# American Institute of Mathematical Sciences

• Previous Article
Existence of anonymous link tolls for decentralizing an oligopolistic game and the efficiency analysis
• JIMO Home
• This Issue
• Next Article
Optimal control of electrorheological clutch described by nonlinear parabolic equation with nonlocal boundary conditions
April  2011, 7(2): 317-345. doi: 10.3934/jimo.2011.7.317

## Sample average approximation method for stochastic complementarity problems with applications to supply chain supernetworks

 1 School of Management Science and Engineering, Dalian University of Technology, Dalian 116023, China 2 School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag-3, Wits-2050, Johannesburg, South Africa 3 School of Mathematical Science, Dalian University of Technology, Dalian 116024, China

Received  November 2009 Revised  January 2011 Published  April 2011

We consider a class of stochastic nonlinear complementarity problems. We propose a new reformulation of the stochastic complementarity problem, that is, a two-stage stochastic mathematical programming model reformulation. Based on this reformulation, we propose a smoothing-based sample average approximation method for stochastic complementarity problem and prove its convergence. As an application, a supply chain super-network equilibrium is modeled as a stochastic nonlinear complementarity problem and numerical results on the problem are reported.
Citation: Mingzheng Wang, M. Montaz Ali, Guihua Lin. Sample average approximation method for stochastic complementarity problems with applications to supply chain supernetworks. Journal of Industrial & Management Optimization, 2011, 7 (2) : 317-345. doi: 10.3934/jimo.2011.7.317
##### References:
 [1] F. Bastin, C. Cirllo and P. L. Toint, Convergence theory for nonconvex stochastic programming with an application to mixed logit,, Mathematical Programming, 108 (2006), 207.  doi: 10.1007/s10107-006-0708-6.  Google Scholar [2] X. Chen and M. Fukushima, Expected residual minimization method for stochastic linear complementarity problems,, Mathematics of Operations Research, 30 (2005), 1022.  doi: 10.1287/moor.1050.0160.  Google Scholar [3] X. Chen, C. Zhang and M. Fukushima, Robust solution of stochastic matrix linear complementarity problems,, Mathematical Programming, 117 (2009), 51.   Google Scholar [4] F. H. Clarke, "Optimization and Nonsmooth Analysis,", SIAM, (1990).   Google Scholar [5] R. W. Cottle, J.-S. Pang and R. Stone, "The linear Complementary Problems,", Academic Press, (1992).   Google Scholar [6] J. Dong, D. Zhang and A. Nagurney, A supply chain network equilibrium model with random demands,, European Journal of Operational Research, 156 (2004), 194.  doi: 10.1016/S0377-2217(03)00023-7.  Google Scholar [7] F. Facchinei and J-S. Pang, "Finite-Dimensional Variational Inequalities and Complementarity Problems,", Springer, (2003).   Google Scholar [8] H. Fang, X. Chen and M. Fukushima, Stochastic $R_0$ Matrix linear complementarity problems,, SIAM Journal on Optimization, 18 (2007), 482.  doi: 10.1137/050630805.  Google Scholar [9] G. Gürkan, A. Y. Özge and S. M. Robinson, Sample-path solutions of stochastic variational inequalities,, Mathematical Programming, 84 (1999), 313.  doi: 10.1007/s101070050024.  Google Scholar [10] G. H. Lin, X. Chen and M. Fukushima, New restricted NCP functions and their applications to stochastic NCP and stochastic MPEC,, Optimization, 56 (2007), 641.  doi: 10.1080/02331930701617320.  Google Scholar [11] G. H. Lin and M. Fukushima, New reformulations for stochastic nonlinear complementarity problems,, Optimization Methods and Software, 21 (2006), 551.  doi: 10.1080/10556780600627610.  Google Scholar [12] F. W. Meng and H. Xu, A regularized sample average approximation method for stochastic mathematical programs with nonsmooth equality constraints,, SIAM Journal on Optimization, 17 (2006), 891.  doi: 10.1137/050638242.  Google Scholar [13] A. Nagurney, "Network Economics: a Variational Inequality Approach,", Kluwer Academic Publishers, (1999).   Google Scholar [14] A. Nagurney, J. Cruz, J. Dong and D. Zhang, Supply chain networks, electronic commence, and supply side and demand side risk,, European Journal of Operational Research, 164 (2005), 120.  doi: 10.1016/j.ejor.2003.11.007.  Google Scholar [15] A. Nagurney and J. Dong, "Super-Networks: Decision-Making for the Information Age,", Edward Elgar Publishers, (2002).   Google Scholar [16] S. M. Robinson, Analysis of sample-path optimization,, Mathematics of Operations Research, 21 (1996), 513.  doi: 10.1287/moor.21.3.513.  Google Scholar [17] A. Ruszcynski and A. Shapiro, Eds., "Stochastic Programming,", Handbooks in OR$&$MS, (2003).   Google Scholar [18] T. Santoso, S. Ahmed and A. Shapiro, A stochastic programming approach for suppy chain network design under uncertainty,, European Journal of Operational Research, 167 (2005), 96.  doi: 10.1016/j.ejor.2004.01.046.  Google Scholar [19] A. Shapiro, Stochastic mathematical programs with equilibrium constraints,, Journal of Optimization Theory and Application, 128 (2006), 223.  doi: 10.1007/s10957-005-7566-x.  Google Scholar [20] A. Shapiro and H. Xu, Stochastic mathematical programs with equilibrium constraints, modeling and sample average approximation,, Optimization, 57 (2008), 395.  doi: 10.1080/02331930801954177.  Google Scholar [21] R. Storn and K. Price, Differential evolution - a simple and efficient heuristic for global optimization over continuous spaces,, Journal of Global Optimization, 11 (1997), 341.  doi: 10.1023/A:1008202821328.  Google Scholar [22] H. Xu and F. Meng, Convergence analysis of sample average approximation methods for a class of stochastic mathematical programs with equality constraints,, Mathematics of Operations Research, 32 (2007), 648.  doi: 10.1287/moor.1070.0260.  Google Scholar [23] C. Zhang and X. Chen, Stochastic nonlinear complementary problem and application to traffic equilibrium under uncertainty,, Journal of Optimization Theory and Applications, 137 (2008), 277.  doi: 10.1007/s10957-008-9358-6.  Google Scholar

show all references

##### References:
 [1] F. Bastin, C. Cirllo and P. L. Toint, Convergence theory for nonconvex stochastic programming with an application to mixed logit,, Mathematical Programming, 108 (2006), 207.  doi: 10.1007/s10107-006-0708-6.  Google Scholar [2] X. Chen and M. Fukushima, Expected residual minimization method for stochastic linear complementarity problems,, Mathematics of Operations Research, 30 (2005), 1022.  doi: 10.1287/moor.1050.0160.  Google Scholar [3] X. Chen, C. Zhang and M. Fukushima, Robust solution of stochastic matrix linear complementarity problems,, Mathematical Programming, 117 (2009), 51.   Google Scholar [4] F. H. Clarke, "Optimization and Nonsmooth Analysis,", SIAM, (1990).   Google Scholar [5] R. W. Cottle, J.-S. Pang and R. Stone, "The linear Complementary Problems,", Academic Press, (1992).   Google Scholar [6] J. Dong, D. Zhang and A. Nagurney, A supply chain network equilibrium model with random demands,, European Journal of Operational Research, 156 (2004), 194.  doi: 10.1016/S0377-2217(03)00023-7.  Google Scholar [7] F. Facchinei and J-S. Pang, "Finite-Dimensional Variational Inequalities and Complementarity Problems,", Springer, (2003).   Google Scholar [8] H. Fang, X. Chen and M. Fukushima, Stochastic $R_0$ Matrix linear complementarity problems,, SIAM Journal on Optimization, 18 (2007), 482.  doi: 10.1137/050630805.  Google Scholar [9] G. Gürkan, A. Y. Özge and S. M. Robinson, Sample-path solutions of stochastic variational inequalities,, Mathematical Programming, 84 (1999), 313.  doi: 10.1007/s101070050024.  Google Scholar [10] G. H. Lin, X. Chen and M. Fukushima, New restricted NCP functions and their applications to stochastic NCP and stochastic MPEC,, Optimization, 56 (2007), 641.  doi: 10.1080/02331930701617320.  Google Scholar [11] G. H. Lin and M. Fukushima, New reformulations for stochastic nonlinear complementarity problems,, Optimization Methods and Software, 21 (2006), 551.  doi: 10.1080/10556780600627610.  Google Scholar [12] F. W. Meng and H. Xu, A regularized sample average approximation method for stochastic mathematical programs with nonsmooth equality constraints,, SIAM Journal on Optimization, 17 (2006), 891.  doi: 10.1137/050638242.  Google Scholar [13] A. Nagurney, "Network Economics: a Variational Inequality Approach,", Kluwer Academic Publishers, (1999).   Google Scholar [14] A. Nagurney, J. Cruz, J. Dong and D. Zhang, Supply chain networks, electronic commence, and supply side and demand side risk,, European Journal of Operational Research, 164 (2005), 120.  doi: 10.1016/j.ejor.2003.11.007.  Google Scholar [15] A. Nagurney and J. Dong, "Super-Networks: Decision-Making for the Information Age,", Edward Elgar Publishers, (2002).   Google Scholar [16] S. M. Robinson, Analysis of sample-path optimization,, Mathematics of Operations Research, 21 (1996), 513.  doi: 10.1287/moor.21.3.513.  Google Scholar [17] A. Ruszcynski and A. Shapiro, Eds., "Stochastic Programming,", Handbooks in OR$&$MS, (2003).   Google Scholar [18] T. Santoso, S. Ahmed and A. Shapiro, A stochastic programming approach for suppy chain network design under uncertainty,, European Journal of Operational Research, 167 (2005), 96.  doi: 10.1016/j.ejor.2004.01.046.  Google Scholar [19] A. Shapiro, Stochastic mathematical programs with equilibrium constraints,, Journal of Optimization Theory and Application, 128 (2006), 223.  doi: 10.1007/s10957-005-7566-x.  Google Scholar [20] A. Shapiro and H. Xu, Stochastic mathematical programs with equilibrium constraints, modeling and sample average approximation,, Optimization, 57 (2008), 395.  doi: 10.1080/02331930801954177.  Google Scholar [21] R. Storn and K. Price, Differential evolution - a simple and efficient heuristic for global optimization over continuous spaces,, Journal of Global Optimization, 11 (1997), 341.  doi: 10.1023/A:1008202821328.  Google Scholar [22] H. Xu and F. Meng, Convergence analysis of sample average approximation methods for a class of stochastic mathematical programs with equality constraints,, Mathematics of Operations Research, 32 (2007), 648.  doi: 10.1287/moor.1070.0260.  Google Scholar [23] C. Zhang and X. Chen, Stochastic nonlinear complementary problem and application to traffic equilibrium under uncertainty,, Journal of Optimization Theory and Applications, 137 (2008), 277.  doi: 10.1007/s10957-008-9358-6.  Google Scholar
 [1] J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008 [2] Ardeshir Ahmadi, Hamed Davari-Ardakani. A multistage stochastic programming framework for cardinality constrained portfolio optimization. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 359-377. doi: 10.3934/naco.2017023 [3] Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192 [4] Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021066 [5] Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 [6] Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399 [7] Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 [8] Seung-Yeal Ha, Dongnam Ko, Chanho Min, Xiongtao Zhang. Emergent collective behaviors of stochastic kuramoto oscillators. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1059-1081. doi: 10.3934/dcdsb.2019208 [9] María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088 [10] Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185 [11] Olena Naboka. On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1933-1956. doi: 10.3934/cpaa.2009.8.1933 [12] Shangzhi Li, Shangjiang Guo. Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2693-2719. doi: 10.3934/dcdsb.2020201 [13] Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175 [14] Longxiang Fang, Narayanaswamy Balakrishnan, Wenyu Huang. Stochastic comparisons of parallel systems with scale proportional hazards components equipped with starting devices. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021004 [15] Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$\alpha$ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113 [16] Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521 [17] Xiaohu Wang, Dingshi Li, Jun Shen. Wong-Zakai approximations and attractors for stochastic wave equations driven by additive noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2829-2855. doi: 10.3934/dcdsb.2020207 [18] Kuan-Hsiang Wang. An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021030 [19] Yunfei Lv, Rong Yuan, Yuan He. Wavefronts of a stage structured model with state--dependent delay. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4931-4954. doi: 10.3934/dcds.2015.35.4931 [20] Baba Issa Camara, Houda Mokrani, Evans K. Afenya. Mathematical modeling of glioma therapy using oncolytic viruses. Mathematical Biosciences & Engineering, 2013, 10 (3) : 565-578. doi: 10.3934/mbe.2013.10.565

2019 Impact Factor: 1.366