Regularization method for stochastic mathematical programs with complementarity constraints

Gui-Hua Lin; Masao Fukushima

ESAIM: Control, Optimisation and Calculus of Variations (2005)

  • Volume: 11, Issue: 2, page 252-265
  • ISSN: 1292-8119

Abstract

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In this paper, we consider a class of stochastic mathematical programs with equilibrium constraints (SMPECs) that has been discussed by Lin and Fukushima (2003). Based on a reformulation given therein, we propose a regularization method for solving the problems. We show that, under a weak condition, an accumulation point of the generated sequence is a feasible point of the original problem. We also show that such an accumulation point is S-stationary to the problem under additional assumptions.

How to cite

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Lin, Gui-Hua, and Fukushima, Masao. "Regularization method for stochastic mathematical programs with complementarity constraints." ESAIM: Control, Optimisation and Calculus of Variations 11.2 (2005): 252-265. <http://eudml.org/doc/245251>.

@article{Lin2005,
abstract = {In this paper, we consider a class of stochastic mathematical programs with equilibrium constraints (SMPECs) that has been discussed by Lin and Fukushima (2003). Based on a reformulation given therein, we propose a regularization method for solving the problems. We show that, under a weak condition, an accumulation point of the generated sequence is a feasible point of the original problem. We also show that such an accumulation point is S-stationary to the problem under additional assumptions.},
author = {Lin, Gui-Hua, Fukushima, Masao},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {stochastic mathematical program with equilibrium constraints; S-stationarity; Mangasarian-Fromovitz constraint qualification; Stochastic mathematical program with equilibrium constraints; Mangasarian-Fromovitz constraint qualification.},
language = {eng},
number = {2},
pages = {252-265},
publisher = {EDP-Sciences},
title = {Regularization method for stochastic mathematical programs with complementarity constraints},
url = {http://eudml.org/doc/245251},
volume = {11},
year = {2005},
}

TY - JOUR
AU - Lin, Gui-Hua
AU - Fukushima, Masao
TI - Regularization method for stochastic mathematical programs with complementarity constraints
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2005
PB - EDP-Sciences
VL - 11
IS - 2
SP - 252
EP - 265
AB - In this paper, we consider a class of stochastic mathematical programs with equilibrium constraints (SMPECs) that has been discussed by Lin and Fukushima (2003). Based on a reformulation given therein, we propose a regularization method for solving the problems. We show that, under a weak condition, an accumulation point of the generated sequence is a feasible point of the original problem. We also show that such an accumulation point is S-stationary to the problem under additional assumptions.
LA - eng
KW - stochastic mathematical program with equilibrium constraints; S-stationarity; Mangasarian-Fromovitz constraint qualification; Stochastic mathematical program with equilibrium constraints; Mangasarian-Fromovitz constraint qualification.
UR - http://eudml.org/doc/245251
ER -

References

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  1. [1] J.R. Birge and F. Louveaux, Introduction to Stochastic Programming. Springer, New York (1997). Zbl0892.90142MR1460264
  2. [2] J.F. Bonnans and A. Shapiro, Optimization problems with perturbations: A guided tour. SIAM Rev. 40 (1998) 228–264. Zbl0915.49021
  3. [3] Y. Chen and M. Florian, The nonlinear bilevel programming problem: Formulations, regularity and optimality conditions. Optimization 32 (1995) 193–209. Zbl0817.90101
  4. [4] R.W. Cottle, J.S. Pang and R.E. Stone, The Linear Complementarity Problem. Academic Press, New York, NY (1992). Zbl0757.90078MR1150683
  5. [5] K. Jittorntrum, Solution point differentiability without strict complementarity in nonlinear programming. Math. Program. Stud. 21 (1984) 127–138. Zbl0571.90080
  6. [6] P. Kall and S.W. Wallace, Stochastic Programming. John Wiley & Sons, Chichester (1994). Zbl0812.90122MR1315300
  7. [7] G.H. Lin, X. Chen and M. Fukushima, Smoothing implicit programming approaches for stochastic mathematical programs with linear complementarity constraints. Technical Report 2003–006, Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto, Japan (2003). 
  8. [8] G.H. Lin and M. Fukushima, A class of stochastic mathematical programs with complementarity constraints: Reformulations and algorithms. Technical Report 2003-010, Department of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto, Japan (2003). Zbl1096.90024
  9. [9] Z.Q. Luo, J.S. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge, UK (1996). Zbl0898.90006MR1419501
  10. [10] H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods. SIAM, Philadelphia (1992). Zbl0761.65002MR1172997
  11. [11] M. Patriksson and L. Wynter, Stochastic mathematical programs with equilibrium constraints. Oper. Res. Lett. 25 (1999) 159–167. Zbl0937.90076
  12. [12] H.S. Scheel and S. Scholtes, Mathematical programs with complementarity constraints: Stationarity, optimality, and sensitivity. Math. Oper. Res. 25 (2000) 1–22. Zbl1073.90557

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