Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

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.

joroyset@nps.edu

Keywords

  • Epi-convergence
  • expectation function
  • stochastic optimization
  • sieve estimators
  • mollifers

Mathematics Subject Classification

90C15, 60F17

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Suggested citation

E. A. Feinberg, P. O. Kasyanov, J. O. Royset. “Epi-Convergence of Expectation Functions under Varying Measures and Integrands.” Journal of Convex Analysis 30 (2023), No. 3, 917–936. https://doi.org/10.68381/jca30043

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