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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References

SHOWING 1-10 OF 33 REFERENCES

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

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.

An Implicit Programming Approach for a Class of Stochastic Mathematical Programs with Complementarity Constraints

This paper investigates the existence, uniqueness, and differentiability of the lower level equilibrium defined by the complementarity constraints, and its dependence using a nonsmooth version of implicit function theorem, and studies the differentiability and convexity of the objective function which implicitly depends upon the lowerlevel equilibrium.

SMOOTHING IMPLICIT PROGRAMMING APPROACHES FOR STOCHASTIC MATHEMATICAL PROGRAMS WITH LINEAR COMPLEMENTARITY CONSTRAINTS

For the lower-level wait-and-see model, a smoothing implicit programming method is proposed and a comprehensive convergence theory is established and it is shown that the two methods possess similar convergence properties.

Convergence theory for nonconvex stochastic programming with an application to mixed logit

This work allows for local SAA minimizers of possibly nonconvex problems and proves, under suitable conditions, almost sure convergence of local second- order solutions of the SAA problem to second-order critical points of the true problem.

Simulation-Based Solution of Stochastic Mathematical Programs with Complementarity Constraints: Sample-Path Analysis

We consider a class of stochastic mathematical programs with complementarity constraints, in which both the objective and the constraints involve limit functions or expectations that need to be

A Regularized Smoothing Newton Method for Box Constrained Variational Inequality Problems with P0-Functions

  • H. Qi
  • Mathematics
    SIAM J. Optim.
  • 2000
Under CD-regularity, this work proves that the proposed regularized smoothing Newton method for the box constrained variational inequality problem with P0-function has a superlinear (quadratic) convergence rate without requiring strict complementarity conditions.

Stochastic convex programming: Kuhn-Tucker conditions

Refinements of necessary optimality conditions in nondifferentiable programming I

In this study, we develop general optimality conditions of both Fritz John and Kuhn-Tucker type for an optimization problem with nondifferentiable data. The already known conditions are sharpened by

On the Rate of Convergence of Optimal Solutions of Monte Carlo Approximations of Stochastic Programs

It is shown that if the corresponding random functions are convex piecewise linear and the distribution is discrete, then an optimal solution of the approximating problem provides an exact optimal solution to the true problem with probability one for sufficiently large sample size.

Stochastic mathematical programs with equilibrium constraints