In this work, we deal with fixed interval scheduling problems with the possibility of random delay of the end of the tasks (FIS). First, we pre- sent the basic deterministic FIS problems and ways to solve them. Next, we introduce the concept of minimax and present two well-known and one new FIS problem under uncertainty, when random task delays are conside- red to belong to a certain uncertainty set. Next, we deal with the solution of previously presented FIS problems for five chosen uncertainty sets. We present both previously achieved and original results. The work concludes with a summary of a numerical study of two problems. First, we explore the possibility of Lagrange relaxation application to the first presented problem. Next we explore the quality of approximation allowing to solve the later of problems as LP. 1

The MAPK cascade is responsible for transmitting information in the cytoplasm of the cell and regulating important fate decisions like cell division and apoptosis. Due to scarce experimental data and limited knowledge about many complex biochemical processes, existing MAPK pathway models, which exhibit bistability, have a significant structural uncertainty. Often, small perturbations of network interactions or components can reduce the bistable region significantly or make it even disappear and small fluctuations of the input can make the system switch back, which reflects its low robustness. However, real biological systems have developed significant robustness through evolution and this robustness should be reflected by the models. The main goal of the present thesis is the development of a methodology for increasing the robustness of biochemical models, which exhibit bistability. Based on modifying existing network interactions or introducing new interactions to the system, several methods for both internal and external robustification are proposed. Internal robustness is addressed through a sensitivity analysis, which deals with a linearization of the model and can be used sequentially to introduce multiple modifications to the model. The methods for external robustness improvement are based on eigenvalue placement and slope modification (drawing on the linear model) and on the identification of feedback structures (nonlinear model). Further, a way to integrate static interaction changes to the nonlinear model, so that these perturbations have only a local impact on its behavior, is proposed. The application of the methods to existing MAPK models shows that, by introducing small modifications, the internal and external robustness of models can be increased significantly and thus provides knowledge about complex dynamics and interactions that play a key role for the inherent robustness of real biological systems. Furthermore, by employing a robustness analysis, stable steady-state branches can be recovered and bistability can be induced.

Cette thèse propose une méthodologie hors ligne pour la robustification de lois de commande prédictives multivariables, se basant sur une problématique d'optimisation convexe d'un paramètre de Youla. Le point de départ de la démarche consiste à synthétiser une loi de commande initiale prédictive multivariable sous forme d'état qui stabilise le système. Le but est de garantir la robustesse en stabilité face à des incertitudes non structurées et d'assurer des performances nominales pour le rejet de perturbations, imposées sous la forme des gabarits temporels sur les sorties. Ce problème d'optimisation est résolu par un formalisme LMI. Le paramètre de Youla obtenu permet de gérer d'une part le compromis entre la robustesse en stabilité et les performances nominales et d'une autre part permet de réduire l'influence du couplage multivariable sur le rejet des perturbations.<br />Le cas de systèmes incertains appartenant à un ensemble donné d'incertitudes polytopiques est également traité. Deux possibilités sont analysées : le correcteur MPC initial stable sur tout le domaine polytopique, le correcteur MPC initial instable sur une partie du domaine incertain considéré. Dans les deux cas, une condition supplémentaire BMI est ajoutée pour chaque sommet du polytope considéré. Il s'agit de deux problèmes d'optimisation non-convexe pour lesquels deux solutions de complexité raisonnable sous une forme LMI sous-optimale sont proposées.<br />Cette technique de robustification est illustrée sur un modèle académique multivariable d'un réacteur. Une application à un robot médical est ensuite détaillée. L'ensemble des stratégies développées pour réduire l'influence des incertitudes non structurées sur le système en respectant les gabarits imposés sur les sorties pour le rejet de perturbations a donné lieu à la mise au point d'un logiciel sous MATLABTM.

This diploma thesis deals with the problem of portfolio optimization in relation to the mean vector and the variance matrix of yields. The emphasis is put on Mar- kowitz model. In the thesis there are explored some possibilities of robustification based on the used parametric set. Beside the classic formulation of the task our focus is also devoted to the cases in which short sales are not allowed. The core of the thesis constitutes of a simulation study that models the impact of errors in the estimation of the input parameters of Markowitz model. It takes into account different types of risk aversions and different approaches to modelling parameter perturbations . Therefore it specifies the hypothesis of the dominating influence of the mean vector estimate which is valid only for a risk lover. 1

<p>Online decision making with switching costs has received considerable attention in many practical problems that face uncertainty in the inputs and key problem parameters. Because of the switching costs that penalize the change of decisions, making good online decisions under such uncertainty is known to be extremely challenging. This thesis aims at providing new online algorithms with strong performance guarantees to address this challenge.</p> <p><br></p> <p>In part 1 and part 2 of this thesis, motivated by Network Functions Virtualization and smart grid, we study competitive online convex optimization with switching costs. Specifically, in part 1, we focus on the setting with an uncertainty set (one type of prediction) and hard infeasibility constraints. We develop new online algorithms that can attain optimized competitive ratios, while ensuring feasibility at all times. Moreover, we design a robustification procedure that helps these algorithms obtain good average-case performance simultaneously. In part 2, we focus on the setting with look-ahead (another type of prediction). We provide the first algorithm that attains a competitive ratio that not only decreases to 1 as the look-ahead window size increases, but also remains upper-bounded for any ratio between the switching-cost coefficient and service-cost coefficient.</p> <p><br></p> <p>In part 3 of this thesis, motivated by edge computing with artificial intelligence, we study bandit learning with switching costs where, in addition to bandit feedback, full feedback can be requested at a cost. We show that, when only 1 arm can be chosen at a time, adding costly full-feedback is not helpful in fundamentally reducing the Θ(<em>T</em>2/3) regret over a time-horizon <em>T</em>. In contrast, when 2 (or more) arms can be chosen at a time, we provide a new online learning algorithm that achieves a significantly smaller regret equal to <em>O</em>(√<em>T</em>), without even using full feedback. To the best of our knowledge, this type of sharp transition from choosing 1 arm to choosing 2 (or more) arms has never been reported in the literature.</p>