Approximating stationary points of stochastic optimization problems in Banach space |
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Authors: | Ramamurthy Balaji Huifu Xu |
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Institution: | a Department of Mathematics and Statistics, University of Hyderabad, Hyderabad 46, India b School of Mathematics, University of Southampton, Highfield Southampton, UK |
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Abstract: | In this paper, we present a uniform strong law of large numbers for random set-valued mappings in separable Banach space and apply it to analyze the sample average approximation of Clarke stationary points of a nonsmooth one stage stochastic minimization problem in separable Banach space. Moreover, under Hausdorff continuity, we show that with probability approaching one exponentially fast with the increase of sample size, the sample average of a convex compact set-valued mapping converges to its expected value uniformly. The result is used to establish exponential convergence of stationary sequence under some metric regularity conditions. |
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Keywords: | Sample average approximation Stationary point Law of large numbers Exponential convergence Metric regularity |
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