京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第18530号 / 情博第534号 / 新制||情||95(附属図書館) / 31416 / 京都大学大学院情報学研究科知能情報学専攻 / (主査)教授 岡部 寿男, 教授 松山 隆司, 教授 阿久津 達也 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

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 solutions optimales locales ou réalisables, on utilise souvent des heuristiques ou des algorithmes 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 déterministe 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 algorithme 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 déterministe 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 littérature 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 modèles : 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 déterministe 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 problème 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. / 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 combinatorial optimization problems. This PhD thesis proposed to solve the deterministic and stochastic optimization problems with metaheuristics. We studied especially Variable 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 deterministic 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 allocation model for uplink wireless multi-cell OFDMA Networks. After transforming the original model into a deterministic one, the proposed VNS is applied on the deterministic model, and find near optimal solutions. Subsequently, several problems either in OFDMA systems or in many other topics in resource allocation can be modeled 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 conservative 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.

Text in Afikaans / In this thesis contributions to the solution of the generalised knapsack problem are given and discussed. Attention is given to problems with functions that are calculable but not necessarily in a closed form. Algorithms and test problems can be used for problems with closed-form functions as well. The focus is on the development of good heuristics and not on exact algorithms. Heuristics must be investigated and good test problems must be designed. A measure of convexity for convex functions is developed and adapted for concave functions. A test problem generator makes use of this measure of convexity to create challenging test problems for the concave, convex and mixed knapsack problems. Four easy-to-interpret characteristics of an S-function are used to create test problems for the S-shaped as well as the generalised knapsack problem. The in uence of the size of the problem and the funding ratio on the speed and the accuracy of the algorithms are investigated. When applicable, the in uence of the interval length ratio and the ratio of concave functions to the total number of functions is also investigated. The Karush-Kuhn-Tucker conditions play an important role in the development of the algorithms. Suf- cient conditions for optimality for the convex knapsack problem with xed interval lengths is given and proved. For the general convex knapsack problem, the key theorem, which contains the stronger necessary conditions, is given and proved. This proof is so powerful that it can be used to proof the adapted key theorems for the mixed, S-shaped and the generalised knapsack problems as well. The exact search-lambda algorithm is developed for the concave knapsack problem with functions that are not in a closed form. This algorithm is used in the algorithms to solve the mixed and S-shaped knapsack problems. The exact one-step algorithm is developed for the convex knapsack problem with xed interval length. This algorithm is O(n). The general convex knapsack problem is solved by using the pivot algorithm which is O(n2). Optimality cannot be proven but in all cases the optimal solution was found and for all practical reasons this problem will be considered as being concluded. A good heuristic is developed for the mixed knapsack problem. Further research can be done on this heuristic as well as on the S-shaped and generalised knapsack problems. / Mathematical Sciences / D. Phil. (Operasionele Navorsing)

Text in Afikaans / In this thesis contributions to the solution of the generalised knapsack problem are given and discussed. Attention is given to problems with functions that are calculable but not necessarily in a closed form. Algorithms and test problems can be used for problems with closed-form functions as well. The focus is on the development of good heuristics and not on exact algorithms. Heuristics must be investigated and good test problems must be designed. A measure of convexity for convex functions is developed and adapted for concave functions. A test problem generator makes use of this measure of convexity to create challenging test problems for the concave, convex and mixed knapsack problems. Four easy-to-interpret characteristics of an S-function are used to create test problems for the S-shaped as well as the generalised knapsack problem. The in uence of the size of the problem and the funding ratio on the speed and the accuracy of the algorithms are investigated. When applicable, the in uence of the interval length ratio and the ratio of concave functions to the total number of functions is also investigated. The Karush-Kuhn-Tucker conditions play an important role in the development of the algorithms. Suf- cient conditions for optimality for the convex knapsack problem with xed interval lengths is given and proved. For the general convex knapsack problem, the key theorem, which contains the stronger necessary conditions, is given and proved. This proof is so powerful that it can be used to proof the adapted key theorems for the mixed, S-shaped and the generalised knapsack problems as well. The exact search-lambda algorithm is developed for the concave knapsack problem with functions that are not in a closed form. This algorithm is used in the algorithms to solve the mixed and S-shaped knapsack problems. The exact one-step algorithm is developed for the convex knapsack problem with xed interval length. This algorithm is O(n). The general convex knapsack problem is solved by using the pivot algorithm which is O(n2). Optimality cannot be proven but in all cases the optimal solution was found and for all practical reasons this problem will be considered as being concluded. A good heuristic is developed for the mixed knapsack problem. Further research can be done on this heuristic as well as on the S-shaped and generalised knapsack problems. / Mathematical Sciences / D. Phil. (Operasionele Navorsing)