Geodesic
Epi-Convergence of Expectation Functions under Varying Measures and Integrands
Journal of convex analysis, Tome 30 (2023) no. 3, pp. 917-936
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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.
Classification : 90C15, 60F17
Mots-clés : Epi-convergence, expectation function, stochastic optimization, sieve estimators, mollifers 
@article{JCA_2023_30_3_JCA_2023_30_3_a7,
     author = {E. A. Feinberg and P. O. Kasyanov and J. O. Royset},
     title = {Epi-Convergence of {Expectation} {Functions} under {Varying} {Measures} and {Integrands}},
     journal = {Journal of convex analysis},
     pages = {917--936},
     year = {2023},
     volume = {30},
     number = {3},
     url = {http://geodesic.mathdoc.fr/item/JCA_2023_30_3_JCA_2023_30_3_a7/}
}
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E. A. Feinberg; P. O. Kasyanov; J. O. Royset. Epi-Convergence of Expectation Functions under Varying Measures and Integrands. Journal of convex analysis, Tome 30 (2023) no. 3, pp. 917-936. http://geodesic.mathdoc.fr/item/JCA_2023_30_3_JCA_2023_30_3_a7/