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Titre : Adequate approach of IPMs for linearly constrained convex optimization Type de document : texte imprimé Auteurs : Raounek Messalti, Auteur ; Manar Zoghbi ; Nawel Boudjellal, Directeur de thèse Editeur : Sétif:UFS Année de publication : 2024 Importance : 1 vol (43 f.) Format : 29 cm Langues : Anglais (eng) Catégories : Thèses & Mémoires:Mathématique Mots-clés : Linearly constrained convex optimization
Weighted primal-dual interior point method
Small-update version
Iteration boundIndex. décimale : 510-Mathématique Résumé :
In this work, a class of primal-dual interior point methods (IPMs) for solving linearly constrained convex optimization problems is presented. It is a weighted short-step primal-dual interior point algorithm. The proposed algorithm uses at each iteration only full-Newton step where is no calculation of the step size with a quadratic rate of convergence. We analyze a small-update version and we obtain the best-known iteration bound which is as good as the bound for the linear optimization analogue.
Finally, some numerical results are reported to show the practical performance of the algorithm with different parameters.Note de contenu : Sommaire
1 Convexanalysisconceptsandconvexoptimization1
1.1 Convexanalysisnotions........................... 1
1.1.1 Convexsetsandfunctions...................... 2
1.1.2 Convexityanddifferentiability.................... 3
1.1.3 Asymptoticnotationsandspeedofconvergence.......... 4
1.2 Mathematicalprogramming......................... 5
1.2.1 Positionofmathematicalproblem.................. 5
1.2.2 Classificationofmathematicalproblems.............. 7
1.2.3 Mainexistenceanduniquenessresultsofanoptimum...... 7
1.2.4 Qualifyingconstraints......................... 8
1.2.5 Optimalityconditions......................... 8
1.2.6 Dualityofamathematicalproblem................. 9
1.3 Linearlyconstrainedconvexprogramming................ 10
1.3.1 LCCOprimalproblem......................... 10
1.3.2 LCCOdualproblem.......................... 11
1.3.3 DualityinLCCOprogramming.................... 12
1.4 SolvingLCCOproblems............................ 13
1.4.1 LCCOproblemsandIPMs....................... 13
1.4.2 Classiccentraltrajectorymethod.................. 16
2 AweightedcentralpathmethodforLCCO19
2.1 Methoddescription.............................. 19
2.1.1 Principle................................. 19
2.1.2 Weighteddirection........................... 21
2.1.3 Algorithmicdescription........................ 22
2.2 Convergenceanalysis............................. 23
2.2.1 Thestrictfeasibilityofiterates.................... 24
2.2.2 Theinfluenceofaweightedfull-Newtonstepontheproximity
measure................................. 26
2.3 Complexityanalysis.............................. 29
3 Numericalimplementation32
3.1 Clarificationsandexamples......................... 32
3.2 Resultsandcommentaries.......................... 36Côte titre : MAM/0729 Adequate approach of IPMs for linearly constrained convex optimization [texte imprimé] / Raounek Messalti, Auteur ; Manar Zoghbi ; Nawel Boudjellal, Directeur de thèse . - [S.l.] : Sétif:UFS, 2024 . - 1 vol (43 f.) ; 29 cm.
Langues : Anglais (eng)
Catégories : Thèses & Mémoires:Mathématique Mots-clés : Linearly constrained convex optimization
Weighted primal-dual interior point method
Small-update version
Iteration boundIndex. décimale : 510-Mathématique Résumé :
In this work, a class of primal-dual interior point methods (IPMs) for solving linearly constrained convex optimization problems is presented. It is a weighted short-step primal-dual interior point algorithm. The proposed algorithm uses at each iteration only full-Newton step where is no calculation of the step size with a quadratic rate of convergence. We analyze a small-update version and we obtain the best-known iteration bound which is as good as the bound for the linear optimization analogue.
Finally, some numerical results are reported to show the practical performance of the algorithm with different parameters.Note de contenu : Sommaire
1 Convexanalysisconceptsandconvexoptimization1
1.1 Convexanalysisnotions........................... 1
1.1.1 Convexsetsandfunctions...................... 2
1.1.2 Convexityanddifferentiability.................... 3
1.1.3 Asymptoticnotationsandspeedofconvergence.......... 4
1.2 Mathematicalprogramming......................... 5
1.2.1 Positionofmathematicalproblem.................. 5
1.2.2 Classificationofmathematicalproblems.............. 7
1.2.3 Mainexistenceanduniquenessresultsofanoptimum...... 7
1.2.4 Qualifyingconstraints......................... 8
1.2.5 Optimalityconditions......................... 8
1.2.6 Dualityofamathematicalproblem................. 9
1.3 Linearlyconstrainedconvexprogramming................ 10
1.3.1 LCCOprimalproblem......................... 10
1.3.2 LCCOdualproblem.......................... 11
1.3.3 DualityinLCCOprogramming.................... 12
1.4 SolvingLCCOproblems............................ 13
1.4.1 LCCOproblemsandIPMs....................... 13
1.4.2 Classiccentraltrajectorymethod.................. 16
2 AweightedcentralpathmethodforLCCO19
2.1 Methoddescription.............................. 19
2.1.1 Principle................................. 19
2.1.2 Weighteddirection........................... 21
2.1.3 Algorithmicdescription........................ 22
2.2 Convergenceanalysis............................. 23
2.2.1 Thestrictfeasibilityofiterates.................... 24
2.2.2 Theinfluenceofaweightedfull-Newtonstepontheproximity
measure................................. 26
2.3 Complexityanalysis.............................. 29
3 Numericalimplementation32
3.1 Clarificationsandexamples......................... 32
3.2 Resultsandcommentaries.......................... 36Côte titre : MAM/0729 Exemplaires (1)
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