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Individual confidence intervals for solutions to expected value formulations of stochastic variational inequalities

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Abstract

Stochastic variational inequalities (SVIs) provide a means for modeling various optimization and equilibrium problems where data are subject to uncertainty. Often the SVI cannot be solved directly and requires a numerical approximation. This paper considers the use of a sample average approximation and proposes three methods for computing confidence intervals for components of the true solution. The first two methods use an “indirect approach” that requires initially computing asymptotically exact confidence intervals for the solution to the normal map formulation of the SVI. The third method directly constructs confidence intervals for the true SVI solution; intervals produced with this method meet a minimum specified level of confidence in the same situations for which the first two methods are applicable. We justify the three methods theoretically with weak convergence results, discuss how to implement these methods, and test their performance using three numerical examples.

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Acknowledgments

Research of Michael Lamm and Shu Lu is supported by National Science Foundation under the Grants DMS-1109099 and DMS-1407241. Research of Amarjit Budhiraja is supported in part by the National Science Foundation (DMS-1004418, DMS-1016441, DMS-1305120) and the Army Research Office (W911NF-10-1-0158). We thank the two anonymous referees for comments and suggestions that have helped to greatly improve the presentation of this paper.

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Authors and Affiliations

  1. Department of Statistics and Operations Research, University of North Carolina at Chapel Hill, B05 Hanes Hall, CB#3260, Chapel Hill, NC, 27599-3260, USA

    Michael Lamm

  2. Department of Statistics and Operations Research, University of North Carolina at Chapel Hill, 355 Hanes Hall, CB#3260, Chapel Hill, NC, 27599-3260, USA

    Shu Lu

  3. Department of Statistics and Operations Research, University of North Carolina at Chapel Hill, 357 Hanes Hall, CB#3260, Chapel Hill, NC, 27599-3260, USA

    Amarjit Budhiraja

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  1. Michael Lamm
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Correspondence to Michael Lamm.

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Lamm, M., Lu, S. & Budhiraja, A. Individual confidence intervals for solutions to expected value formulations of stochastic variational inequalities. Math. Program. 165, 151–196 (2017). https://doi.org/10.1007/s10107-016-1046-y

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