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Variants of Deterministic and Stochastic Nonlinear Optimization Problems Chen Wang Variants of Deterministic and Stochastic Nonlinear Optimization Problems Chen Wang To cite this version: Chen Wang. Variants of Deterministic and Stochastic Nonlinear Optimization Problems. Data Struc- tures and Algorithms [cs.DS]. Université Paris Sud - Paris XI, 2014. English. NNT : 2014PA112294. tel-01127048 HAL Id: tel-01127048 https://tel.archives-ouvertes.fr/tel-01127048 Submitted on 6 Mar 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. UNIVERSITÉ PARIS-SUD ECOLE DOCTORALE N◦:427 INFORMATIQUE DE PARIS-SUD Laboratoire: Laboratoire de Recherche en Informatique Thèse de doctorat par Chen WANG Variants of Deterministic and Stochastic Nonlinear Optimization Problems Date de soutenance: 31/10/2014 Composition du jury: Directeur de thèse : M. Abdel Lisser Rapporteurs : Mme. Janny Leung M. Alexandre Caminada Examinateurs : M. Sylvain Conchon M. Pablo Adasme 2 Acknowledgments Foremost, I would like to thank my supervisor, Prof. Abdel Lisser for providing me with the opportunity to complete my PhD thesis at the Paris Sud University. I am very grateful for his patient guidance to my research and warm help in my daily life. Special thanks to Prof. Janny Leung and Prof. Alexandre Caminada for agreeing to review my thesis and their invaluable comments and suggestions. Thanks also to Prof. Sylvain Conchon and Dr. Pablo Adasme for being part of my defense jury and their precious suggestions. I appreciate Dr. Pablo Adasme and Chuan Xu for our fruitful collaboration. I would like to thank all colleagues in research group for their hospitality and help. I also wish to express my appreciation to my friends Jianqiang Cheng, Weihua Yang, Yandong Bai, Weihua He et al. for sharing the good time in the lab and this beautiful city. Finally, I am really grateful to my parents for their infinite love, encouragement and support over the past three years in the foreign country. 3 4 Abstract Combinatorial optimization problems are generally NP-hard problems, so they can only rely on heuristic or approximation algorithms to find a local optimum or a feasible solution. During the last decades, more general solving techniques have been proposed, namely metaheuristics which can be applied to many types of combinato- rial optimization problems. This PhD thesis proposed to solve the deterministic and stochastic optimization problems with metaheuristics. We studied especially Vari- able Neighborhood Search (VNS) and choose this algorithm to solve our optimization problems since it is able to find satisfying approximated optimal solutions within a reasonable computation time. Our thesis starts with a relatively simple determin- istic combinatorial optimization problem: Bandwidth Minimization Problem. The proposed VNS procedure offers an advantage in terms of CPU time compared to the literature. Then, we focus on resource allocation problems in OFDMA systems, and present two models. The first model aims at maximizing the total bandwidth channel capacity of an uplink OFDMA-TDMA network subject to user power and subcarrier assignment constraints while simultaneously scheduling users in time. For this problem, VNS gives tight bounds. The second model is stochastic resource al- location model for uplink wireless multi-cell OFDMA Networks. After transforming the original model into a deterministic one, the proposed VNS is applied on the de- terministic model, and find near optimal solutions. Subsequently, several problems either in OFDMA systems or in many other topics in resource allocation can be mod- eled as hierarchy problems, e.g., bi-level optimization problems. Thus, we also study stochastic bi-level optimization problems, and use robust optimization framework to deal with uncertainty. The distributionally robust approach can obtain slight conser- vative solutions when the number of binary variables in the upper level is larger than the number of variables in the lower level. Our numerical results for all the problems studied in this thesis show the performance of our approaches. Keyword: Variable Neighborhood Search, Bandwidth minimization problem, Resource allocation problem of OFDMA network, Bi-level programming. 5 6 Résumé Les problèmes d’optimisation combinatoire sont généralement réputés NP-difficiles, donc il n’y a pas d’algorithmes efficaces pour les résoudre. Afin de trouver des solu- tions optimales locales ou réalisables, on utilise souvent des heuristiques ou des algo- rithmes approchés. Les dernières décennies ont vu naitre des méthodes approchées connues sous le nom de métaheuristiques, et qui permettent de trouver une solution approchées. Cette thèse propose de résoudre des problèmes d’optimisation détermin- iste et stochastique à l’aide de métaheuristiques. Nous avons particulièrement étudié la méthode de voisinage variable connue sous le nom de VNS. Nous avons choisi cet al- gorithme pour résoudre nos problèmes d’optimisation dans la mesure où VNS permet de trouver des solutions de bonne qualité dans un temps CPU raisonnable. Le premier problème que nous avons étudié dans le cadre de cette thèse est le problème déter- ministe de largeur de bande de matrices creuses. Il s’agit d’un problème combinatoire difficile, notre VNS a permis de trouver des solutions comparables à celles de la littéra- ture en termes de qualité des résultats mais avec temps de calcul plus compétitif. Nous nous sommes intéressés dans un deuxième temps aux problèmes de réseaux mobiles appelés OFDMA-TDMA. Nous avons étudié le problème d’affectation de ressources dans ce type de réseaux, nous avons proposé deux mod¨¨les : Le premier modèle est un modèle déterministe qui permet de maximiser la bande passante du canal pour un réseau OFDMA à débit monodirectionnel appelé Uplink sous contraintes d’énergie utilisée par les utilisateurs et des contraintes d’affectation de porteuses. Pour ce problème, VNS donne de très bons résultats et des bornes de bonne qualité. Le deuxième modèle est un problème stochastique de réseaux OFDMA d’affectation de ressources multi-cellules. Pour résoudre ce problème, on utilise le problème déter- ministe équivalent auquel on applique la méthode VNS qui dans ce cas permet de trouver des solutions avec un saut de dualité très faible. Les problèmes d’allocation de ressources aussi bien dans les réseaux OFDMA ou dans d’autres domaines peuvent aussi être modélisés sous forme de problèmes d’optimisation bi-niveaux appelés aussi problèmes d’optimisation hiérarchique. Le dernier problème étudié dans le cadre de cette thèse porte sur les problèmes bi-niveaux stochastiques. Pour résoudre le prob- 7 lème lié à l’incertitude dans ce problème, nous avons utilisé l’optimisation robuste plus précisément l’approche appelée "distributionnellement robuste". Cette approche donne de très bons résultats légèrement conservateurs notamment lorsque le nombre de variables du leader est très supérieur à celui du suiveur. Nos expérimentations ont confirmé l’efficacité de nos méthodes pour l’ensemble des problèmes étudiés. Mots clés: Recherche à Voisinage Variable, problème de minimisation de la largeur de bande de matrices, problème d’allocation de ressource dans les réseaux OFDMA, problèmes bi-niveaux. 8 Contents 1 Introduction 17 2 Metaheuristics 25 2.1 Single solution based metaheuristics . 27 2.1.1 Simulated Annealing . 28 2.1.2 Tabu Search . 33 2.1.3 Greedy Randomized Adaptive Search Procedure . 41 2.1.4 Variable Neighborhood Search . 44 2.2 Population based metaheuristics . 51 2.2.1 Genetic Algorithm . 51 2.2.2 Scatter Search . 60 2.3 Improvement of metaheuristics . 65 2.3.1 Algorithm Parameters . 65 2.3.2 General Strategy . 67 2.4 Evaluation of metaheuristics . 68 2.5 Conclusions . 69 3 Bandwidth Minimization Problem 71 3.1 Formulations . 72 3.1.1 Matrix bandwidth minimization problem . 72 3.1.2 Graph bandwidth minimization problem . 73 3.1.3 Equivalence between graph and matrix versions . 73 3.2 Solution methods . 76 9 3.2.1 Exact algorithms . 76 3.2.2 Heuristic algorithms . 76 3.2.3 Metaheuristics . 77 3.3 The VNS approach for bandwidth minimization problem . 92 3.3.1 Initial solution . 92 3.3.2 Shaking . 93 3.3.3 Local search . 94 3.3.4 Move or not . 95 3.4 Numerical results . 96 3.5 Conclusions . 98 4 Wireless Network 101 4.1 Orthogonal Frequency Division Multiplexing (OFDM) . 103 4.1.1 Development and application . 103 4.1.2 OFDM characteristics . 104 4.2 Orthogonal Frequency Division Multiplexing Access (OFDMA) . 105 4.2.1 Background of OFDMA . 107 4.2.2 OFDMA resource allocation method . 110 4.2.3 Research status of algorithms . 120 4.3 Scheduling in wireless OFDMA-TDMA networks using variable neigh- borhood search metaheuristic . 123 4.3.1 Introduction . 124 4.3.2 Problem formulation . 127 4.3.3 The VNS approach . 128 4.3.4 Numerical results . 131 4.3.5 Conclusions . 135 4.4 Stochastic resource allocation for uplink wireless multi-cell OFDMA networks . 136 4.4.1 Introduction . 137 4.4.2 System description and problem formulation . 140 10 4.4.3 Deterministic equivalent formulation . 143 4.4.4 Variable neighborhood search procedure . 145 4.4.5 Numerical results . 147 4.4.6 Conclusions . 152 4.5 Conclusions . 154 5 Bi-level Programming Problem 155 5.1 Definition . 157 5.2 Formulation . 158 5.3 Application . 163 5.4 Characteristics . 165 5.4.1 Complexity . 165 5.4.2 Optimality condition . 165 5.5 Methods . 167 5.5.1 Extreme-point approach . 167 5.5.2 Branch-and-bound algorithm . 168 5.5.3 Complementary pivoting approach . 169 5.5.4 Descent method . 169 5.5.5 Penalty function method .
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