# A Regularized Sample Average Approximation Method for Stochastic Mathematical Programs with Nonsmooth Equality Constraints

@article{Meng2006ARS, title={A Regularized Sample Average Approximation Method for Stochastic Mathematical Programs with Nonsmooth Equality Constraints}, author={Fanwen Meng and Huifu Xu}, journal={SIAM J. Optim.}, year={2006}, volume={17}, pages={891-919} }

We investigate a class of two stage stochastic programs where the second stage problem is subject to nonsmooth equality constraints parameterized by the first stage variant and a random vector. We consider the case when the parametric equality constraints have more than one solution. A regularization method is proposed to deal with the multiple solution problem, and a sample average approximation method is proposed to solve the regularized problem. We then investigate the convergence of…

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## 29 Citations

### Convergence Analysis of Sample Average Approximation Methods for a Class of Stochastic Mathematical Programs with Equality Constraints

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- 2007

A uniform Strong Law of Large Numbers for random compact set-valued mappings is derived and used to investigate the convergence of Karush-Kuhn-Tucker points of SAA programs as the sample size increases.

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A detailed stability analysis is carried out of the approximated problems, including continuity and local Lipschitz continuity of optimal value functions and outer semicontinuity and continuity of the set of optimal solutions and stationary points.

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It is shown under some moderate conditions that the statistical estimators obtained from solving the penalized SAA problems converge almost surely to its true counterpart as the sample size increases.

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Sample average approximation method is one of the well-behaved methods in the stochastic optimization.
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The authors formulate the problems as constrained optimization problems and then propose a sample average approximation method for solving the problems and investigate the limiting behavior of the optimal values and the optimal solutions of the approximation problems.

### A smooth penalty-based sample average approximation method for stochastic complementarity problems

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Under a set of mild assumptions, it is proven that the sequences of solution and multiplier obtained by the proposed algorithm converge to the Kuhn-Tucker pair of the original problem with probability one as the sample size increases.

### Smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications

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A smoothing scheme for a general class of nonsmooth stochastic problems is considered and the convergence of stationary points of the smoothed sample average approximation problem as sample size increases is investigated and an error bound on approximate stationary points is obtained.

### Quantitative stability of two-stage stochastic linear variational inequality problems with fixed recourse

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ABSTRACT This paper focus on the quantitative stability of a class of two-stage stochastic linear variational inequality problems whose second stage problems are stochastic linear complementarity…

### Convergence of Stationary Points of Sample Average Two-Stage Stochastic Programs: A Generalized Equation Approach

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It is shown under moderate conditions that an accumulation point of the SAA stationary points satisfies a relaxed stationary condition for the true problem and further that, with probability approaching one exponentially fast with increasing sample size, a stationary point of SAA converges to the set of relaxed stationary points.

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