BONNANS Frederic

Using an extension of Varadhan's formula to Randers manifolds, we notice that Randers distances may be approximated by a logarithmic transformation of a linear second-order partial differential equation. Following an idea introduced by Crane, Weischedel, and Wardetzky in the case of Riemannian distances, we study a numerical method for approximating Randers distances which involves a discretization of this linear equation. We propose to use Selling's formula, which originates from the theory of low-dimensional lattice geometry, to build a monotone and linear finite-difference scheme. By injecting the logarithmic transformation in this linear scheme, we are able to prove convergence of this numerical method to the Randers distance, as well as consistency to the order two thirds far from the boundary of the considered domain. We explain how this method may be used to approximate optimal transport distances, how has been previously done in the Riemannian case.

We propose and investigate a general class of discrete time and finite state space mean field game (MFG) problems with potential structure. Our model incorporates interactions through a congestion term and a price variable. It also allows hard constraints on the distribution of the agents. We analyze the connection between the MFG problem and two optimal control problems in duality. We present two families of numerical methods and detail their implementation: (i) primal-dual proximal methods (and their extension with nonlinear proximity operators), (ii) the alternating direction method of multipliers (ADMM) and a variant called ADM-G. We give some convergence results. Numerical results are provided for two examples with hard constraints.

A new approach to estimate traffic energy consumption via traffic data aggregation in (speed,acceleration) probability distributions is proposed. The aggregation is done on each segment composing the road network. In order to reduce data occupancy, clustering techniques are used to obtain meaningful classes of traffic conditions. Different times of the day with similar speed patterns and traffic behavior are thus grouped together in a single cluster. Different energy consumption models based on the aggregated data are proposed to estimate the energy consumption of the vehicles in the road network. For validation purposes, a microscopic traffic simulator is used to generate the data and compare the estimated energy consumption to the reference one. A thorough sensitivity analysis with respect to the parameters of the proposed method (i.e. number of clusters, size of the distributions support, etc.) is also conducted in simulation. Finally, a real-life scenario using floating car data is analyzed to evaluate the applicability and the robustness of the proposed method.

We design adaptive finite differences discretizations, which are degenerate elliptic and second order consistent, of linear and quasi-linear partial differential operators featuring both a first order term and an anisotropic second order term. Our approach requires the domain to be discretized on a Cartesian grid, and takes advantage of techniques from the field of low-dimensional lattice geometry. We prove that the stencil of our numerical scheme is optimally compact, in dimension two, and that our approach is quasi-optimal in terms of the compatibility condition required of the first and second order operators, in dimension two and three. Numerical experiments illustrate the efficiency of our method in several contexts.

This thesis deals with the design of an Energy Management System (EMS), taking into account the traffic constraints, for a hybrid electric vehicle. Currently, EMSs are usually classified into two categories: those proposing a real-time architecture seeking a local optimum, and those seeking a global optimum, which is more costly in terms of computation time and therefore more appropriate for offline use. This thesis is based on the fact that energy consumption can be accurately modeled using probability distributions on speed and acceleration. In order to reduce the size of the data, a classification is proposed, based on the Wasserstein distance, where the barycenters of the classes can be computed using Sinkhorn iterations or the Alternate Stochastic Gradient method. This traffic modeling allowed an offline optimization to determine the optimal control (the torque of the electric motor) that minimizes the fuel consumption of the hybrid vehicle on a road segment. Continuing on, a two-level algorithm took advantage of this information to optimize fuel consumption over the entire route. The upper level of optimization, being deterministic, is fast enough for a real time implementation. The relevance of the traffic model and the bi-level method is illustrated using traffic data generated by a simulator, but also using real data collected near Lyon (France). Finally, an extension of the bi-level method to the eco-routing problem is considered, using an augmented graph to determine the load state during the optimal path.

This work proposes a new approach to optimize the consumption of a hybrid electric vehicle taking into account the traffic conditions. The method is based on a bi-level decomposition in order to make the implementation suitable for online use. The offline lower level computes cost maps thanks to a stochastic optimization that considers the influence of traffic, in terms of speed/acceleration probability distributions. At the online upper level, a deterministic optimization computes the ideal state of charge at the end of each road segment, using the computed cost maps. Since the high computational cost due to the uncertainty of traffic conditions has been managed at the lower level, the upper level is fast enough to be used online in the vehicle. Errors due to discretization and computation in the proposed algorithm have been studied. Finally, we present numerical simulations using actual traffic data, and compare the proposed bi-level method to a deterministic optimization with perfect information about traffic conditions. The solutions show a reasonable over-consumption compared with deterministic optimization, and manageable computational times for both the offline and online parts.

A new approach to estimate traffic energy consumption via traffic data aggregation in (speed,acceleration) probability distributions is proposed. The aggregation is done on each segment composing the road network. In order to reduce data occupancy, clustering techniques are used to obtain meaningful classes of traffic conditions. Different times of the day with similar speed patterns and traffic behavior are thus grouped together in a single cluster. Different energy consumption models based on the aggregated data are proposed to estimate the energy consumption of the vehicles in the road network. For validation purposes, a microscopic traffic simulator is used to generate the data and compare the estimated energy consumption to the reference one. A thorough sensitivity analysis with respect to the parameters of the proposed method (i.e. number of clusters, size of the distributions support, etc.) is also conducted in simulation. Finally, a real-life scenario using floating car data is analyzed to evaluate the applicability and the robustness of the proposed method.

We consider the task of solving an aircraft trajectory optimization problem where the system dynamics have been estimated from recorded data. Additionally, we want to avoid optimized trajectories that go too far away from the domain occupied by the data, since the model validity is not guaranteed outside this region. This motivates the need for a proximity indicator between a given trajectory and a set of reference trajectories. In this presentation, we propose such an indicator based on a parametric estimator of the training set density. We then introduce it as a penalty term in the optimal control problem. Our approach is illustrated with an aircraft minimal consumption problem and recorded data from real flights. We observe in our numerical results the expected trade-off between the consumption and the penalty term.

An extension of the bi-level optimization for the energy management of hybrid electric vehicles (HEVs) proposed in Le Rhun et al. (2019a) to the eco-routing problem is presented. Using the knowledge of traffic conditions over the entire road network, we search both the optimal path and state of charge trajectory. This problem results in finding the shortest path on a weighted graph whose nodes are (position, state of charge) pairs for the vehicle, the edge cost being evaluated thanks to the cost maps from optimization at the 'micro' level of a bi-level decomposition. The error due to the discretization of the state of charge is proven to be linear if the cost maps are Lipschitz. The classical A * algorithm is used to solve the problem, with a heuristic based on a lower bound of the energy needed to complete the travel. The eco-routing method is validated by numerical simulations and compared to the fastest path on a synthetic road network.

This work proposes a new approach to optimize the consumption of a hybrid electric vehicle taking into account the traffic conditions. The method is based on a bi-level decomposition in order to make the implementation suitable for online use. The offline lower level computes cost maps thanks to a stochastic optimization that considers the influence of traffic, in terms of speed/acceleration probability distributions. At the online upper level, a deterministic optimization computes the ideal state of charge at the end of each road segment, using the computed cost maps. Since the high computational cost due to the uncertainty of traffic conditions has been managed at the lower level, the upper level is fast enough to be used online in the vehicle. Errors due to discretization and computation in the proposed algorithm have been studied. Finally, we present numerical simulations using actual traffic data, and compare the proposed bi-level method to a deterministic optimization with perfect information about traffic conditions. The solutions show a reasonable over-consumption compared with deterministic optimization, and manageable computational times for both the offline and online parts.

Electricity grids have to absorb an increasing production of renewable energy in a decentralized way. Their optimal management leads to specific problems. We study in this thesis the mathematical formulation of such problems as multi-step stochastic optimization problems. We analyze more specifically the time and space decomposition of such problems. In the first part of this manuscript, Time Decomposition for the Optimization of Domestic Microgrid Management, we apply stochastic optimization methods to small microgrid management. We compare different optimization algorithms on two examples: the first one considers a domestic microgrid equipped with a battery and a micro-cogeneration plant. The second one considers another domestic microgrid, this time equipped with a battery and solar panels. In the second part, Temporal and spatial decomposition of large optimization problems, we extend the previous studies to larger microgrids, with different units and storages connected together. The frontal solution of such large problems by Dynamic Programming proves impractical. We propose two original algorithms to overcome this problem by mixing a temporal decomposition with a spatial decomposition --- by prices or by resources. In the last part, Contributions to the Stochastic Dual Dynamic Programming algorithm, we focus on the emph{Stochastic DualDynamic Programming} (SDDP) algorithm which is currently a reference method for solving multi-time step stochastic optimization problems. We study a new stopping criterion for this algorithm based on a dual version of SDDP, which allows to obtain a deterministic upper bound for the primal problem.

This paper addresses the problem of identifying structured nonlinear dynamical systems, with the goal of using the learned dynamics in model-based reinforcement learning problems. We present in this setting a new class of scalable multi-task estimators which promote sparsity, while preserving the dynamics structure and leveraging available physical insight. An implementation leading to consistent feature selection is suggested, allowing to obtain accurate models. An additional regularizer is also proposed to help in recovering realistic hidden representations of the dynamics. We illustrate our method by applying it to an aircraft trajectory optimization problem. Our numerical results based on real flight data from 25 medium haul aircraft, totaling 8 millions observations, show that our approach is competitive with existing methods for this type of application.

We consider the task of solving an aircraft trajectory optimization problem where the system dynamics have been estimated from recorded data. Additionally, we want to avoid optimized trajectories that go too far away from the domain occupied by the data, since the model validity is not guaranteed outside this region. This motivates the need for a proximity indicator between a given trajectory and a set of reference trajectories. In this presentation, we propose such an indicator based on a parametric estimator of the training set density. We then introduce it as a penalty term in the optimal control problem. Our approach is illustrated with an aircraft minimal consumption problem and recorded data from real flights. We observe in our numerical results the expected trade-off between the consumption and the penalty term.

In this paper, we tackle the problem of quantifying the closeness of a newly observed curve to a given sample of random functions, supposed to have been sampled from the same distribution. We define a probabilistic criterion for such a purpose, based on the marginal density functions of an underlying random process. For practical applications, a class of estimators based on the aggregation of multivariate density estimators is introduced and proved to be consistent. We illustrate the effectiveness of our estimators, as well as the practical usefulness of the proposed criterion, by applying our method to a dataset of real aircraft trajectories.

The hydro-offer problem consists of computing optimal bidding conditions to maximize the expected profit of a hydro producer participating in an electricity market. It combines the decision making process of the trader and the hydro dispatcher into a single stochastic optimization problem. It is a sequential decision making problem, and can be formulated as a multi-stage stochastic program.These models can be difficult to solve when the value function is not concave. In this thesis, we study some of the limitations of the hydro-bidding problem and propose a new stochastic optimization method called the Mixed-Integer Dynamic Approximation Scheme (MIDAS). MIDAS solves nonconvex stochastic programs with monotonic value functions. It works similarly to Stochastic Dual Dynamic Programming (SDDP), but instead of using hyperplanes, it uses step functions to create an outer approximation of the value function. MIDAS converges "almost surely" to (T+1)ε optimal solution when continuous state variables, and to the exact optimal solution when integer state variables.We use MIDAS to solve three types of hydro-bidding problems that are nonconvex. The first hydro-bidding model we solve for integer state variables because the outputs are discrete. In this model, we show that MIDAS constructions offer that are better than SDDP. The next hydro-bidding model uses autoregressive price processes instead of a Markov chain. The last hydro-bidding model incorporates headwater effects, where the power generation function depends on the reservoir storage level and the turbine water flow rate. In all these models, we demonstrate the convergence of MIDAS in finite iterations.The convergence time of MIDAS is higher than SDDP because subproblems is the mixed-integer programs (MIP). For hydraulic auction models with continuous state variables, its computation time depends on the value of the δ. If the δ is large, then it reduces the convergence computation time but it also increases the optimality error ε.In order to speed up MIDAS, we introduced two heuristics. The first heuristic is a step function selection heuristic, which is similar to the "cut selection" scheme in SDDP. This heuristic improves the solution time by up to 64%. The second heuristic iteratively solves MIP subproblems in MIDAS using smaller MIPs, rather than as a single large MIP. This heuristic improves the solution time by up to 60%. By applying both heuristics, we were able to use MIDAS to solve a hydro-bidding problem with 4 reservoirs, 4 stations and integer state variables.

Four new Maximum Likelihood based approaches for aircraft dynamics identification are presented and compared. The motivation is the need of accurate dynamic models for minimizing aircraft fuel consumption using optimal control techniques. A robust method for building aerodynamic models is also suggested. All these approaches were validated using real flight data from 25 different aircraft.

We analyze an algorithm for solving stochastic control problems, based on Pontryagin's maximum principle, due to Sakawa and Shindo in the deterministic case and extended to the stochastic setting by Mazliak. We assume that either the volatility is an affine function of the state, or the dynamics are linear. We obtain a monotone decrease of the cost functions as well as, in the convex case, the fact that the sequence of controls is minimizing, and converges to an optimal solution if it is bounded. In a specific case we interpret the algorithm as the gradient plus projection method and obtain a linear convergence rate to the solution.

This paper deals with numerical solutions to an optimal multiple stopping problem. The corresponding dynamic programing (DP) equation is a variational inequality satisfied by the value function in the viscosity sense. The convergence of the numerical scheme is shown by viscosity arguments. An optimal quantization method is used for computing the conditional expectations arising in the DP equation. Numerical results are presented for the price of swing option and the behavior of the value function.

Mixed Integer Dynamic Approximation Scheme (MIDAS) is a new sampling-based algorithm for solving finite-horizon stochastic dynamic programs with monotonic Bellman functions. MIDAS approximates these value functions using step functions, leading to stage problems that are mixed integer programs. We provide a general description of MIDAS, and prove its almost-sure convergence to an ε-optimal policy when the Bellman functions are known to be continuous, and the sampling process satisfies standard assumptions.

No summary available.

We present our contribution on the control and optimization of electricity production. The first part concerns the optimization of the management of a micro grid. We formulate the management program as an optimal control problem in continuous time, then we solve this problem by dynamic programming using a solver developed for this purpose: BocopHJB. We show that this type of formulation can be extended to a stochastic modeling. We end this part with the adaptive weights algorithm, which allows a management of the micro network battery integrating its aging. The algorithm exploits the two time scale structure of the control problem. The second part concerns networked market models, and in particular those of electricity. We introduce an incentive mechanism to decrease the market power of energy producers, to the benefit of the consumer. We study some mathematical properties of the optimization problems faced by market agents (producers and regulators). The last chapter studies the existence and uniqueness of Nash equilibria in pure strategies of a class of Bayesian games to which some network market models belong. For some simple cases, an equilibrium computation algorithm is proposed. An appendix gathers a documentation on the numerical solver BocopHJB.

The original Bocop package implements a local optimization method. The optimal control problem is approximated by a finite dimensional optimization problem (NLP) using a time discretization (the direct transcription approach). The NLP problem is solved by the well known software Ipopt, using sparse exact derivatives computed by Adol-C. The second package BocopHJB implements a global optimization method. Similarly to the Dynamic Programming approach, the optimal control problem is solved in two steps. First we solve the Hamilton-Jacobi-Bellman equation satisfied by the value fonction of the problem. Then we simulate the optimal trajectory from any chosen initial condition. The computational effort is essentially taken by the first step, whose result, the value fonction, can be stored for subsequent trajectory simulations.

In their paper, Carmona and Touzi [8] studied an optimal multiple stopping time problem in a market where the price process is continuous. In this article, we generalize their results when the price process is allowed to jump. Also, we generalize the problem associated to the valuation of swing options to the context of jump diffusion processes. We relate our problem to a sequence of ordinary stopping time problems. We characterize the value function of each ordinary stopping time problem as the unique viscosity solution of the associated Hamilton–Jacobi–Bellman variational inequality.

The 4th cover page states: "Optimization problems with combinatorial aspects, due to the presence of integer decision variables, are used in all sectors of economic life (investment, management of human resources or equipment, energy production planning) but also in technology (design of integrated circuits, optimization of telecommunication networks or on-line services). This book introduces the main principles of solving such problems, based on the theory of convex functions, duality in optimization, polyhedra and linear programming, methods of flow, dynamic programming, separation and evaluation, or integrity cuts. This overview includes two more advanced chapters, on applications in combinatorics of optimization under matrix positivity constraints (SDP optimization), and on interior point algorithms for convex quadratic programming. While relying on a rigorous mathematical analysis, this book presents numerous examples. In particular, a chapter of answers to a selection of exercises, as well as thirty or so problem statements with corrections, extend the course and provide illustrations from various fields of application.

In this report, given a reference feasible trajectory of an optimal control problem, we say that the quadratic growth property for bounded strong solutions holds if the cost function of the problem has a quadratic growth over the set of feasible trajectories with a bounded control and with a state variable sufficiently close to the reference state variable. Our sufficient second-order optimality conditions in Pontryagin form ensure this property and ensure a fortiori that the reference trajectory is a bounded strong solution. Our proof relies on a decomposition principle, which is a particular second-order expansion of the Lagrangian of the problem.

This thesis is divided into two parts. In the first part, we study deterministic optimal control problems with constraints and we focus on sensitivity analysis issues. The point of view we adopt is that of abstract optimization. Necessary and sufficient second order optimality conditions play a crucial role and are also studied as such. In this thesis, we are interested in strong solutions. In general, we use this generic term to refer to locally optimal L1-norm controls. By reinforcing the notion of local optimality used, we expect to obtain stronger results. Two tools are used in an essential way: a relaxation technique, which consists in using several controls simultaneously, and a decomposition principle, which is a particular second-order Taylor expansion of the Lagrangian. Chapters 2 and 3 deal with necessary and sufficient second-order optimality conditions for strong solutions of pure, mixed and final state constrained problems. In Chapter 4, we perform a sensitivity analysis for relaxed problems with constraints on the final state. In chapter 5, we perform a sensitivity analysis for a nuclear power generation problem. In the second part, we study stochastic optimal control problems with probability constraints. We study a dynamic programming approach, in which the probability level is seen as an additional state variable. In this framework, we show that the sensitivity of the value function with respect to the probability level is constant along the optimal trajectories. This analysis allows us to develop numerical methods for continuous time problems. These results are presented in Chapter 6, in which we also study an application to asset-liability management.

This paper develops a theory of singular arc, and the corresponding second order necessary and sufficient conditions, for the optimal control of a semilinear parabolic equation with scalar control applied on the r.h.s. We obtain in particular an extension of Kelley's condition, and the characterization of a quadratic growth property for a weak norm.

In this article, we compute a second-order expansion of the value function of a family of relaxed optimal control problems with final-state constraints, parameterized by a perturbation variable. The sensitivity analysis is performed for controls that we call R-strong solutions. They are optimal solutions with respect to the set of feasible controls with a uniform norm smaller than a given R and having an associated trajectory in a small neighborhood for the uniform norm. In this framework, relaxation enables us to consider a wide class of perturbations and therefore to derive sharp estimates of the value function.

We give an overview of the shooting technique for solving deterministic optimal control problems. This approach allows to reduce locally these problems to a finite dimensional equation. We first recall the basic idea, in the case of unconstrained or control constrained problems, and show the link with second-order optimality conditions and the analysis or discretization errors. Then we focus on two cases that are now better undestood: state constrained problems, and affine control systems. We end by discussing extensions to the optimal control of a parabolic equation.

The thesis deals with optimal control problems where the dynamics are given by differential equations with memory. For these optimization problems, optimality conditions are established. Second order optimality conditions constitute an important part of the results of the thesis. In the case - without memory - of ordinary differential equations, the standard optimality conditions are strengthened by involving only the Lagrange multipliers for which the Pontryaguine principle is satisfied. This restriction to a subset of the multipliers represents a challenge in establishing the necessary conditions and allows the sufficient conditions to ensure local optimality in a stronger sense. The standard conditions are further extended to the case - with memory - of integral equations. The pure constraints on the state of the previous problem have been preserved and require a specific study of the integral dynamics. Another form of memory in the equation of state of an optimal control problem comes from a modeling work with therapeutic optimization as a medical application in mind. The population dynamics of cancer cells under the action of a treatment is reduced to differential equations with time delays. The asymptotic behavior in long time of the age-structured model is also studied.

In this thesis we focus on optimal control problems for affine systems in one part of the control. First, we give a second order necessary condition for the case where the system is affine in all controls. We have bounds on the controls and a bang-singular solution. A sufficient condition is given for the case of a scalar control. We then propose a shooting algorithm and a sufficient condition for its local quadratic convergence. This condition guarantees the stability of the optimal solution and implies that the algorithm converges locally quadratically for the perturbed problem, in some cases. We present numerical tests that validate our method. Then, we study an affine system in a part of the orders. We obtain necessary and sufficient conditions of the second order. Then, we propose a shooting algorithm and we show that the mentioned sufficient condition guarantees that this algorithm converges locally quadratically. Finally, we study a planning problem for a hydro-thermal power plant. We analyze, by means of the necessary conditions obtained by Goh, the possible appearance of singular arcs.

Abstract: This thesis is divided into two parts. In the first part, we focus on deterministic optimal control problems and study interior approximations for two model problems with non-negativity constraints on the control. The first model is an optimal control problem with a quadratic cost function and dynamics governed by an ordinary differential equation. For a general class of interior penalty functions, we show how to compute the principal term of the pointwise state and adjoint state expansion. Our main argument is based on the following fact: if the optimal control for the initial problem satisfies the strict complementarity conditions for the Hamiltonian except at a finite number of times, the estimates for the penalized optimal control problem can be obtained from the estimates for an associated stationary problem. Our results provide several types of approximation quality measures for the penalization technique: error estimates for the control, error estimates for the state and adjoint state and also error estimates for the value function. The second model is the optimal control problem of a semi-linear elliptic equation with homogeneous Dirichlet conditions at the edge, the control being distributed on the domain and positive. The approach is the same as for the first model, i.e. we consider a family of penalized problems, whose solution defines a central trajectory that converges to the solution of the initial problem. In this way, we can extend the results, obtained in the framework of differential equations, to the optimal control of semi-linear elliptic equations. In the second part we focus on stochastic optimal control problems. First, we consider a linear quadratic stochastic problem with non-negativity constraints on the control and we extend the error estimates for the logarithmic penalty approximation. The proof relies on the stochastic Pontriaguine principle and a duality argument. Next, we consider a general stochastic control problem with convex constraints on the control. The so-called variational approach allows us to obtain a first and second order development for the state and the cost function, around a local minimum. With these developments we can show general first order optimality conditions and, under a geometric assumption on the set of constraints, second order necessary conditions are also established.

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No summary available.

We study the convergence of an anti-dissipative scheme, the UltraBee, for Hamilton Jacobi Bellman equations in dimension 1. Two solution methods using this scheme are proposed. The first one combines the UltraBee with a grid adaptation, the second one uses a hollow storage. The latter is applied to the problem of atmospheric reentry. Finally, some theoretical extensions are given.

This thesis is concerned with the optimal (deterministic) control problem of an ordinary differential equation subject to one or more constraints on the state, of any order, in the case where the strong Legendre-Clebsch condition is satisfied. The Pontryaguine minimum principle provides a well-known necessary optimality condition. In this thesis, we first obtain a second order sufficient optimality condition that is as close as possible to the second order necessary condition and characterizes the quadratic growth. This condition allows us to obtain a characterization of the well-posedness of the shooting algorithm in the presence of constraints on the state. Then we perform a stability and sensitivity analysis of the solutions when the problem data is perturbed. For constraints of order greater than or equal to two, we obtain for the first time a stability result for the solutions making no assumption on the structure of the trajectory. Moreover, results on the structural stability of Pontryaguine extremals are given. Finally, these results on the one hand on the shooting algorithm and on the other hand on the stability analysis allow us to propose, for constraints on the state of order one and two, a homotopy algorithm whose novelty is to automatically determine the structure of the trajectory and to initialize the associated shooting parameters.

This thesis is concerned with the solution of large non-differentiable optimization problems, most often resulting from a Lagrangian relaxation of a difficult problem. This technique is commonly used to solve linear integer problems or complex convex problems. The obtained dual problem is non-differentiable -possibly separable- and can be solved by a bundle algorithm. Chapter 2 proposes a literature review of non-differentiable optimization methods. In some situations, the dual problem can itself be very difficult to solve and require adapted strategies. For example, when the number of dualized constraints is very high, an explicit dualization may be impossible or the updating of dual variables may fail. In chapter 3, we study the convergence properties when a dynamic Lagrangian relaxation is performed: only a subset of constraints is dualized at each iteration, which allows to reduce the dimension of the dual problem. Another limit of Lagrangian relaxation can appear when the dual function is separable into a large number of sub-functions, or when these sub-functions remain difficult to evaluate. A natural strategy is then to take advantage of the separable reading by performing dual iterations having evaluated only a subset of the subfunctions. In Chapter 4, we propose to use a beam method in this incremental context. Finally, Chapter 5 presents numerical applications on power generation management problems.

In this thesis, we study the problem of physical risk management in power generation for the weekly horizon. First, we focus on the integration of the hydraulic supply hazard in the local management of a hydraulic valley. This approach is conducted using robust optimization and linear decision rules. The results of multiple simulation modes show that these approaches allow a significant reduction of spillages compared to deterministic models applied in operation, with a small increase in cost. The second issue addressed is the active management of the generation margin, defined as the difference between total supply and total demand, taking into account the contingencies affecting the power system. The aim is to determine the optimal decisions to be taken, according to a certain economic criterion, in order to hedge against a too high risk of not satisfying the demand in at least 99% of the situations. For this purpose, a novel open-loop formulation, based on the stochastic process of production margin and constraints in probability is proposed. For the purpose of this formulation, we generate scenarios using more realistic techniques than in operation. Finally, a less anticipatory resolution is studied by applying the heuristic "Stochastic Programming with Piecewise Constant Decision Rules" introduced by Thénié and Vial. The first results are very encouraging in comparison with the open loop models.

The purpose of this thesis is to study the numerical approximations of different HJB equations associated with stochastic optimal control problems. In the first part, the theory of error estimation has been extended to a differential game problem and to the case of the problem with impulses. The latter has been numerically implemented. In all this part the control set is bounded. Then, in the second part of the thesis, we studied stochastic control problems coming from finance and whose set of controls is not bounded, in particular problems of over-coverage.

No summary available.

No summary available.

The first part of this thesis deals with a robustness study of corrective predictor interior point algorithms, as well as a decomposition approach of this method for solving multiflot problems. In the second part, we focus on the global security problem whose objective is to determine a multiflot (which transports any request from its origin node to its destination node respecting Kirchhoff's law) and the least cost investment in nominal and reserve capacity that ensures nominal routing and guarantees its survival by global rerouting. In our model, the routings and capacities are fractionable. PSG is then formulated as a large linear problem with several levels of coupling. Its particular structure calls for the use of decomposition algorithms. We propose four methods using the column generation technique. The first two are based on proximal techniques. Their main task is to solve independent quadratic subproblems. The third algorithm is inspired by the interior point approach described in the first part. Finally, we integrate a path elimination procedure into an adaptation of an interior point solver. We report numerical results obtained by testing these algorithms on real data provided by the CNET.

New interior point methods are playing an increasingly important role in the optimization of large systems. In this thesis we study in a first part, from a theoretical and numerical point of view, an extension of an interior point algorithm for convex and non convex quadratic programming. This extension uses the idea of the confidence region which can be made explicit through an affine transformation. Under certain assumptions we prove results on the global convergence and on the convergence speed of the algorithm. We also give a practical version of this algorithm, based on a generalization of Lanczos' method for solving indefinite linear systems. This one gives very encouraging results in practice. In the second part, we study from a theoretical point of view an extension of another interior point algorithm for nonlinear optimization with linear constraints. This extension uses the idea of reducing a potential function after an affine transformation of the admissible set. Results on the global convergence and on the complexity of the algorithm are given.