Plusieurs problèmes d'ordonnancement cyclique ont été étudiés dans la littérature. Cependant, la plupart de ces travaux considèrent que les paramètres sont connus avec certitude et ne prennent pas en compte les différents aléas qui peuvent survenir. Par ailleurs, un ordonnancement optimal pour un problème déterministe peut très vite devenir le pire ordonnancement en présence d'incertitude. Parmi les incertitudes que nous pouvons rencontrer dans les problèmes d'ordonnancement, la variation des durées des tâches par rapport au valeurs estimées, pannes des machines, incorporation de nouvelles tâches qui ne sont pas considérées au départ, etc. Dans cette thèse, nous étudions des problèmes d'ordonnancement cyclique où les durées des tâches sont affectées par des incertitudes. Ces dernières sont décrites par un ensemble d'incertitude où les durées des tâches sont supposées appartenir à des intervalles et le nombre de déviations par rapport aux valeurs nominales est contrôlé par un paramètre appelé budget d'incertitude. Nous étudions deux problèmes en particulier. Le premier est le problème d'ordonnancement cyclique de base (BCSP). Nous formulons celui-ci comme un problème d'optimisation robuste bi-niveau et, à partir des propriétés de cette formulation, nous proposons différents algorithmes pour le résoudre. Le deuxième problème considéré est le problème du jobshop cyclique. De manière similaire au BSCP, nous proposons une formulation en termes de problème d'optimisation bi-niveau et, en exploitant les algorithmes développés pour le problème d'ordonnancement cyclique de base, nous développons un algorithme de Branch-and-Bound pour le résoudre. Afin d'évaluer l'efficacité de notre méthode nous l'avons comparé à des méthodes de décomposition qui existent dans la littérature pour ce type de problèmes. Enfin, nous avons étudié une version du problème du jobshop cyclique où les durées des tâches prennent des valeurs dans des intervalles d'une manière uniforme et dont l'objectif est de minimiser la valeur moyenne du temps de cycle. Pour résoudre ce problème nous avons adopté un algorithme de Branch-and-Bound où chaque sous-problème de l'arbre de recherche consiste à calculer le volume d'un polytope. Enfin, pour montrer l'efficacité de chacune de ses méthodes, des résultats numériques sont présentés. / Several studies on cyclic scheduling problems have been presented in the literature. However, most of them consider that the problem parameters are deterministic and do not consider possible uncertainties on these parameters. However, the best solution for a deterministic problem can quickly become the worst one in the presence of uncertainties, involving bad schedules or infeasibilities. Many sources of uncertainty can be encountered in scheduling problems, for example, activity durations can decrease or increase, machines can break down, new activities can be incorporated, etc. In this PhD thesis, we focus on scheduling problems that are cyclic and where activity durations are affected by uncertainties. More precisely, we consider an uncertainty set where each task duration belongs to an interval, and the number of parameters that can deviate from their nominal values is bounded by a parameter called budget of uncertainty. This parameter allows us to control the degree of conservatism of the resulting schedule. In particular, we study two cyclic scheduling problems. The first one is the basic cyclic scheduling problem (BCSP). We formulate the problem as a two-stage robust optimization problem and, using the properties of this formulation, we propose three algorithms to solve it. The second considered problem is the cyclic jobshop problem (CJSP). As for the BCSP, we formulate the problem as two-stage robust optimization problem and by exploiting the algorithms proposed for the robust BCSP we propose a Branch-and-Bound algorithm to solve it. In order to evaluate the efficiency of our method, we compared it with classical decomposition methods for two-stage robust optimization problems that exist in the literature. We also studied a version of the CJSP where each task duration takes uniformly values within an interval and where the objective is to minimize the mean value of the cycle time. In order to solve the problem, we adapted the Branch-and-Bound algorithm where in each node of the search tree, the problem to be solved is the computation of a volume of a polytope. Numerical experiments assess the efficiency of the proposed methods.

This thesis describes a robust and efficient design of Student-t based Robust Spatio-Temporal Prediction, namely, St-RSTP, to provide estimation based on observations over spatio-temporal neighbors. It is crucial to many applications in geographical information systems, medical imaging, urban planning, economy study, and climate forecasting. The proposed St-RSTP is more resilient to outliers or other small departures from model assumptions than its ancestor, the Spatio-Temporal Random Effects (STRE) model. STRE is a statistical model with linear order complexity for processing large scale spatiotemporal data. However, STRE has been shown sensitive to outliers or anomaly observations. In our design, the St-RSTP model assumes that the measurement error follows Student's t-distribution, instead of a traditional Gaussian distribution. To handle the analytical intractable inference of Student's t model, we propose an approximate inference algorithm in the framework of Expectation Propagation (EP). Extensive experimental evaluations, based on both simulation and real-life data sets, demonstrated the robustness and the efficiency of our Student-t prediction model compared with the STRE model. / Master of Science

Regression analysis is one of the most extensively used statistical tools applied across different fields of science, with linear regression being its most well-known method. How- ever, the traditional procedure to obtain the linear model estimates, the least squares approach, is highly sensitive to even slight departures from the assumed modelling frame- work. This is especially pronounced when atypical values occur in the observed data. This lack of stability of the least squares approach is a serious problem in applications. Thus, the focus of this thesis lies in assessing the available robust alternatives to least squares estimation, which are not so easily affected by any outlying values. First, we introduce the linear regression model theory and derive the least squares method. Then, we char- acterise different types of unusual observations and outline some fundamental robustness measures. Next, we define and examine the robust alternatives to the classical estimation in the linear regression models. Finally, we conduct a comprehensive simulation study comparing the performance of robust methods under different scenarios. 1

付記する学位プログラム名: デザイン学大学院連携プログラム / 京都大学 / 0048 / 新制・課程博士 / 博士(工学) / 甲第21761号 / 工博第4578号 / 新制||工||1713(附属図書館) / 京都大学大学院工学研究科航空宇宙工学専攻 / (主査)教授 藤本 健治, 教授 泉田 啓, 教授 太田 快人 / 学位規則第4条第1項該当 / Doctor of Philosophy (Engineering) / Kyoto University / DGAM

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Modern large data sets inevitably contain outliers that deviate from the model assumptions. However, many widely used estimators, such as maximum likelihood estimators and least squared estimators, perform weakly with the existence of outliers. Alternatively, many statistical modeling approaches have matrices as the parameters. We consider penalized estimators for matrix-valued parameters with a focus on their robustness properties in the presence of outliers. We propose a general framework for robust modeling with matrix-valued parameters by minimizing robust loss functions with penalization. However, there are challenges to this approach in both computation and theoretical analysis. To tackle the computational challenges from the large size of the data, non-smoothness of robust loss functions, and the slow speed of matrix operations, we propose to apply the Frank-Wolfe algorithm, a first-order algorithm for optimization on a restricted region with low computation burden per iteration. Theoretically, we establish finite-sample error bounds under high-dimensional settings. We show that the estimation errors are bounded by small terms and converge in probability to zero under mild conditions in a neighborhood of the true model. Our method accommodates a broad classes of modeling problems using robust loss functions with penalization. Concretely, we study three cases: matrix completion, multivariate regression, and network estimation. For all cases, we illustrate the robustness of the proposed method both theoretically and numerically. / Statistics

Solution-path convex clustering is combined with concave penalties by Ma and Huang (2017) to reduce clustering bias. Their method was introduced in the setting of single-response regression to handle heterogeneity. Such heterogeneity may come from either the regression intercepts or the regression slopes. The procedure, realized by the alternating direction method of multipliers (ADMM) algorithm, can simultaneously identify the grouping structure of observations and estimate regression coefficients. In the first part of our work, we extend this procedure to multi-response regression. We propose models to solve cases with heterogeneity in either the regression intercepts or the regression slopes. We combine the existing gadgets of the ADMM algorithm and group-wise concave penalties to find solutions for the model. Our work improves model performance in both clustering accuracy and estimation accuracy. We also demonstrate the necessity of such extension through the fact that by utilizing information in multi-dimensional space, the performance can be greatly improved. In the second part, we introduce robust solutions to our proposed work. We introduce two approaches to handle outliers or long-tail distributions. The first is to replace the squared loss with robust loss, among which are absolute loss and Huber loss. The second is to characterize and remove outliers' effects by a mean-shift vector. We demonstrate that these robust solutions outperform the squared loss based method when outliers are present, or the underlying distribution is long-tailed. / Statistics