Mathematical Sciences: Approximation in Stochastic Programming and Other Variational Problems

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9625787 Wets The unifying theme of this project is the convergence, in particular the epi- convergence, of integral functions as they arise in stochastic optimization, but also in other variational problems: optimal control problems, the calculus of variations, mathematical statistics, etc.. The main reason for concentrating on approximation questions is that although much is known about the properties of integral functionals and much progress has been made in the design of efficient and reliable optimization routines for finite dimensional linear and nonlinear optimization problems, the use of these routines to solve infinite dimensional problems has been hampered by serious shortcomings in what we know about the approximation of infinite dimensional optimization problems, in particular by finite optimization problems. Stochastic optimization models provide tools that support decision making under uncertainty. These are ``decision'' problems whose data is only known in a statistical sense, i.e., there is uncertainty about the values to assign to some of the parameters of the problem, and at best one might have some statistical information about these parameters. Typical stochastic optimization problems are: the choice of a flexible manufacturing plan when the demand is not known with certainty, asset/liability management problems, policy setting for pollution controls that are affected by atmospheric conditions, the design of structures (such as bridges, skyscrapers) that are going to be subjected to (randomly distributed) seismic shocks, etc.. Mathematically, stochastic optimization problems are very difficult to solve because one must, in one way or another, take into account all possible outcomes of these random parameters. That is why the design of an approximation theory for such problems is essential, and that is what is at the core of this project.