Opportunisme et traitement des contraintes dans MADS

RÉSUMÉ: Dans le domaine de l’optimisation de boites noires, l’utilisateur n’a pas d’expressions analytiques de la fonction objectif et des contraintes. De fait, il n’a pas accès aux gradients. Le gradient est une information importante en optimisation étant donné qu’il permet de fournir une direction de montée de la fonction. De plus, pour récupérer les différentes valeurs de la fonction objectif et des contraintes, des simulations informatiques ou des tests en laboratoires doivent être effectués. Ceci rajoute de nombreuses difficultés supplémentaires : temps de calculs importants pour récupérer les données, bruitage des données et certaines simulations peuvent échouer. Pour résoudre ce genre de problèmes, des algorithmes ont été développés. Parmi eux, MADS a été proposé par Audet et Dennis en 2006. C’est un algorithme itératif de recherche directe qui évalue des points de proche en proche sur un treillis. Il offrait à la base un traitement rudimentaire des contraintes, en associant une valeur infinie à tous points non -réalisables. Il a depuis été étoffé pour offrir un traitement plus souple à des contraintes de plus en plus hétéroclites. Cette thèse propose trois nouvelles fonctionnalités à l’algorithme MADS. Premièrement, MADS calcule des modèles des contraintes afin d’ordonner les points du plus prometteur au moins prometteur. Cependant, un traitement adéquat des contraintes binaires, qui ne retournent que deux valeurs, manque dans MADS. Pour pallier cette absence, des modèles des contraintes binaires seront proposés en utilisant des outils de régression, issus de la classification supervisée. Deuxièmement, ces mêmes outils permettent de proposer un ordonnancement nouveau des points à évaluer quand aucune fonction substitut n’est accessible dans MADS. Les points qui ont le plus de chance d’être réalisables seront évalués en premier pour favoriser la recherche de solutions réalisables de qualité. Cette stratégie sera comparée à une méthode favorisant les points les plus éloignés des points déjà évalués et à la méthode par défaut dans ce cas dans MADS, qui favorise les points qui sont le plus dans la direction du dernier succès par rapport au centre du treillis. Enfin, il peut être noté que la mise à l’échelle des contraintes choisie par l’utilisateur au moment de définir le problème a un impact sur le fonctionnement de MADS. MADS propose un traitement de mise à l’échelle des variables en entrée de la boite noire, mais rien pour les contraintes en sortie. Cette thèse propose une façon de les mettre à l’échelle, de sorte qu’elles prennent des valeurs de même ordre de grandeur. Cela permettra qu’elles aient globalement la même importance.----------ABSTRACT: In the field of blackbox optimization, the user does not have access to the analytical expressions of the objective function and of the constraints. Thus, there is no access to the gradient. But the gradient is an important piece of information since it gives an increasing direction of the function. Moreover, in order to obtain those values, computer simulations or experiments in laboratory have to be done. This adds further difficulties: heavy computational times to get the data, noisy data and some simulations may fail. To solve this kind of problems, algorithms have been developed. Among them, MADS has been proposed by Audet and Dennis in 2006. It is a direct search iterative algorithm that evaluates points on a mesh. At first, it offered a basic management of the constraints by associating an infinite value to all infeasible elements. Since then, more flexible ways have been proposed to handle various types of constraints. There are for example models for most of the constraints in order to sort points from the most to the least promising. However, in MADS, there is no specific management of binary constraints, which can return only two different values. Thus, models of binary constraints will be offered using tools of regression from supervised classification. Those tools also offer new ordering methods to sort the points that need to be evaluated when no models are available in MADS. The points which are the most likely to be feasible will be evaluated first in order to look most likely for feasible solutions. This strategy will be compared to one evaluating first the elements the furthest from the ones already evaluated and to the default, in that situation, in MADS which sorts the points according to the direction of last success. Finally, it should be pointed out that the scaling of the constraints provided by the user chosen while defining the problem has an impact on MADS’s behaviour. MADS deals with the scaling of the input variables of the blackbox, but nothing is done for the constraints in the output. This thesis offers to handle the scaling of the output so that they take values of about the same range so that they have more or less the same influence

Ergodic control of a heterogeneous population and application to electricity pricing

We consider a control problem for a heterogeneous population composed of customers able to switch at any time between different contracts, depending not only on the tariff conditions but also on the characteristics of each individual. A provider aims to maximize an average gain per time unit, supposing that the population is of infinite size. This leads to an ergodic control problem for a "mean-field" MDP in which the state space is a product of simplices, and the population evolves according to a controlled linear dynamics. By exploiting contraction properties of the dynamics in Hilbert's projective metric, we show that the ergodic eigenproblem admits a solution. This allows us to obtain optimal strategies, and to quantify the gap between steady-state strategies and optimal ones. We illustrate this approach on examples from electricity pricing, and show in particular that the optimal policies may be cyclic-alternating between discount and profit taking stages

Quadratic Regularization of Unit-Demand Envy-Free Pricing Problems and Application to Electricity Markets

We consider a profit-maximizing model for pricing contracts as an extension of the unit-demand envy-free pricing problem: customers aim to choose a contract maximizing their utility based on a reservation bill and multiple price coefficients (attributes). A classical approach supposes that the customers have deterministic utilities; then, the response of each customer is highly sensitive to price since it concentrates on the best offer. A second approach is to consider logit model to add a probabilistic behavior in the customers' choices. To circumvent the intrinsic instability of the former and the resolution difficulties of the latter, we introduce a quadratically regularized model of customer's response, which leads to a quadratic program under complementarity constraints (QPCC). This allows to robustify the deterministic model, while keeping a strong geometrical structure. In particular, we show that the customer's response is governed by a polyhedral complex, in which every polyhedral cell determines a set of contracts which is effectively chosen. Moreover, the deterministic model is recovered as a limit case of the regularized one. We exploit these geometrical properties to develop an efficient pivoting heuristic, which we compare with implicit or non-linear methods from bilevel programming. These results are illustrated by an application to the optimal pricing of electricity contracts on the French market.Comment: 37 pages, 9 figures; adding a section on the pricing of electricity contract

Electron energy loss spectroscopy determination of Ti oxidation state at the (001) LaAlO3/SrTiO3 interface as a function of LaAlO3 growth conditions

At the (001) interface between the two band-insulators LaAlO3 and SrTiO3, a high-mobility electron gas may appear, which has been the object of numerous works over the last four years. Its origin is a subject of debate between the interface polarity and unintended doping. Here we use electron energy loss 'spectrum images', recorded in cross-section in a scanning transmission electron microscope, to analyse the Ti3+ ratio, characteristic of extra electrons. We find an interface concentration of Ti3+ that depends on growth conditions.Comment: 6 page

Economic Analysis of Summer Fallow Management to Reduce Take-All and N-Leaching in a Wheat Crop Rotation

This paper addresses the question of summer cover crop adoption by farmers in presence of a risk of yield loss due to take-all disease and climate variability. To analyse the public incentives needed to encourage farmers to adopt summer cover crops as a means of reducing N leaching, we combine outputs from an economic, an epidemiological and an agronomic model. The economic model is a simple model of choice under uncertainty. The farmer is assumed to choose among a range of summer fallow managements and input uses on the basis of the expected utility criterion (HARA assumption) in presence of both climate and take all risks. The epidemiological model proposed by Ennaïfar et al. (2007) is used to determine the impact of take all on yields and N-uptake. The crop-soil model (STICS) is used to\ud compute yield developments and N-leaching under various management options and climatic conditions. These models are calibrated to match the conditions prevailing in Grignon, located in the main wheatgrowing\ud area in France. Eight management systems are examined: 4 summer fallow managements: 'wheat volunteers' (WV), 'bare soil' (BS), 'early mustard' (EM), 'late mustard' (LM), and 2 input intensities. We show that the optimal systems are BS (WV) when the take-all risk is (not) taken into account by agents. We then compute the minimum payment to each system such that it emerges in the optimum. We thus derive the required amounts of transfer needed to trigger catch crop adoption. The results of the Monte Carlo sensitivity analysis show that the ranking of management systems is robust over a wide range of input parameters

Hemisynthesis and Structural and Chromatic Characterization of Delphinidin 3-O-Glucoside–Vescalagin Hybrid Pigments

[EN] During red wine maturation in the presence of oak wood, reactions involving anthocyanins and ellagitannins might affect wine organoleptic properties such as color and astringency. In this work, the condensation reaction between myrtillin (delphinidin 3-O-glucoside) and vescalagin has been performed to determine the behavior of this anthocyanin in this kind of reaction and to assess the possible impact of such a reaction in wine color modulation. Two different hybrid pigments have been hemisynthetized and characterized by HPLC-DAD-MS and NMR spectroscopy. These pigments have been identified as 1-deoxyvescalagin-(1β→8)-myrtillin (major) and 1-deoxyvescalagin-(1β→6)-myrtillin (minor). The minor pigment could be formed both by the condensation reaction and by a regioisomerization process from the major pigment. Moreover, the chromatic properties of these pigments have been studied and compared to those of myrtillin. The hybrid pigments showed an important bathochromic shift (ca. 20 nm) in the maximum absorbance wavelength and lower molar absorption coefficients