A regularized sample average approximation method for stochastic mathematical programs with nonsmooth equality constraints

SIAM Journal on Optimization

Volumn 17, Issue 3, 2006, Pages 891-919

  Meng, Fanwen ^a,b   Xu, Huifu ^a  

^a UNIVERSITY OF SOUTHAMPTON   (United Kingdom)

^b NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

Author keywords

Convergence of stationary points; Karush Kuhn Tucker conditions; P0 variational inequality; Regularization methods; Sample average approximation

Indexed keywords

CONSTRAINT THEORY; NUMERICAL METHODS; PROBLEM SOLVING; VECTORS;

RANDOM VECTORS; REGULARIZATION METHODS; SAMPLE AVERAGE APPROXIMATION; VARIATIONAL INEQUALITY;

STOCHASTIC PROGRAMMING;

EID: 34547948140     PISSN: 10526234     EISSN: None     Source Type: Journal    
DOI: 10.1137/050638242     Document Type: Article

References (36)

1. Z. ARTSTEIN AND R. A. VITALE, A strong law of large numbers for random compact sets, Ann. Probability, 3 (1975), pp. 879-882.
2. Z. ARTSTEIN AND S. HART, Law of large numbers for random sets and allocation processes, Math. Oper. Res., 6 (1981), pp. 485-492.
3. F. BASTIN, C. CIRILLO, AND P. TOINT, Convergence theory for nonconvex stochastic programming with an application to mixed logit, Math. Program., 108 (2006), pp. 207-234.
4. S. I. BIRBIL, G. GÜRKAN, AND O. LISTES, Simulation-Based Solution of Stochastic Mathematical Programs with Complementarity Constraints: Sample-Path Analysis, working paper, Center for Economic Research, Tilburg University, Tilburg, The Netherlands, 2004.
5. J. BIRGE AND F. LOUVEAUX, Introduction to Stochastic Programming, Springer-Verlag, New York, 1997.
6. J. F. BONNANS AND A. SHAPIRO, Perturbation Analysis of Optimization Problems, Springer-Verlag, New York, 2000.
7. B. CHEN AND P. T. HARKER, A non-interior-point continuation method for linear complementarity problems, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 1168-1190.
8. X. CHEN, L. QI, AND D. SUN, Global and superlinear convergence of the smoothing Newton's method and its application to general box constrained variational inequalities, Math. Comp., 67 (1998), pp. 519-540.
9. S. CHRISTIANSEN, M. PATRIKSSON, AND L. WYNTER, Stochastic bilevel programming in structural optimization, Struct. Multidiscip. Optim., 21 (2001), pp. 361-371.
10. F. H. CLARKE, Optimization and Nonsmooth Analysis, John Wiley, New York, 1983.
11. F. FACCHINEI AND J.-S. PANG, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer-Verlag, New York, 2003.
12. S. A. GABRIEL AND J. J. MORÉ, Smoothing of mixed complementarity problems, in Complemenarity and Variational Problems: State of the Art, M. C. Ferris and J. S. Pang, eds., SIAM, Philadelphia, 1997, pp. 105-116.
13. J. L. HIGLE AND S. SEN, Stochastic Decomposition: A Statistical Method for Large Scale Stochastic Linear Programming, Kluwer Academic Publishers, Boston, 1996.
14. J.-B. HIRIART-URRUTY, Refinements of necessary optimality conditions in nondifferentiable programming I, Appl. Math. Optim., 5 (1979), pp. 63-82.
15. T. HOMEM-DE-MELLO, Estimation of derivatives of nonsmooth performance measures in regenerative systems, Math. Oper. Res., 26 (2001), pp. 741-768.
16. C. KANZOW, Some noninterior continuation methods for linear complementarity problems, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 851-868.
17. G.-H. LIN, X. CHEN, AND M. FUKUSHIMA, Smoothing Implicit Programming Approaches for Stochastic Mathematical Programs with Linear Complementarity Constraints, http://www.amp.i.kyoto-u. ac.jp/tecrep/index-e.html (2003).
18. M. PATRIKSSON AND L. WYNTER, Stochastic mathematical programs with equilibrium constraints, Oper. Res. Lett., 25 (1999), pp. 159-167.
19. 0-functions, SIAM J. Optim., 10 (1999), pp. 315-330.
20. L. QI, D. SUN, AND G. ZHOU, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities, Math. Program., 87 (2000), pp. 1-35.
21. D. RALPH AND H. XU, Implicit smoothing and its application to optimization with piecewise smooth equality constraints, J. Optim. Theory Appl., 124 (2005), pp. 673-699.
22. S. M. ROBINSON, Analysis of sample-path optimization, Math. Oper. Res., 21 (1996), pp. 513-528.
23. R. T. ROCKAFELLAR AND R. J.-B. WETS, Stochastic convex programming: Kuhn-Tucker conditions, J. Math. Econom., 2 (1975), pp. 349-370.
24. R. T. ROCKAFELLAR AND R.J.-B. WETS, Variational Analysis, Springer-Verlag, Berlin, 1998.
25. R. Y. RUBINSTEIN AND A. SHAPIRO, Discrete Events Systems: Sensitivity Analysis and Stochastic Optimization by the Score Function Methods, John Wiley and Sons, New York, 1993.
26. A. RUSZCZYŃSKI, Decomposition methods, in Stochastic Programming, Handbooks Oper. Res. Management Sci. 10, A. Ruszczyński and A. Shapiro, eds., North-Holland, Amsterdam, 2003, pp. 141-211.
27. A. RUSZCZYŃSKI AND A. SHAPIRO, Stochastic Programming, Handbooks Oper. Res. Management Sci. 10, North-Holland, Amsterdam, 2003.
28. T. SANTOSO, S. AHMED, M. GOETSCHALCKX, AND A. SHAPIRO, A stochastic programming approach for supply chain network design under uncertainty, European J. Oper. Res., 167 (2005), pp. 96-115.
29. S. SCHOLTES, Introduction to Piecewise Smooth Equations, Habilitation, University of Karlsruhe, Karlsruhe, Germany, 1994.
30. A. SHAPIRO, Stochastic mathematical programs with equilibrium constraints, J. Optim. Theory Appl., 128 (2006), pp. 223-243.
31. A. SHAPIRO AND T. HOMEM-DE-MELLO, On the rate of convergence, of optimal solutions of Monte Carlo approximations of stochastic programs, SIAM J. Optim., 11 (2000), pp. 70-86.
32. A. SHAPIRO AND H. XU, Stochastic Mathematical Programs with Equilibrium Constraints, Modeling and Sample Average Approximation, http://www.optimization-online.org/DB_ HTML/2005/01/1046.html (2005).
33. R.J.-B. WETS, Stochastic programming, in Optimization, Handbooks Oper. Res. Management Sci. 1, G. L. Nemhauser, A. Rinnooy, and M. Todd, eds., North-Holland, Amsterdam, 1989, pp. 573-629.
34. H. XU, An MPCC approach for stochastic Stackelberg-Nash- Cournot equilibrium, Optimization, 54 (2005), pp. 27-57.
35. H. XU, An implicit programming approach for a class of stochastic mathematical programs with complementarity constraints, SIAM J. Optim., 16 (2006), pp. 670-696.
36. H. XU AND F. MENG, Convergence analysis of sample, average approximation methods for a class of stochastic mathematical programs with equality constraints, Math. Oper. Res., to appear.