Stochastic Variational Problems: Optimization and Equilibrium

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0205699WetsThis research proposal is centered around approximation issues in stochastic programming, in particular as they arise in two quite challenging problems: equilibria problems in a stochastic environment and recourse problems involving partial differential equations. The "stochastic" equilibrium problem adds a new level of difficulty; rather than just optimizing one must find a mechanism to determine a price system under which the optimization takes place. Such equilibria have been derived by relying on fixed point theorems. Thirdly, because it is only possible to solve discretized versions of stochastic optimization problems, it is of paramount importance to investigate thoroughly approximation issues. Not only the question of approximating the stochastic process that describes the uncertainty but also how to improve the construction of this process from the available data and to analyze the effect this will have on the solution of the stochastic program.The field of stochastic programming provides mathematical tools for solving and analyzing models for decision making under uncertainty. This project will be concerned with two significant and difficult applications and with approximation issues: -- A problem in groundwater remediation which was selected because it requires both theoretical and computational developments. It is a stochastic optimization problem where the state of the system is obtained by solving a partial differential equation whose coefficients are rapidly oscillating (heterogeneous media) and stochastic (uncertainty about the media composition). The possibility of deriving a homogenized version of this problem will also be investigated. -- Walras equilibrium problem in an uncertain environment. This problem is selected because it adds a dimension to stochastic optimization in that one must also find price systems (setting up an "equilibrium") under which this stochastic optimization must take place. -- Approximation issues in stochastic programming. The question of having a reliable estimate for the parameters of a dynamic stochastic programming problem is raised. It is expected that a more comprehensive, approach which makes use of all the information available, rather than just the collected data will result in more reliable solutions for stochastic programming problems.