Penalized Sample Average Approximation Methods for Stochastic Mathematical Programs with Complementarity Constraints

Published Online:https://doi.org/10.1287/moor.1110.0513

References

  • Aumann R. J. Integrals of set-valued functions. J. Math. Anal. Appl. (1965) 12(1):1–12CrossrefGoogle Scholar
  • Birbil S. I., Gürkan G., Listes O. Solving stochastic mathematical programs with complementarity constraints using simulation. Math. Oper. Res. (2006) 31(4):739–760LinkGoogle Scholar
  • Burke J. V. An exact penalization viewpoint of constrained optimization. SIAM J. Control Optim. (1991) 29(4):968–998CrossrefGoogle Scholar
  • Castaing C., Valadier M.Convex Analysis and Measurable Multifunctions (1977) 580(Springer-Verlag, Berlin/Heidelberg) Lecture Notes in MathematicsCrossrefGoogle Scholar
  • Christiansen S., Patriksson M., Wynter L. Stochastic bilevel programming in structural optimization. Structual Multidisciplinary Optim. (2001) 21(5):361–371CrossrefGoogle Scholar
  • Clarke F. H.Optimization and Nonsmooth Analysis (1983) (John Wiley & Sons, New York) Google Scholar
  • Dontchev A. L., Lewis A. S., Rockafellar R. T. The radius of metric regularity. Trans. AMS (2003) 355(2):493–517CrossrefGoogle Scholar
  • Hiriart-Urruty J.-B., Lemaréchal C.Convex Analysis and Minimization Algorithms: II (1993) (Springer-Verlag, Berlin/Heidelberg) CrossrefGoogle Scholar
  • Ioffe A. D., Outrata J. V. On metric and calmness qualification conditions in subdifferential calculus. Set-Valued Anal. (2008) 16(2–3):199–227CrossrefGoogle Scholar
  • Kiewiel K. C.Methods of Descent for Nondifferentiable Optimization (1985) 1133(Springer-Verlag, New York) Lecture Notes in MathematicsCrossrefGoogle Scholar
  • Lemarchal C., Lemarechal C., Mifflin R. Bundle methods in nonsmooth optimization. Nonsmooth Optimization (1978) (Pergamon Press, Oxford, UK) 79–102Google Scholar
  • Lin G.-H., Chen X., Fukushima M. Solving stochastic mathematical programs with equilibrium constraints via approximation and smoothing implicit programming with penalization. Math. Programming (2009) 116(1):343–368CrossrefGoogle Scholar
  • Lin G.-H., Xu H., Fukushima M. Monte Carlo and quasi-Monte Carlo sampling methods for a class of stochastic mathematical programs with equilibrium constraints. Math. Methods Oper. Res. (2008) 67(3):423–441CrossrefGoogle Scholar
  • Liu G., Ye J. J., Zhu J. Partial exact penalty for mathematical program with equilibrium constraints. Set-Valued Anal. (2008) 16(5–6):785–804CrossrefGoogle Scholar
  • Lucet Y., Ye J. J. Erratum: Sensitivity analysis of the value function for optimization problems with variational inequality constraints. SIAM J. Control Optim. (2002) 41(4):1315–1319CrossrefGoogle Scholar
  • Luo Z. Q., Pang J.-S., Ralph D.Mathematical Programs with Equilibrium Constraints (1996) (Cambridge University Press, Cambridge, UK) CrossrefGoogle Scholar
  • Luo Z. Q., Pang J.-S., Ralph D., Migdalas A., Pardas P., Värbrand P. Piece-wise sequential quadratic programming for mathematical programs with nonlinear complementarity constraints. Multilevel Optimization: Algorithms Complexity and Application (1998) (Kluwer Academic Publishers, Dordrecht, The Netherlands) 209–229CrossrefGoogle Scholar
  • Meng F., Xu H. Exponential convergence of sample average approximation methods for a class of stochastic mathematical programs with complementarity constraints. J. Comput. Math. (2006) 24(6):733–748Google Scholar
  • Mordukhovich B. S. Complete characterization of openness, metric regularity, and Lipschitzian properties of multifunctions. Trans. Amer. Math. Soc. (1993) 340(1):1–35CrossrefGoogle Scholar
  • Mordukhovich B. S. Generalized differential calculus for nonsmooth and set-valued mappings. J. Math. Anal. Appl. (1994) 183:250–288CrossrefGoogle Scholar
  • Mordukhovich B. S.Variational Analysis and Generalized Differentiation I: Basic Theory (2006) 330(Springer-Verlag, Berlin/Heidelberg) A Series of Comprehensive Studies in MathematicsCrossrefGoogle Scholar
  • Outrata J. V. Optimality conditions for a class of mathematical programs with equilibrium constraints. Math. Oper. Res. (1999) 24(3):627–644LinkGoogle Scholar
  • Outrata J., Kočvara M., Zowe J.Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications and Numerical Constraints (1998) (Kluwer Publishers, Dordrecht, The Netherlands) CrossrefGoogle Scholar
  • Patriksson M., Wynter L. Stochastic mathematical programs with equilibrium constraints. Oper. Res. Lett. (1999) 25(4):159–167CrossrefGoogle Scholar
  • Pillo G. D., Grippo L. Exact penalty functions in constrained optimization. SIAM J. Optim. (1989) 27(6):1333–1360CrossrefGoogle Scholar
  • Rachev S. T., Römisch W. Quantitative stability in stochastic programming: The method of probability metrics. Math. Oper. Res. (2002) 27(4):792–818LinkGoogle Scholar
  • Robinson S. M. Analysis of sample-path optimization. Math. Oper. Res. (1996) 21(3):513–528LinkGoogle Scholar
  • Rockafellar R. T., Wets R. J.-B.Variational Analysis (1998) (Springer-Verlag, Berlin/Heidelberg) CrossrefGoogle Scholar
  • Ruszczyński A., Shapiro A., Ruszczyński A., Shapiro A. Stochastic programming. Handbook in Operations Research and Management Science (2003) 10(North-Holland Publishing Company, Amsterdam) Google Scholar
  • Scholtes S. Convergence properties of a regularization sheme for mathematical programs with complementarity constraints. SIAM J. Optim. (2001) 11(4):918–936CrossrefGoogle Scholar
  • Shapiro A. Asymptotic analysis of stochastic programs. Ann. Oper. Res. (1991) 30(1):169–186CrossrefGoogle Scholar
  • Shapiro A. Stochastic mathematical programs with equilibrium constraints. J. Optim. Theory Appl. (2006) 128(1):223–243CrossrefGoogle Scholar
  • Shapiro A., Xu H. Stochastic mathematical programs with equilibrium constraints, modeling and sample average approximation. Optim. (2008) 57(3):395–418CrossrefGoogle Scholar
  • Tomasgard A., Smeers Y., Midthun K. Capacity booking in a transportation network with stochastic demand. Proc. 20th Internat. Sympos. Math. Programming (2009) ChicagoGoogle Scholar
  • Werner A. S. Bilevel stochastic programming problems: Analysis and application to telecommunications. (2004) . Doctoral dissertation, Norwegian University of Science and Technology, Trondheim, NorwayGoogle Scholar
  • Werner A. S., Wang Q. Resale in vertically separated markets: Profit and consumer surplus implications. Proc. 20th Internat. Sympos. Math. Programming (2009) ChicagoGoogle Scholar
  • Xu H. Uniform exponential convergence of sample average random functions under general sampling with applications in stochastic programming. J. Math. Anal. Appl. (2010) 368(2):692–710CrossrefGoogle Scholar
  • Xu H., Meng F. Convergence analysis of sample average approximation methods for a class of stochastic mathematical programs with equality constraints. Math. Oper. Res. (2007) 32(3):648–668LinkGoogle Scholar
  • Ye J. J. Constraint qualifications and necessary optimality conditions for optimization problems with variational inequality constraints. SIAM J. Optim. (2000) 10(4):943–962CrossrefGoogle Scholar
  • Ye J. J. Necessary and sufficient optimality conditions for mathemtical programs with equilibrium constraints. J. Math. Anal. Appl. (2005) 307(1):305–369CrossrefGoogle Scholar
  • Ye J. J., Zhu D. L., Zhu Q. J. Exact penalization and necessary optimality conditions for generalized bilevel programming problems. SIAM J. Optim. (1997) 7(2):481–507CrossrefGoogle Scholar
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