Mathematical Sciences: Approximation in Stochastic Programming and Other Variational Problems

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9300930 Wets The unifying theme of this research is the epi-convergence of integral functionals as they arise in variational problems, in particular stochastic optimization, i.e., problems involving random integral functionals. The P.I. has identified the following questions that will be investigated: the preservation of epi-convergence under addition and epi-addition, the epi- convergence of integral functions with discontinuous integrands, the development of a quantitative theory for the convergence of integral functionals. In the case of random integral functionals, the P.I. will develop epi-consistency results for random lower semicontinuous functions with and without the epi- iid assumption, design implementable procedures to estimate the reliability level of approximate solutions and exploit the theory of large deviations to obtain convergence rates. %%% Optimization problems (such as the allocation of scare resources, optimal engineering design, planning of logistic support operations, etc.) cannot be solved without resorting to approximations. To validate an approximation scheme one should demonstrate that a refinement of the approximation yields a better solution (convergence questions) and, whenever possible, allows for the calculation of error bounds so that one can have an estimate of the error that may occur (convergence rate). The P.I. plans to investigate such questions, in particular in connection with so-called stochastic optimization problems (that model decision making under uncertainty) a class of problems that is very difficult to solve, but are very important as far as applications are concerned. ***