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Accueil du site > Séminaires > Probabilités Statistiques et réseaux de neurones > Statistical solutions to Stochastic Programs

Vendredi 26 janvier 2007 à 11h00

Statistical solutions to Stochastic Programs

Carlos Bouza (Université La Havane, Cuba),

Résumé : Stochastic optimization is of use in many applications when some of the coefficients are unknown and of random nature. The touchstone example is given by a decision-maker that must make decisions under un-certainty. Generally he/she should model the problem as an optimization one. Uncertainty may modeled by the randomness of the studied phenomena. A probability space , , is determined and the problem to be solved is Min g(x)=EP [g(x,w]x, w. It may be quite difficult to write down the theoretical model. A more complicated task is to compute the solution because the calculation of a numerical solution generally relies on rather complicated theoretical models. Even the solution of simple linear programming problems needs of the computation of the optimal solutions of non-linear models. The contemporary existence of large computing capacities has permitted to develop software packages for solving a large collection of stochastic programs. We consider the approach of Morita, H., H. Ishii and T. Nishida (1989). [ Stochastic Programming with estimated objective. Technologic Reports of the Osaka University, Vol. 39, 1-7] . We consider the effect of using an approximation to the upper bound of the Minimax optimal solution in Stochastic Linear Programming following Bouza, C.N. (1992) [ Bounding the expected approximation error in Stochastic Linear Programming with Completely Fixed Recourses In : System Modeling and Optimization. Lecture Notes on Control’. Springer, Berlin] It can be computed accepting the normality or by using a re-sampling method. Simulated Annealing is introduced for looking for the second stage solution. Least Absolute Deviation is proposed as an alternative to Least Squares for computing the initial solution. When the realizations of the stochastic phenomena are described through random variables, which follows a certain distribution function (DF) F. The existence of changes in the nature of the phenomena or the influence of some external shocks can introduce changes in the stochastic description of the problem. It may be modeled by considering that another DF H contaminates the underlying distribution F. The use of an F based model may conduct to incorrect decisions. Tis is the basic problem stated in. We may consider the case in which the random variables are generated by a distribution F=F+(1-) H, where measures the contamination level and it belongs to the interval [0,1]. It is of inters to know its value. We propose a solution in which the relation permits to identify for each observation. An estimate of the function determined by the above quoted relation for each observation permits consider each of them as a random variable with expectation equal to . The interest in estimating is illustrated by analyzing the Two-stage Stochastic optimization problem. We use for illustrating the problem quoted in Bouza, C.N. (2001) : Investigation of Burn-in-time problems with unknown failure time distribution. J. of Statistics and Management Sc.37, 1-7. The proposals are compared through Monte Carlo Simulation experiments.

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