comments:......................................................................... 45
1.2 Writing long expressions:......................................................... 45
1.3 Reading data in a program (Inputs): ............................................. 46
1.4 Writing data in a program (Outputs): ............................................46
2. Logical expressions: ............................................................ 47
2.1 Logical operations: ........................................................... 47
2.2 Matrix comparison: ............................................................. 50
3. Flow control structures: ........................................................ 51
3.1 The if statement: ............................................................ 52
3.2 The switch statement:........................................................ 54
3.3 The for statement:......................................................... 55
3.4 The while statement: ..................................................... 56
Example of a for loop in MATLAB: ........................................... 57
Replacing the for loop with a while loop in MATLAB: ......................... 57
4. Summary the control structures: ............................................. 58
5. Summary exercise: ..........................................................59
6. The functions: .............................................................. 60
6.1 Creating a function in an M-Files: ......................................... 60
6.2 Comparison between a program is a function: ................................. 61
6.3 Exercices with Solution:...................................................... 64
7. Polynomials: .................................................................... 78
7.1 Polynomials in MATLAB: ......................................................... 78
7.2 Polynomial zeros:................................................................ 78
7.3 Polynomial operations:............................................................ 78
Chapter V: Graphs and data visualization in Matlab....................................... 80
1. The plot function: .................................................................81
2. Change the appearance of a curve: .................................................. 83
3. Annotation of a figure: .............................................................. 84
4. Draw multiple curves in the same figure: ........................................... 84
4.1 The hold command:................................................................. 84
4.2 Use plot with multiple arguments: ................................................. 85
4.3 Using matrices as argument for the plot function: ............................ 86
4.4 Using the fplot function: ................................................86
5. Manipulating the axes of a figure: ......................................... 87
6. Other types of graphs: ....................................................... 89
7. Transfer figures to a Word document: ...................................... 92
8. Figure Editor: .................................................. 93
Exercice: .................................................................. 94
9. Symbolic Calculation:...........................................................95
9.1 Calling the symbolic toolbox: ......................................... 95
9.2 Expanding and transforming expressions: ............................ 95
9.3 Derivatives and integrals of a function: .................................... 95
9.4 Taylor series expansion of a function: .............................................. 95
TP3................................................................................... 96
Exercice 01: ........................................................................ 96
Exercice 02: ............................................................................ 94
Exercice 03: ................................................................ 96
Exercice 04: ........................................................ 95
Exercice 05: .................................................. 97
Exercice 06: ................................................... 97
Exercice 07: ..................................................................... 98
Solution TP 3 ....................................................... 99
Command Catalogue ........................................................ 108
Content of the subject ........................................... 114
Bibliography ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 117Côte titre : PM/0036 En ligne : http://dspace.univ-setif.dz:8888/jspui/handle/123456789/5317 Programming Tools PT 2 : (MATLAB) [texte imprimé] / Assma Leulmi, Auteur . - 2025 . - 1 vol (117 p.) ; 29 cm.
Langues : Français (fre)
Catégories : Publications pédagogiques:Mathématiaue P/P Mots-clés : MATLAB Vectors Matrices Graphs Data visualization : Matlab Index. décimale : 510-Mathématique Note de contenu :
Table of Contents
Introduction ....................................................................... 1
Chapter I: Introduction to the Matlab environment ................................... 3
1. Introduction and Opening a Matlab Session: ....................................... 3
1. General: .......................................................................... 3
2. MATLAB environnement:.............................................................. 4
A. Editor/Debugger: .................................................................. 6
B. M-Files:............................................................................ 6
D. Help: ............................................................................. 8
2.1 First interaction with MATLAB:.................................................... 9
2.2 The numbers in MATLAB:............................................................ 11
2.3 The main constants, functions and commands:........................................14
2.4 The priority of operations in an expression: ...................................... 15
TP1........................................................................................ 17
Exercice 01: ............................................................................ 17
Exercice 02: ............................................................................. 17
Exercice 03: ............................................................................. 18
Exercice 04: ............................................................................. 18
Solution TP 1 ............................................................................ 21
Chapter II: Vectors and Matrices ........................................................ 21
1. The vectors: .......................................................................... 21
1.1 Referencing and access to vector elements: ........................................... 23
1.2 Element-by-element operations for vectors: ............................................24
1.3 The linspace function: ............................................................... 25
2. The matrices: ......................................................................... 26
2.1 Referencing and access to matrix elements: ........................................... 28
2.2 Automatic generation of matrices:..................................................... 30
Example 1: ............................................................................... 32
Example 2: ............................................................................... 32
2.3 Basic operations on the matrices: .................................................... 32
2.4 Useful functions for matrix processing:............................................... 33
TP2......................................................................................... 36
Exercice 01: ............................................................................36
Exercice 02: ............................................................................. 36
Exercice 03: .............................................................................. 36
Exercice 04: .............................................................................. 37
Exercice 05: .............................................................................37
Exercice 06: ............................................................................37
Exercice 07: .......................................................................... 38
Solution TP 2 .......................................................................... 39
Chapter III: Introduction to programming with Matlab................................. 45
1. General: ......................................................................... 45
1.1 Comments:......................................................................... 45
1.2 Writing long expressions:......................................................... 45
1.3 Reading data in a program (Inputs): ............................................. 46
1.4 Writing data in a program (Outputs): ............................................46
2. Logical expressions: ............................................................ 47
2.1 Logical operations: ........................................................... 47
2.2 Matrix comparison: ............................................................. 50
3. Flow control structures: ........................................................ 51
3.1 The if statement: ............................................................ 52
3.2 The switch statement:........................................................ 54
3.3 The for statement:......................................................... 55
3.4 The while statement: ..................................................... 56
Example of a for loop in MATLAB: ........................................... 57
Replacing the for loop with a while loop in MATLAB: ......................... 57
4. Summary the control structures: ............................................. 58
5. Summary exercise: ..........................................................59
6. The functions: .............................................................. 60
6.1 Creating a function in an M-Files: ......................................... 60
6.2 Comparison between a program is a function: ................................. 61
6.3 Exercices with Solution:...................................................... 64
7. Polynomials: .................................................................... 78
7.1 Polynomials in MATLAB: ......................................................... 78
7.2 Polynomial zeros:................................................................ 78
7.3 Polynomial operations:............................................................ 78
Chapter V: Graphs and data visualization in Matlab....................................... 80
1. The plot function: .................................................................81
2. Change the appearance of a curve: .................................................. 83
3. Annotation of a figure: .............................................................. 84
4. Draw multiple curves in the same figure: ........................................... 84
4.1 The hold command:................................................................. 84
4.2 Use plot with multiple arguments: ................................................. 85
4.3 Using matrices as argument for the plot function: ............................ 86
4.4 Using the fplot function: ................................................86
5. Manipulating the axes of a figure: ......................................... 87
6. Other types of graphs: ....................................................... 89
7. Transfer figures to a Word document: ...................................... 92
8. Figure Editor: .................................................. 93
Exercice: .................................................................. 94
9. Symbolic Calculation:...........................................................95
9.1 Calling the symbolic toolbox: ......................................... 95
9.2 Expanding and transforming expressions: ............................ 95
9.3 Derivatives and integrals of a function: .................................... 95
9.4 Taylor series expansion of a function: .............................................. 95
TP3................................................................................... 96
Exercice 01: ........................................................................ 96
Exercice 02: ............................................................................ 94
Exercice 03: ................................................................ 96
Exercice 04: ........................................................ 95
Exercice 05: .................................................. 97
Exercice 06: ................................................... 97
Exercice 07: ..................................................................... 98
Solution TP 3 ....................................................... 99
Command Catalogue ........................................................ 108
Content of the subject ........................................... 114
Bibliography ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 117Côte titre : PM/0036 En ligne : http://dspace.univ-setif.dz:8888/jspui/handle/123456789/5317 Exemplaires (1)
Code-barres Cote Support Localisation Section Disponibilité PM/0036 PM/0036 imprimé / autre Bibliothèque des sciences Anglais Disponible
DisponibleA Theoretical and Numerical Study of a New Hybrid Conjugate Gradient Method for Nonlinear Programming / Lamya Mendil
Titre : A Theoretical and Numerical Study of a New Hybrid Conjugate Gradient Method for Nonlinear Programming Type de document : document électronique Auteurs : Lamya Mendil, Auteur ; Chalabia Tebbal ; Assma Leulmi, Directeur de thèse Editeur : Sétif:UFS Année de publication : 2024 Importance : 1 vol (56 f.) Format : 29 cm Langues : Anglais (eng) Catégories : Thèses & Mémoires:Mathématique Mots-clés : Unconstrained nonlinear optimization
Conjugate gradient method
Hybrid method
Inexact line search
The convergenceIndex. décimale : 510-Mathématique Résumé : The conjugate gradient method is one of the oldest methods for solving nonlinear unconstrained optimization problems especially in large size. This memory present a new hybrid conjugate gradient method based on the convex combination of NHS (modified HS) and MLS (modified LS) methods. The proposed algorithm satisfies the sufficient descent condition and converges globally under the usual and strong Wolfe line search assumptions. To illustrate the effectiveness of this method a numerical study is achieved. Note de contenu : Sommaire
Introductioniii
1 Preliminariesandbasicconcepts1
1.1GeneraldeÂ…nitionsonunconstrainedoptimization..........1
1.2Existenceanduniquenessresults....................4
1.3Conditionsofoptimality........................5
1.3.1Necessaryconditions(NC)...................5
1.3.2Su¢cientConditions(SC)...................6
1.4Descentdirectionmethod.......................6
1.4.1Iterativealgorithm.......................7
2 Exactandinexactlinesearch9
2.1Purposeoflinesearch.........................9
2.2Safetyinterval..............................10
2.3Basicalgorithm.............................10
2.4Exactlinesearchmethods.......................10
2.5Inexactlinesearchmethods......................11
2.5.1Armijorule(1966).......................12
2.5.2Goldstein-Pricerule(1969)...................13
2.5.3Wolferule(1969)........................14
3 Theconjugategradientmethod18
3.1Gradientmethod............................18
3.1.1Gradientmethodalgorithm..................19
3.1.2convergenceofthegradientmethod..............19
3.2Theconjugategradientmethodfornonlinearfunctions.......20
3.2.1Generalprinciple........................20
3.2.2Somevariantsofthenonlinearconjugategradientmethod.21
3.2.3NonlinearconjugategradientalgorithmwithstrongWolfeÂ’s
inexactlinesearch.......................21
3.3Convergenceresultsfortheconjugategradientmethod.......22
3.3.1Conditions C1 and C2 (Lipschitzandboundness)......22
3.3.2ZoutendijkÂ’sTheorem.....................23
3.3.3UsingZoutendijkÂ’stheoremtodemonstrateglobalconvergence23
4 Hybridmethodofnonlinearconjugategradient27
4.1HybridmethodsbasedonthemodiÂ…cationofclassicalconjugate
gradientmethods............................27
4.1.1ThemodiÂ…cationofPRPmethod...............27
4.1.2ThemodiÂ…cationofFRmethod................28
4.1.3ThemodiÂ…cationofHSmethod................28
4.1.4ThemodiÂ…cationofLSmethod................29
4.2Hybridnonlinearconjugategradientmethodsbasedonconvexcom-
binations.................................29
4.2.1Generalprinciple........................29
4.3AnewhybridCGmethodbasedonconvexcombination......30
4.3.1Theproposedalgorithm....................30
4.3.2Thesu¢cientdescentconditionandtheglobalconvergence.31
5 Numericalapplications 35Côte titre : MAM/0727 En ligne : http://dspace.univ-setif.dz:8888/jspui/bitstream/123456789/5451/1/mam0727.pdf A Theoretical and Numerical Study of a New Hybrid Conjugate Gradient Method for Nonlinear Programming [document électronique] / Lamya Mendil, Auteur ; Chalabia Tebbal ; Assma Leulmi, Directeur de thèse . - [S.l.] : Sétif:UFS, 2024 . - 1 vol (56 f.) ; 29 cm.
Langues : Anglais (eng)
Catégories : Thèses & Mémoires:Mathématique Mots-clés : Unconstrained nonlinear optimization
Conjugate gradient method
Hybrid method
Inexact line search
The convergenceIndex. décimale : 510-Mathématique Résumé : The conjugate gradient method is one of the oldest methods for solving nonlinear unconstrained optimization problems especially in large size. This memory present a new hybrid conjugate gradient method based on the convex combination of NHS (modified HS) and MLS (modified LS) methods. The proposed algorithm satisfies the sufficient descent condition and converges globally under the usual and strong Wolfe line search assumptions. To illustrate the effectiveness of this method a numerical study is achieved. Note de contenu : Sommaire
Introductioniii
1 Preliminariesandbasicconcepts1
1.1GeneraldeÂ…nitionsonunconstrainedoptimization..........1
1.2Existenceanduniquenessresults....................4
1.3Conditionsofoptimality........................5
1.3.1Necessaryconditions(NC)...................5
1.3.2Su¢cientConditions(SC)...................6
1.4Descentdirectionmethod.......................6
1.4.1Iterativealgorithm.......................7
2 Exactandinexactlinesearch9
2.1Purposeoflinesearch.........................9
2.2Safetyinterval..............................10
2.3Basicalgorithm.............................10
2.4Exactlinesearchmethods.......................10
2.5Inexactlinesearchmethods......................11
2.5.1Armijorule(1966).......................12
2.5.2Goldstein-Pricerule(1969)...................13
2.5.3Wolferule(1969)........................14
3 Theconjugategradientmethod18
3.1Gradientmethod............................18
3.1.1Gradientmethodalgorithm..................19
3.1.2convergenceofthegradientmethod..............19
3.2Theconjugategradientmethodfornonlinearfunctions.......20
3.2.1Generalprinciple........................20
3.2.2Somevariantsofthenonlinearconjugategradientmethod.21
3.2.3NonlinearconjugategradientalgorithmwithstrongWolfeÂ’s
inexactlinesearch.......................21
3.3Convergenceresultsfortheconjugategradientmethod.......22
3.3.1Conditions C1 and C2 (Lipschitzandboundness)......22
3.3.2ZoutendijkÂ’sTheorem.....................23
3.3.3UsingZoutendijkÂ’stheoremtodemonstrateglobalconvergence23
4 Hybridmethodofnonlinearconjugategradient27
4.1HybridmethodsbasedonthemodiÂ…cationofclassicalconjugate
gradientmethods............................27
4.1.1ThemodiÂ…cationofPRPmethod...............27
4.1.2ThemodiÂ…cationofFRmethod................28
4.1.3ThemodiÂ…cationofHSmethod................28
4.1.4ThemodiÂ…cationofLSmethod................29
4.2Hybridnonlinearconjugategradientmethodsbasedonconvexcom-
binations.................................29
4.2.1Generalprinciple........................29
4.3AnewhybridCGmethodbasedonconvexcombination......30
4.3.1Theproposedalgorithm....................30
4.3.2Thesu¢cientdescentconditionandtheglobalconvergence.31
5 Numericalapplications 35Côte titre : MAM/0727 En ligne : http://dspace.univ-setif.dz:8888/jspui/bitstream/123456789/5451/1/mam0727.pdf Exemplaires (1)
Code-barres Cote Support Localisation Section Disponibilité MAM/0727 MAM/0727 Mémoire Bibliothèque des sciences Anglais Disponible
Disponible
