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. BirbilG. 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. LiuH. 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. LiuH. 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. LiuH. 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. LiuW. 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

show all references

References:
[1]

S. I. BirbilG. 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. LiuH. 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. LiuH. 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. LiuH. 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. LiuW. 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

[1]

Yu Zhou, Xinfeng Dong, Yongzhuang Wei, Fengrong Zhang. A note on the Signal-to-noise ratio of $ (n, m) $-functions. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020117

[2]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[3]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[4]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[5]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020264

[6]

Xin-Guang Yang, Lu Li, Xingjie Yan, Ling Ding. The structure and stability of pullback attractors for 3D Brinkman-Forchheimer equation with delay. Electronic Research Archive, 2020, 28 (4) : 1395-1418. doi: 10.3934/era.2020074

[7]

Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020275

[8]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441

 Impact Factor: 

Metrics

  • PDF downloads (108)
  • HTML views (360)
  • Cited by (0)

Other articles
by authors

[Back to Top]