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Journal of Convex Analysis 30 (2023), No. 3, 917--936
Copyright Heldermann Verlag 2023

Epi-Convergence of Expectation Functions under Varying Measures and Integrands

Eugene A. Feinberg
Dept. of Applied Mathematics and Statistics, Stony Brook University, New York, U.S.A.

Pavlo O. Kasyanov
Institute for Applied System Analysis, National Technical University, Kyiv, Ukraine

Johannes O. Royset
Operations Research Department, Naval Postgraduate School, Monterey, U.S.A.

For expectation functions on metric spaces, we provide sufficient conditions for epi-convergence under varying probability measures and integrands, and examine applications in the area of sieve estimators, mollifier smoothing, PDE-constrained optimization, and stochastic optimization with expectation constraints. As a stepping stone to epi-convergence of independent interest, we develop parametric Fatou's lemmas under mild integrability assumptions. In the setting of Suslin metric spaces, the assumptions are expressed in terms of Pasch-Hausdorff envelopes. For general metric spaces, the assumptions shift to semicontinuity of integrands also on the sample space, which then is assumed to be a metric space.

Keywords: Epi-convergence, expectation function, stochastic optimization, sieve estimators, mollifers.

MSC: 90C15, 60F17.

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