Numerical algorithms for efficiently solving optimal control problems are important for commonly used advanced control strategies, such as model predictive control (MPC), but can also be useful for advanced estimation techniques, such as moving horizon estimation (MHE). In MPC, the control input is computed by solving a constrained finite-time optimal control (CFTOC) problem on-line, and in MHE the estimated states are obtained by solving an optimization problem that often can be formulated as a CFTOC problem. Common types of optimization methods for solving CFTOC problems are interior-point (IP) methods, sequential quadratic programming (SQP) methods and active-set (AS) methods. In these types of methods, the main computational effort is often the computation of the second-order search directions. This boils down to solving a sequence of systems of equations that correspond to unconstrained finite-time optimal control (UFTOC) problems. Hence, high-performing second-order methods for CFTOC problems rely on efficient numerical algorithms for solving UFTOC problems. Developing such algorithms is one of the main focuses in this thesis. When the solution to a CFTOC problem is computed using an AS type method, the aforementioned system of equations is only changed by a low-rank modification between two AS iterations. In this thesis, it is shown how to exploit these structured modifications while still exploiting structure in the UFTOC problem using the Riccati recursion. Furthermore, direct (non-iterative) parallel algorithms for computing the search directions in IP, SQP and AS methods are proposed in the thesis. These algorithms exploit, and retain, the sparse structure of the UFTOC problem such that no dense system of equations needs to be solved serially as in many other algorithms. The proposed algorithms can be applied recursively to obtain logarithmic computational complexity growth in the prediction horizon length. For the case with linear MPC problems, an alternative approach to solving the CFTOC problem on-line is to use multiparametric quadratic programming (mp-QP), where the corresponding CFTOC problem can be solved explicitly off-line. This is referred to as explicit MPC. One of the main limitations with mp-QP is the amount of memory that is required to store the parametric solution. In this thesis, an algorithm for decreasing the required amount of memory is proposed. The aim is to make mp-QP and explicit MPC more useful in practical applications, such as embedded systems with limited memory resources. The proposed algorithm exploits the structure from the QP problem in the parametric solution in order to reduce the memory footprint of general mp-QP solutions, and in particular, of explicit MPC solutions. The algorithm can be used directly in mp-QP solvers, or as a post-processing step to an existing solution. / Numeriska algoritmer för att effektivt lösa optimala styrningsproblem är en viktig komponent i avancerade regler- och estimeringsstrategier som exempelvis modellprediktiv reglering (eng. model predictive control (MPC)) och glidande horisont estimering (eng. moving horizon estimation (MHE)). MPC är en reglerstrategi som kan användas för att styra system med flera styrsignaler och/eller utsignaler samt ta hänsyn till exempelvis begränsningar i styrdon. Den grundläggande principen för MPC och MHE är att styrsignalen och de estimerade variablerna kan beräknas genom att lösa ett optimalt styrningsproblem. Detta optimeringsproblem måste lösas inom en kort tidsram varje gång som en styrsignal ska beräknas eller som variabler ska estimeras, och således är det viktigt att det finns effektiva algoritmer för att lösa denna typ av problem. Två vanliga sådana är inrepunkts-metoder (eng. interior-point (IP)) och aktivmängd-metoder (eng. active-set (AS)), där optimeringsproblemet löses genom att lösa ett antal enklare delproblem. Ett av huvudfokusen i denna avhandling är att beräkna lösningen till dessa delproblem på ett tidseffektivt sätt genom att utnyttja strukturen i delproblemen. Lösningen till ett delproblem beräknas genom att lösa ett linjärt ekvationssystem. Detta ekvationssystem kan man exempelvis lösa med generella metoder eller med så kallade Riccatirekursioner som utnyttjar strukturen i problemet. När man använder en AS-metod för att lösa MPC-problemet så görs endast små strukturerade ändringar av ekvationssystemet mellan varje delproblem, vilket inte har utnyttjats tidigare tillsammans med Riccatirekursionen. I denna avhandling presenteras ett sätt att utnyttja detta genom att bara göra små förändringar av Riccatirekursionen för att minska beräkningstiden för att lösa delproblemet. Idag har behovet av  parallella algoritmer för att lösa MPC och MHE problem ökat. Att algoritmerna är parallella innebär att beräkningar kan ske på olika delar av problemet samtidigt med syftet att minska den totala verkliga beräkningstiden för att lösa optimeringsproblemet. I denna avhandling presenteras parallella algoritmer som kan användas i både IP- och AS-metoder. Algoritmerna beräknar lösningen till delproblemen parallellt med ett förutbestämt antal steg, till skillnad från många andra parallella algoritmer där ett okänt (ofta stort) antal steg krävs. De parallella algoritmerna utnyttjar problemstrukturen för att lösa delproblemen effektivt, och en av dem har utvärderats på parallell hårdvara. Linjära MPC problem kan också lösas genom att utnyttja teori från multiparametrisk kvadratisk programmering (eng. multiparametric quadratic programming (mp-QP)) där den optimala lösningen beräknas i förhand och lagras i en tabell, vilket benämns explicit MPC. I detta fall behöver inte MPC problemet lösas varje gång en styrsignal beräknas, utan istället kan den förberäknade optimala styrsignalen slås upp. En nackdel med mp-QP är att det krävs mycket plats i minnet för att spara lösningen. I denna avhandling presenteras en strukturutnyttjande algoritm som kan minska behovet av minne för att spara lösningen, vilket kan öka det praktiska användningsområdet för mp-QP och explicit MPC.

This thesis is devoted to the mathematical analysis of models describing the energy of defects in crystalline solids via variational methods. The first part of this work studies a discrete model describing the energy of a point defect in a one dimensional chain of atoms. We derive an expansion of the ground state energy using Gamma-convergence, following previous work on similar models [BDMG99,BC07,SSZ11]. The main novelty here is an explicit characterisation of the first order limit as the solution of a variational problem in an infinite lattice. Analysing this variational problem, we prove a regularity result for the perturbation caused by the defect, and demonstrate the order of the next term in the expansion. The second main topic is a discrete model describing screw dislocations in body centred cubic crystals. We formulate an anti plane lattice model which describes the energy difference between deformations and, using the framework defined in [AO05], provide a kinematic description of the Burgers vector, which is a key geometric quantity used to describe dislocations. Apart from the anti plane restriction, this model is invariant under all the natural symmetries of the lattice and in particular allows for the creation and annihilation of dislocations. The energy difference formulation enables us to provide a clear definition of what it means to be a stable deformation. The main results of the analysis of this model are then first, a proof that deformations with unit net Burgers vector exist as globally stable states in an infinite body, and second, that deformations containing multiple screw dislocations exist as locally stable states in both infinite bodies and finite convex bodies. To prove the former result, we establish coercivity with respect to the elastic strain, and exploit a concentration compactness principle. In the latter case, we use a form of the inverse function theorem, proving careful estimates on the residual and stability of an ansatz which combines continuum linear elasticity theory with an atomistic core correction.

The thesis concerns the mathematical study of the calculus of variations and its application to liquid crystals. In the first chapter we examine vectorial problems in the calculus of variations with an additional pointwise constraint so that any admissible function <strong>n</strong> ε W^1,1(ΩM), and M is a manifold of suitable regularity. We formulate necessary and sufficient conditions for any given state <strong>n</strong> to be a strong or weak local minimiser of I. This is achieved using a nearest point projection mapping in order to use the more classical results which apply in the absence of a constraint. In the subsequent chapters we study various static continuum theories of liquid crystals. More specifically we look to explain a particular cholesteric fingerprint pattern observed by HP Labs. We begin in Chapter 2 by focusing on a specific cholesteric liquid crystal problem using the theory originally derived by Oseen and Frank. We find the global minimisers for general elastic constants amongst admissible functions which only depend on a single variable. Using the one-constant approximation for the Oseen-Frank free energy, we then show that these states are global minimisers of the three-dimensional problem if the pitch of the cholesteric liquid crystal is sufficiently long. Chapter 3 concerns the application of the results from the first chapter to the situations investigated in the second. The local stability of the one-dimensional states are quantified, analytically and numerically, and in doing so we unearth potential shortcomings of the classical Oseen-Frank theory. In Chapter 4, we ascertain some equivalence results between the continuum theories of Oseen and Frank, Ericksen, and Landau and de Gennes. We do so by proving lifting results, building on the work of Ball and Zarnescu, which relate the regularity of line and vector fields. The results prove to be interesting as they show that for a director theory to respect the head to tail symmetry of the liquid crystal molecules, the appropriate function space for the director field is S BV² (Ω,S<sup>2,/sup>). We take this idea and in the final chapter we propose a mathematical model of liquid crystals based upon the Oseen-Frank free energy but using special functions of bounded variation. We establish the existence of a minimiser, forms of the Euler-Lagrange equation, and find solutions of the Euler-Lagrange equation in some simple cases. Finally we use our proposed model to re-examine the same problems from Chapter 2. By doing so we extend the analysis we were able to achieve using Sobolev spaces and predict the existence of multi-dimensional minimisers consistent with the known experimental properties of high-chirality cholesteric liquid crystals.

Deux stratégies de contrôle ont été successivement mises en œuvre pour réduire la traînée et l’énergie cinétique turbulente d’un canal plan en régime turbulent (Re[tau]=180) par soufflage/aspiration aux parois. L’objectif principal était de prouver qu’une simulation aux grandes échelles (LES) pouvait être utilisée de manière pertinente comme modèle réduit des équations de Navier-Stokes et ainsi diminuer fortement les coûts numériques. Une approche heuristique dite de contrôle par opposition a d’abord été employée. Les résultats montrent que l'efficacité énergétique est maximale pour une position du plan de détection différente de celle qui correspond au maximum de réduction de traînée. Par ailleurs, nos résultats confirment que la réduction de traînée diminue avec l'augmentation du nombre de Reynolds. Par la suite, une procédure de contrôle optimal a été utilisée en considérant différentes fonctionnelles objectif (traînée, énergie cinétique au temps terminal, énergie cinétique moyen). Pour Re[tau]=100, le contrôle est parvenu à relaminariser complètement l’écoulement (réduction de traînée de l'ordre de 50 %) en prenant comme fonctionnelle coût l’énergie cinétique au temps terminal. Pour cette même fonctionnelle coût, une réduction importante de traînée de l'ordre de 55 % est encore obtenue à Re[tau] =180 mais sans atteindre la relaminarisation. Nos résultats confirment que pour minimiser la traînée de l’écoulement, il est plus efficace de considérer comme objectif l’énergie cinétique que directement la traînée. Enfin, il est essentiel pour la convergence de la minimisation que le système optimal soit résolu sur un horizon temporel suffisamment long / Two control strategies were successively implemented to reduce the drag and the turbulent kinetic energy of a plane channel flow in turbulent regime (Re[tau]=180). Wall transpiration (unsteady blowing/suction) with zero net mass flux is used as the control. The main objective was to prove that a large eddy simulation (LES) could be relevant as a reduced-order model of the Navier-Stokes equations and thus strongly reduce the numerical costs. A heuristic approach known as opposition control was initially employed. The results show that the energetic efficiency is maximum for a position of the detection plane different from that which corresponds to the maximum of drag reduction. In addition, our results confirm that the drag reduction decreases with the increase of the Reynolds number. Then, an optimal control procedure was used by considering different cost functional (drag, terminal turbulent kinetic energy, mean turbulent kinetic energy). At Re[tau] =100, control managed to fully relaminarize the flow (drag reduction of about 50%) by considering as cost functional the terminal kinetic energy. For this same cost functional, an important drag reduction of about 55% is still obtained at Re[tau] =180 but without reaching the relaminarization. Our results show that to minimize the flow drag, it is more effective to consider the kinetic energy as cost functional than directly the drag. Lastly, it is essential for the convergence of the minimization that the optimality system is solved on a sufficiently long time horizon

Dans cette thèse, nous traitons deux problèmes de contrôle optimal stochastique. Chaque problème correspond à une Partie de ce document. Le premier problème traité est très précis, il s'agit de la valorisation des contrats optionnels de vente de type Américain (dit Put Américain) en présence de dividendes discrets (Partie I). Le deuxième est plus général, puisqu'il s'agit dans un cadre discret en temps de prouver l'existence d'un principe de programmation dynamique sous des contraintes en probabilités (Partie II). Bien que les deux problèmes soient assez distincts, le principe de programmation dynamique est au coeur de ces deux problèmes. La relation entre la valorisation d'un Put Américain et un problème de frontière libre a été prouvée par McKean. La frontière de ce problème a une signification économique claire puisqu'elle correspond à tout instant à la borne supérieure de l'ensemble des prix d'actifs pour lesquels il est préférable d'exercer tout de suite son droit de vente. La forme de cette frontière en présence de dividendes discrets n'avait pas été résolue à notre connaissance. Sous l'hypothèse que le dividende est une fonction déterministe du prix de l'actif à l'instant précédant son versement, nous étudions donc comment la frontière est modifiée. Au voisinage des dates de dividende, et dans le modèle du Chapitre 3, nous savons qualifier la monotonie de la frontière, et dans certains cas quantifier son comportement local. Dans le Chapitre 3, nous montrons que la propriété du smooth-fit est satisfaite à toute date sauf celles de versement des dividendes. Dans les deux Chapitres 3 et 4, nous donnons des conditions pour garantir la continuité de cette frontière en dehors des dates de dividende. La Partie II est originellement motivée par la gestion optimale de la production d'une centrale hydro-electrique avec une contrainte en probabilité sur le niveau d'eau du barrage à certaines dates. En utilisant les travaux de Balder sur la relaxation de Young des problèmes de commande optimale, nous nous intéressons plus spécifiquement à leur résolution par programmation dynamique. Dans le Chapitre 5, nous étendons au cadre des mesures de Young des résultats dûs à Evstigneev. Nous établissons alors qu'il est possible de résoudre par programmation dynamique certains problèmes avec des contraintes en espérances conditionnelles. Grâce aux travaux de Bouchard, Elie, Soner et Touzi sur les problèmes de cible stochastique avec perte contrôlée, nous montrons dans le Chapitre 6 qu'un problème avec contrainte en espérance peut se ramener à un problème avec des contraintes en espérances conditionnelles. Comme cas particulier, nous prouvons ainsi que le problème initial de la gestion du barrage peut se résoudre par programmation dynamique / In this thesis, we address two problems of stochastic optimal control. Each problem constitutes a different Part in this document. The first problem addressed is very precise, it is the valuation of American contingent claims and more specifically the American Put in the presence of discrete dividends (Part I). The second one is more general, since it is the proof of the existence of a dynamic programming principle under expectation constraints in a discrete time framework (Part II). Although the two problems are quite distinct, the dynamic programming principle is at the heart of these two problems. The relationship between the value of an American Put and a free boundary problem has been proved by McKean. The boundary of this problem has a clear economic meaning since it corresponds at all times to the upper limit of the asset price above which the holder of such an option would exercise immediately his right to sell. The shape of the boundary in the presence of discrete dividends has not been solved to the best of our knowledge. Under the assumption that the dividend is a deterministic function of asset prices at the date just before the dividend payment, we investigate how the boundary is modified. In the neighborhood of dividend dates and in the model of Chapter 3, we know what the monotonicity of the border is, and we quantify its local behavior. In Chapter 3, we show that the smooth-fit property is satisfied at any date except for those of the payment of dividends. In both Chapters 3 and 4, we are able to give conditions to guarantee the continuity of the border outside dates of dividend. Part II was originally motivated by the optimal management of the production of an hydro-electric power plant with a probability constraint on the reservoir level on certain dates. Using Balder'sworks on Young's relaxation of optimal control problems, we focus more specifically on their resolution by dynamic programming. In Chapter 5, we extend results of Evstigneev to the framework of Young measures. We show that dynamic programming can be used to solve some problems with conditional expectations constraints. Through the ideas of Bouchard, Elie, Soner and Touzi on stochastic target problems with controlled loss, we show in Chapter 6 that a problem with expectation constraints can be reduced to a problem with conditional expectation constraints. Finally, as a special case, we show that the initial problem of dam management can be solved by dynamic programming

Nous introduisons un modèle mathématique d'évolution d'une maladie cancéreuse à l'échelle de l'organisme, prenant en compte les métastases ainsi que leur taille et permettant de simuler l'action de plusieurs thérapies telles que la chirurgie, la chimiothérapie ou les traitements anti-angiogéniques. Le problème mathématique est une équation de renouvellement structurée en dimension deux. Son analyse mathématique ainsi que l'analyse fonctionnelle d'un espace de Sobolev sous-jacent sont effectuées. Existence, unicité, régularité et comportement asymptotique des solutions sont établis dans le cas autonome. Un schéma numérique lagrangien est introduit et analysé, permettant de prouver l'existence de solutions dans le cas non-autonome. L'effet de la concentration de la donnée au bord en une masse de Dirac est aussi envisagé.Le potentiel du modèle est ensuite illustré pour des problématiques cliniques telles que l'échec des anti-angiogéniques, les protocoles temporels d'administration pour la combinaison d'une chimiothérapie et d'un anti-angiogénique et les chimiothérapies métronomiques. Pour tenter d'apporter des réponses mathématiques à ces problèmes cliniques, un problème de contrôle optimal est formulé, analysé et simulé. / We introduce a mathematical model for the evolution of a cancer disease at the organism scale, taking into account for the metastases and their sizes as well as action of several therapies such as primary tumor surgery, chemotherapy and anti-angiogenic therapy. The mathematical problem is a renewal equation with bi-dimensional structuring variable. Mathematical analysis and functional analysis of an underlying Sobolev space are performed. Existence, uniqueness, regularity and asymptotic behavior of the solutions are proven in the autonomous case. A lagrangian numerical scheme is introduced and analyzed. Convergence of this scheme proves existence in the non-autonomous case. The effect of concentration of the boundary data into a Dirac mass is also investigated.Possible applications of the model are numerically illustrated for clinical issues such as the failure of anti-angiogenic monotherapies, scheduling of combined cytotoxic and anti-angiogenic therapies and metronomic chemotherapies. In order to give mathematical answers to these clinical problems an optimal control problem is formulated, analyzed and simulated.

An assessment study of a novel approach is presented that combines discrete state-space Dynamic Programming and Pontryagin’s Maximum Principle for online optimal control of hybrid electric vehicles (HEV). In addition to electric energy storage and gear, kinetic energy and travel time are considered states in this paper. After presenting the corresponding model using a parallel HEV as an example, a benchmark method with Dynamic Programming is introduced which is used to show the solution quality of the novel approach. It is illustrated that the proposed method yields a close-to-optimal solution by solving the optimal control problem over one hundred thousand times faster than the benchmark method. Finally, a potential online usage is assessed by comparing solution quality and calculation time with regard to the quantization of the state space.

Sistemas Lineares com Saltos Markovianos (SLSMs) são estudados desde a década de 1960 e vêm ganhando visibilidade desde então, com diversas aplicações dentre as quais Finanças, Robótica e Engenharias diversas. Um problema de regulação trata de controlar o SLSM buscando fazer sua trajetória se aproximar de zero. Quando os saltos markovianos são observados, o problema é simples e bem resolvido, muito diferente de quando não se observam os saltos. Neste trabalho é estudado um algoritmo da literatura utilizado para resolver o problema de regulação sem observação dos saltos, chamado Método Variacional (MV). Sendo um dos melhores métodos para o dado problema, enfrenta dificuldades de cunho numérico. Neste trabalho se procura analisar e melhorar o condicionamento dos subproblemas envolvidos, de forma a favorecer a convergência do método. São testadas abordagens diferentes usando precondicionadores e comparados os resultados, permitindo concluir que três das cinco abordagens é que trouxeram os melhores resultados. Por se tratar de sistemas lineares do tipo Ax = b, as abordagens de condicionamento podem ser adaptadas para outros problemas semelhantes. / Markov Jump Linear Systems (MJLSs) have been studied since the decade of 1960 and they are gaining visibility ever since, due to a wide range of applications, such as Finance, Robotics, several Engeneerings among others. The so called regulation problem is to control the MJLS seeking to make its trajectory to approach zero. When markovian jumps are observed, the problem is simple and the solution given as closed formulas, which is quite different from the situation when jumps are not observed. We study an algorithm available in literature called Variational Method (VM). Even though it is one of the best methods to solve the problem, it has some numerical difficulties. We analyse its performance and propose some ideas aiming at the ill-conditioning of the subproblems involved, in order to improve the convergence of the method. Different approaches are tested using preconditioners and the results are compared, indicating that three approaches of the five tested ones are promising for convergence improvement. Because the subproblems are linear systems of type Ax = b, these approaches can be adapted to similar problems.

The Linear Quadratic Regulator (LQR) is an optimal control approach which aims to drive states of a linear system to its origin through the minimization of a quadratic cost functional. Such an approach has been widely successful for both theoretical and practical applications. However, when such controllers are subject to uncertainties, optimal closed-loop performance cannot be obtained since robustness properties are no longer guaranteed. Guaranteed Cost Controllers (GCC) presents robust asymptotic stability and provides a guaranteed upper bound to a quadratic cost function. Such method addresses the lack of performance guarantees of the LQR. Meanwhile, Model Predictive Control (MPC) is a class of optimization-based control algorithms that use an explicit model of the controlled system to predict its future states. The MPC can be as a generalization of the LQR for constrained linear systems. Therefore, it equally suffers from a lack of robustness guarantees when the system is subject to uncertainties. Robust MPC (RMPC) approaches were proposed to address MPCs poor closed-loop performance subject to uncertainties. Its objective is to obtain a control input sequence that simultaneously minimizes a cost function and guarantees the feasibility of system states and control inputs, for a system subject to the worst-case disturbance within an uncertainty set. Autonomous vehicles have gained increasing interest from both the industry and research communities in recent years. An essential aspect in the design of automotive control systems is to ensure the controller is stable and has acceptable performance within the entire operational envelope which it is designed to operate. In the case of autonomous vehicles, where there is no human driver as a fallback, it is of utmost importance to ensure the safe operations of the control system and its capability to avoid saturating the handling limits of the vehicle. In this thesis, we propose Guaranteed Cost Controller approaches for both unconstrained and constrained linear systems subject to multiplicative structured norm-bounded uncertainties and present the application of such a controller to the lateral control problem of autonomous vehicles up to the tire saturation limits. / O Regulador Quadrático Linear (Linear Quadratic Regulator, LQR) é uma abordagem de controle ótimo que visa conduzir estados de um sistema linear à sua origem através da minimização de um custo funcional quadrático. Tal abordagem tem sido amplamente bem sucedida para aplicações teóricas e práticas. No entanto, não é possível obter o desempenho ótimo de malha fechada quando esses controladores são sujeitos a incertezas no sistema em decorrência de suas propriedades de robustez não serem garantidas. Controladores de Custo Garantido (Guaranteed Cost Control, GCC) visam abordar a falta de garantia de desempenho do LQR, neste caso. Esses controladores apresentam estabilidade assintótica robusta e fornecem um custo garantido de pior caso para uma função de custo quadrático. O Controle Preditivo de Modelo (Model Predictive Control, MPC) é uma classe de algoritmos de controle baseados em otimização que usa um modelo explícito do sistema controlado para prever seus estados futuros. Uma possível interpretação do MPC é uma generalização do LQR para sistemas lineares com restrições de estado e entrada de controle. Portanto, essa abordagem sofre igualmente da falta de garantias de robustez quando o sistema é sujeito a incertezas. As abordagens de MPC Robustas (Robust MPC, RMPC) foram propostas para abordar o desempenho de malha fechada do MPC sujeito a incertezas no sistema. Seu objetivo é obter uma sequência de entrada de controle que minimize simultaneamente uma função de custo e garanta que os estados do sistema e as entradas de controle estão contidos dentro das restrições para um sistema sujeito à pior das perturbações dentro de um conjunto admissível de incertezas. Pesquisas voltadas para veículos autônomos ganharam crescente interesse nos últimos anos, tanto da indústria automobilística quanto da comunidade acadêmica. Um aspecto essencial no projeto de sistemas de controle automotivo é a garantia de estabilidade e desempenho do controlador dentro de todo o envelope operacional ao qual ele foi projetado para operar. No caso de veículos autônomos, onde não há motoristas humanos para lidar com casos de falha do sistema, é de suma importância assegurar as operações seguras do sistema de controle e sua capacidade de evitar a saturação dos limites de manuseio do veículo. Nesta tese, propomos abordagens GCC para sistemas lineares restritos e irrestritos, sujeitos a incertezas estruturadas contidas por norma e apresentamos a aplicação de tais controladores ao problema de controle lateral de veículos autônomos até os limites de saturação dos pneus.

This report investigates the possibilities to model the causal effect of treatment within the insurance domain to increase return on investment of sales through telemarketing. In order to capture the causal effect, two or more subgroups are required where one group receives control treatment. Two different uplift models model the causal effect of treatment, Class Transformation Method, and Modeling Uplift Directly with Random Forests. Both methods are evaluated by the Qini curve and the Qini coefficient. To model the causal effect of treatment, the comparison with a control group is a necessity. The report attempts to find the optimal treatment group allocation in order to maximize the precision in the difference between the treatment group and the control group. Further, the report provides a rule of thumb that ensure that the control group is of sufficient size to be able to model the causal effect. If has provided the data material used to model uplift and it consists of approximately 630000 customer interactions and 60 features. The total uplift in the data set, the difference in purchase rate between the treatment group and control group, is approximately 3%. Uplift by random forest with a Euclidean distance splitting criterion that tries to maximize the distributional divergence between treatment group and control group performs best, which captures 15% of the theoretical best model. The same model manages to capture 77% of the total amount of purchases in the treatment group by only giving treatment to half of the treatment group. With the purchase rates in the data set, the optimal treatment group allocation is approximately 58%-70%, but the study could be performed with as much as approximately 97%treatment group allocation.