# American Institute of Mathematical Sciences

December  2018, 8(4): 451-460. doi: 10.3934/naco.2018028

## Quantitative stability analysis of stochastic mathematical programs with vertical complementarity constraints

 School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

Received  December 2017 Revised  August 2018 Published  September 2018

Fund Project: The work is supported by NSFC grant 11571056.

This paper studies the quantitative stability of stochastic mathematical programs with vertical complementarity constraints (SMPVCC) with respect to the perturbation of the underlying probability distribution. We first show under moderate conditions that the optimal solution set-mapping is outer semiconitnuous and optimal value function is Lipschitz continuous with respect to the probability distribution. We then move on to investigate the outer semiconitnuous of the M-stationary points by employing the reformulation of stationary points and some stability results on the stochastic generalized equations. The particular focus is given to discrete approximation of probability distributions, where both cases that the sample is chosen in a fixed procedure and random procedure are considered. The technical results lay a theoretical foundation for approximation schemes to be applied to solve SMPVCC.

Citation: Yongchao Liu. Quantitative stability analysis of stochastic mathematical programs with vertical complementarity constraints. Numerical Algebra, Control & Optimization, 2018, 8 (4) : 451-460. doi: 10.3934/naco.2018028
##### References:
 [1] S. I. Birbil, G. Gürkan and O. Listes, Solving stochastic mathematical programs with complementarity constraints using simulation, Math. Oper. Res., 31 (2006), 739-760.  doi: 10.1287/moor.1060.0215.  Google Scholar [2] F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, V. Ⅰ-Ⅱ, Springer, 2003.  Google Scholar [3] A. L. Gibbs and F. E. Su, On choosing and bounding probability metrics, Inter. stat. Rev., 70 (2002), 419-435.   Google Scholar [4] H. Gfrerer and J. J. Ye, New constraint qualifications for mathematical programs with equilibrium constraints via variational analysis, SIAM J. Optim., 27 (2017), 842-865.  doi: 10.1137/16M1088752.  Google Scholar [5] Y. C. Liang and G. H. Lin, Stationarity conditions and their reformulations for mathematical programs with vertical complementarity constraints, J. Optim.Theorey Appl., 154 (2012), 54-70.  doi: 10.1007/s10957-012-9992-x.  Google Scholar [6] Y. Liu, H. Xu and G. H. Lin, Stability analysis of two stage stochastic mathematical programs with complementarity constraints via NLP-regularization, SIAM J. Optim., 21 (2011), 609-705.  doi: 10.1137/100785685.  Google Scholar [7] Y. Liu, H. Xu and G. H. Lin, Stability analysis of one stage stochastic mathematical programs with complementarity constraints, J. Optim. Theory Appl., 152 (2012), 573-555.  doi: 10.1007/s10957-011-9903-6.  Google Scholar [8] Y. Liu, H. Xu and J. J. Ye, Penalized sample average approximation methods for stochastic mathematical programs with complementarity constraints, Math. Oper. Res., 36 (2011), 670-694.  doi: 10.1287/moor.1110.0513.  Google Scholar [9] Y. Liu, W. Römisch and H. Xu, Quantitative stability analysis of stochastic generalized equations, SIAM J. Optim., 24 (2014), 467-497.  doi: 10.1137/120880434.  Google Scholar [10] Z. Q. Luo, J. S. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge, United Kingdom, 1996. doi: 10.1017/CBO9780511983658.  Google Scholar [11] J. V. Outrata, M. Kocvara and J. Zowe, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications and Numerical Results, Kluwer Academic Publishers, Boston, 1998. doi: 10.1007/978-1-4757-2825-5.  Google Scholar [12] J. S. Pang, Error bound in mathematical programming, Math. Prog., 79 (1997), 299-332.  doi: 10.1007/BF02614322.  Google Scholar [13] M. Patriksson and L. Wynter, Stochastic mathematical programs with equilibrium constraints, Oper. Res. Lett., 25 (1999), 159-167.  doi: 10.1016/S0167-6377(99)00052-8.  Google Scholar [14] G. Ch. Pflug and A. Pichler, Multistage Stochastic Optimization, Springer Series in Operations Research and Financial Engineering, Springer, 2014. doi: 10.1007/978-3-319-08843-3.  Google Scholar [15] S. T. Rachev, Probability Metrics and the Stability of Stochastic Models, John Wiley and Sons, West Sussex, England, 1991.  Google Scholar [16] W. Römisch, Stability of stochastic programming problems, in Stochastic Programming, Handbooks in Operations Research and Management Science, 10, (eds. A. Ruszczynski and A. Shapiro), Elsevier, (2003), 483-554.  Google Scholar [17] H. Scheel and S. Scholtes, Mathematical programs with complementarity constraints: Stationarity, optimality, and sensivity, Math. Oper. Res., 25 (2000), 1-22.  doi: 10.1287/moor.25.1.1.15213.  Google Scholar [18] A. Shapiro, Stochastic mathematical programs with equilibrium constraints, J. Optim. Theory Appl., 128 (2006), 223-243.  doi: 10.1007/s10957-005-7566-x.  Google Scholar [19] A. Shapiro and H. Xu, Stochastic mathematical programs with equilibrium constraints, modeling and sample average approximation, Optimization, 57 (2008), 395-418.  doi: 10.1080/02331930801954177.  Google Scholar [20] H. Xu, Y. Liu, and H. Sun, Distributionally robust optimization with matrix moment constraints: Lagrange duality and cutting plane method, Math. Prog. , to appear. doi: 10.1007/s10107-017-1143-6.  Google Scholar [21] J. J. Ye, Necessary and sufficient conditions for mathematical programs with equilibrium constraints, J. Math. Anal. Appl., 307 (2005), 350-369.  doi: 10.1016/j.jmaa.2004.10.032.  Google Scholar [22] J. J. Ye and X. Y. Ye, Necessary optimality conditions for optimization problems with variational inequality constraints, Math. Oper. Res., 22 (1997), 977-997.  doi: 10.1287/moor.22.4.977.  Google Scholar

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##### References:
 [1] S. I. Birbil, G. Gürkan and O. Listes, Solving stochastic mathematical programs with complementarity constraints using simulation, Math. Oper. Res., 31 (2006), 739-760.  doi: 10.1287/moor.1060.0215.  Google Scholar [2] F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, V. Ⅰ-Ⅱ, Springer, 2003.  Google Scholar [3] A. L. Gibbs and F. E. Su, On choosing and bounding probability metrics, Inter. stat. Rev., 70 (2002), 419-435.   Google Scholar [4] H. Gfrerer and J. J. Ye, New constraint qualifications for mathematical programs with equilibrium constraints via variational analysis, SIAM J. Optim., 27 (2017), 842-865.  doi: 10.1137/16M1088752.  Google Scholar [5] Y. C. Liang and G. H. Lin, Stationarity conditions and their reformulations for mathematical programs with vertical complementarity constraints, J. Optim.Theorey Appl., 154 (2012), 54-70.  doi: 10.1007/s10957-012-9992-x.  Google Scholar [6] Y. Liu, H. Xu and G. H. Lin, Stability analysis of two stage stochastic mathematical programs with complementarity constraints via NLP-regularization, SIAM J. Optim., 21 (2011), 609-705.  doi: 10.1137/100785685.  Google Scholar [7] Y. Liu, H. Xu and G. H. Lin, Stability analysis of one stage stochastic mathematical programs with complementarity constraints, J. Optim. Theory Appl., 152 (2012), 573-555.  doi: 10.1007/s10957-011-9903-6.  Google Scholar [8] Y. Liu, H. Xu and J. J. Ye, Penalized sample average approximation methods for stochastic mathematical programs with complementarity constraints, Math. Oper. Res., 36 (2011), 670-694.  doi: 10.1287/moor.1110.0513.  Google Scholar [9] Y. Liu, W. Römisch and H. Xu, Quantitative stability analysis of stochastic generalized equations, SIAM J. Optim., 24 (2014), 467-497.  doi: 10.1137/120880434.  Google Scholar [10] Z. Q. Luo, J. S. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints, Cambridge University Press, Cambridge, United Kingdom, 1996. doi: 10.1017/CBO9780511983658.  Google Scholar [11] J. V. Outrata, M. Kocvara and J. Zowe, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications and Numerical Results, Kluwer Academic Publishers, Boston, 1998. doi: 10.1007/978-1-4757-2825-5.  Google Scholar [12] J. S. Pang, Error bound in mathematical programming, Math. Prog., 79 (1997), 299-332.  doi: 10.1007/BF02614322.  Google Scholar [13] M. Patriksson and L. Wynter, Stochastic mathematical programs with equilibrium constraints, Oper. Res. Lett., 25 (1999), 159-167.  doi: 10.1016/S0167-6377(99)00052-8.  Google Scholar [14] G. Ch. Pflug and A. Pichler, Multistage Stochastic Optimization, Springer Series in Operations Research and Financial Engineering, Springer, 2014. doi: 10.1007/978-3-319-08843-3.  Google Scholar [15] S. T. Rachev, Probability Metrics and the Stability of Stochastic Models, John Wiley and Sons, West Sussex, England, 1991.  Google Scholar [16] W. Römisch, Stability of stochastic programming problems, in Stochastic Programming, Handbooks in Operations Research and Management Science, 10, (eds. A. Ruszczynski and A. Shapiro), Elsevier, (2003), 483-554.  Google Scholar [17] H. Scheel and S. Scholtes, Mathematical programs with complementarity constraints: Stationarity, optimality, and sensivity, Math. Oper. Res., 25 (2000), 1-22.  doi: 10.1287/moor.25.1.1.15213.  Google Scholar [18] A. Shapiro, Stochastic mathematical programs with equilibrium constraints, J. Optim. Theory Appl., 128 (2006), 223-243.  doi: 10.1007/s10957-005-7566-x.  Google Scholar [19] A. Shapiro and H. Xu, Stochastic mathematical programs with equilibrium constraints, modeling and sample average approximation, Optimization, 57 (2008), 395-418.  doi: 10.1080/02331930801954177.  Google Scholar [20] H. Xu, Y. Liu, and H. Sun, Distributionally robust optimization with matrix moment constraints: Lagrange duality and cutting plane method, Math. Prog. , to appear. doi: 10.1007/s10107-017-1143-6.  Google Scholar [21] J. J. Ye, Necessary and sufficient conditions for mathematical programs with equilibrium constraints, J. Math. Anal. Appl., 307 (2005), 350-369.  doi: 10.1016/j.jmaa.2004.10.032.  Google Scholar [22] J. J. Ye and X. Y. Ye, Necessary optimality conditions for optimization problems with variational inequality constraints, Math. Oper. Res., 22 (1997), 977-997.  doi: 10.1287/moor.22.4.977.  Google Scholar
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