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October  2019, 15(4): 1653-1675. doi: 10.3934/jimo.2018116

Asymptotic convergence of stationary points of stochastic multiobjective programs with parametric variational inequality constraint via SAA approach

1. 

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

2. 

School of Information and Control Engineering, Qingdao University of Technology, Qingdao 266033, China

* Corresponding author: Fanyun Meng

Received  February 2017 Revised  May 2018 Published  August 2018

Fund Project: The research is partially supported by Huzhou science and technology plan on No.2016GY03, the Natural Science Foundation of China, Grant 11601061.

We consider the sample average approximation method for a stochastic multiobjective programming problem constrained by parametric variational inequalities. The first order necessary conditions for the original problem and its sample average approximation (SAA) problem are established under constraint qualifications. By graphical convergence of set-valued mappings, the stationary points of the SAA problem converge to the stationary points of the true problem when the sample size tends to infinity. Under proper assumptions, the convergence of optimal solutions of SAA problems is also obtained. The numerical experiments on stochastic multiobjective optimization problems with variational inequalities are given to illustrate the efficiency of SAA estimators.

Citation: Liping Pang, Fanyun Meng, Jinhe Wang. Asymptotic convergence of stationary points of stochastic multiobjective programs with parametric variational inequality constraint via SAA approach. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1653-1675. doi: 10.3934/jimo.2018116
References:
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Z. Artstein and R. A. Vitale, A strong law of large numbers for random compact sets, Ann. Probab., 3 (1975), 879-882. doi: 10.1214/aop/1176996275. Google Scholar

[2]

H. Bonnel and J. Collonge, Stochastic optimization over a Pareto set associated with a stochastic multi-objective optimization problem, J. Optim. Theory Appl., 162 (2014), 405-427. doi: 10.1007/s10957-013-0367-8. Google Scholar

[3]

R. CaballeroE. CerdáM. M. MuñozL. Rey and I. M. Stancu-Minasian, Efficient solution concepts and their relations in stochastic multiobjective programming, J. Optim. Theory Appl., 110 (2001), 53-74. doi: 10.1023/A:1017591412366. Google Scholar

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R. CaballeroE. CerdáM. M. Muñoz and L. Rey, Stochastic approach versus multiobjective approach for obtaining efficient solutions in stochastic multiobjective programming problems, European Journal of Operational Research, 158 (2004), 633-648. doi: 10.1016/S0377-2217(03)00371-0. Google Scholar

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A. ChenJ. KimS. Lee and and Y. C. Kim, Stochastic multi-objective models for network design problem, Expert Systems with Applications, 37 (2010), 1608-1619. doi: 10.1016/j.eswa.2009.06.048. Google Scholar

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G. Y. Chen, X. X. Huang and X. Q. Yang, Vector Optimization Set-valued and Variational Analysis, Springer, Berlin, 2005. Google Scholar

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F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, Inc., New York, 1983. Google Scholar

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K. Deb and A. Sinha, An efficient and accurate solution methodology for bilevel multi-objective programming problems using a hybrid evolutionary-local-search algorithm, Evolutionary Computation, 18 (2010), 403-449. doi: 10.1162/EVCO_a_00015. Google Scholar

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J. Fliege and H. F. Xu, Stochastic multiobjective optimization: Sample average approximation and applications, J. Optim. Theory Appl., 151 (2011), 135-162. doi: 10.1007/s10957-011-9859-6. Google Scholar

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M. Fukushima, Fundamentals of Nonlinear Optimization, Asakura Shoten, Tokyo. 2001.Google Scholar

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A. Göpfer, C. Tammer, H. Riahi and C. Zălinescu, Variational Methods in Partially Ordered Spaces, Springer, New York, 2003. Google Scholar

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W. J. Gutjahr and A. Pichler, Stochastic multi-objective optimization: A survey on non-scalarizing methods, Ann. Oper. Res., 236 (2016), 475-499. doi: 10.1007/s10479-013-1369-5. Google Scholar

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R. HenrionA. Jourani and J. Outrata, On the calmness of a class of multifunction, SIAM J. Optimization, 13 (2002), 603-618. doi: 10.1137/S1052623401395553. Google Scholar

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J. Jahn, Vector Optimization, Springer, Berlin, 2004. doi: 10.1007/978-3-540-24828-6. Google Scholar

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S. Kim and J. H. Ryu, The sample avreage approximation method for multi-objective stochastic optimization, Proceeding of the 2011 Winter Simulation Conference, (2011), 4026-4037. doi: 10.1109/WSC.2011.6148092. Google Scholar

[16]

G. H. LinD. L. Zhang and Y. C. Liang, Stochastic multiobjective problems with complementarity constraints and applications in healthcare management, European Journal of Operational Research, 226 (2013), 461-470. doi: 10.1016/j.ejor.2012.11.005. Google Scholar

[17]

G. H. LinX. J. Chen and M. Fukushima, Solving stochastic mathematical programs with equilibrium constraints via approximation and smoothing implicit programming with penalization, Math. Program. Ser. B, 116 (2009), 343-368. doi: 10.1007/s10107-007-0119-3. Google Scholar

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K. Massimiliano and R. Daris, Multi-Objective stochastic optimization programs for a non-life insurance company under solvency constraints, Risk, 3 (2015), 390-419. doi: 10.3390/risks3030390. Google Scholar

[19]

K. Miettinen, Nonlinear Multiobjective Optimization, Kluwer Academic Publishers, Boston, MA, 1999. Google Scholar

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B. S. Mordukhovich, Equilibrium problems with equilibrium constraints via multiobjective optimization, Optimization Methods and Software, 19 (2004), 479-492. doi: 10.1080/1055678042000218966. Google Scholar

[21]

B. Mordukhovich, Variational Analysis and Generalized Differentiation I, Springer, Berlin, 2006. Google Scholar

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B. S. Mordukhovich and J. V. Outrata, Coderivative analysis of quasi-variational inequalities with applications to stability and optimization, SIAM J. Optim., 18 (2007), 752-777. doi: 10.1137/060652889. Google Scholar

[23]

B. S. Mordukhovich, Multiojective optimization problems with equilibrium constraints, Mathematical Programming, 117 (2009), 331-354. doi: 10.1007/s10107-007-0172-y. Google Scholar

[24]

L. P. PangF. Y. MengS. Chen and D. Li, Optimality condition for multi-objective optimization problem constrained by parameterized variational inequalities, Set-Valued and Variational Analysis, 22 (2014), 285-298. doi: 10.1007/s11228-014-0277-4. Google Scholar

[25]

M. Patriksson and L. Wynter, Stochastic mathematical programs with equilibrium constraints, Operations Reaserch Letters, 25 (1999), 159-167. doi: 10.1016/S0167-6377(99)00052-8. Google Scholar

[26]

R. T. Rockafellar and R. J. B. Wets, Variational Analysis, Springer, Berlin, 1998 doi: 10.1007/978-3-642-02431-3. Google Scholar

[27]

E. RoghanianS. J. Sadjadi and M. B. Aryanezhad, A probabilistic bi-level linear multi-objective programming problem to supply chain planning, Applied Mathematics and Computation, 188 (2007), 786-800. doi: 10.1016/j.amc.2006.10.032. Google Scholar

[28]

A. Shapiro, Stochastic programming with equilibrium constraints, J. Optim. Theory Appl., 128 (2006), 223-243. doi: 10.1007/s10957-005-7566-x. Google Scholar

[29]

A. Shapiro and H. F. Xu, Stochastic mathematical programs with equilibrium constraints modeling and sample average approximation, Optimization, 57 (2008), 395-418. doi: 10.1080/02331930801954177. Google Scholar

[30]

R. Slowinski and J. Teghem, Stochastic Versus Fuzzy Approaches to Multiobjective Mathematical Programming Under Uncertainty, Kluwer Academic, Dordrecht, 1990. doi: 10.1007/978-94-009-2111-5. Google Scholar

[31]

H. F. Xu and J. J. Ye, Approximating stationary points of stochastic mathematical programs with equilibrium constraints via sample averaging, Set-Valued Analysis, 19 (2011), 283-309. doi: 10.1007/s11228-010-0160-x. Google Scholar

[32]

H. F. Xu and J. J. Ye, Necessary conditions for two-stage stochastic programs with equilibrium constraints, SIAM J. Optim, 20 (2010), 1685-1715. doi: 10.1137/090748974. Google Scholar

[33]

H. F. Xu, Uniform exponential convergence of sample average random functions under general sampling with applications in stochastic programming, Journal of Mathematical Analysis and Applications, 368 (2010), 692-710. doi: 10.1016/j.jmaa.2010.03.021. Google Scholar

[34]

J. J. Ye and Q. J. Zhu, Multiobjective optimization problems with variational inequality constraints, Mathematical Program, 96 (2003), 139-160. doi: 10.1007/s10107-002-0365-3. Google Scholar

[35]

J. ZhangL. W. Zhang and L. P. Pang, On the convergence of coderivative of SAA solution mapping for a parametric stochastic variational inequality, Set-Valued Ana, 20 (2012), 75-109. doi: 10.1007/s11228-011-0181-0. Google Scholar

show all references

References:
[1]

Z. Artstein and R. A. Vitale, A strong law of large numbers for random compact sets, Ann. Probab., 3 (1975), 879-882. doi: 10.1214/aop/1176996275. Google Scholar

[2]

H. Bonnel and J. Collonge, Stochastic optimization over a Pareto set associated with a stochastic multi-objective optimization problem, J. Optim. Theory Appl., 162 (2014), 405-427. doi: 10.1007/s10957-013-0367-8. Google Scholar

[3]

R. CaballeroE. CerdáM. M. MuñozL. Rey and I. M. Stancu-Minasian, Efficient solution concepts and their relations in stochastic multiobjective programming, J. Optim. Theory Appl., 110 (2001), 53-74. doi: 10.1023/A:1017591412366. Google Scholar

[4]

R. CaballeroE. CerdáM. M. Muñoz and L. Rey, Stochastic approach versus multiobjective approach for obtaining efficient solutions in stochastic multiobjective programming problems, European Journal of Operational Research, 158 (2004), 633-648. doi: 10.1016/S0377-2217(03)00371-0. Google Scholar

[5]

A. ChenJ. KimS. Lee and and Y. C. Kim, Stochastic multi-objective models for network design problem, Expert Systems with Applications, 37 (2010), 1608-1619. doi: 10.1016/j.eswa.2009.06.048. Google Scholar

[6]

G. Y. Chen, X. X. Huang and X. Q. Yang, Vector Optimization Set-valued and Variational Analysis, Springer, Berlin, 2005. Google Scholar

[7]

F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, Inc., New York, 1983. Google Scholar

[8]

K. Deb and A. Sinha, An efficient and accurate solution methodology for bilevel multi-objective programming problems using a hybrid evolutionary-local-search algorithm, Evolutionary Computation, 18 (2010), 403-449. doi: 10.1162/EVCO_a_00015. Google Scholar

[9]

J. Fliege and H. F. Xu, Stochastic multiobjective optimization: Sample average approximation and applications, J. Optim. Theory Appl., 151 (2011), 135-162. doi: 10.1007/s10957-011-9859-6. Google Scholar

[10]

M. Fukushima, Fundamentals of Nonlinear Optimization, Asakura Shoten, Tokyo. 2001.Google Scholar

[11]

A. Göpfer, C. Tammer, H. Riahi and C. Zălinescu, Variational Methods in Partially Ordered Spaces, Springer, New York, 2003. Google Scholar

[12]

W. J. Gutjahr and A. Pichler, Stochastic multi-objective optimization: A survey on non-scalarizing methods, Ann. Oper. Res., 236 (2016), 475-499. doi: 10.1007/s10479-013-1369-5. Google Scholar

[13]

R. HenrionA. Jourani and J. Outrata, On the calmness of a class of multifunction, SIAM J. Optimization, 13 (2002), 603-618. doi: 10.1137/S1052623401395553. Google Scholar

[14]

J. Jahn, Vector Optimization, Springer, Berlin, 2004. doi: 10.1007/978-3-540-24828-6. Google Scholar

[15]

S. Kim and J. H. Ryu, The sample avreage approximation method for multi-objective stochastic optimization, Proceeding of the 2011 Winter Simulation Conference, (2011), 4026-4037. doi: 10.1109/WSC.2011.6148092. Google Scholar

[16]

G. H. LinD. L. Zhang and Y. C. Liang, Stochastic multiobjective problems with complementarity constraints and applications in healthcare management, European Journal of Operational Research, 226 (2013), 461-470. doi: 10.1016/j.ejor.2012.11.005. Google Scholar

[17]

G. H. LinX. J. Chen and M. Fukushima, Solving stochastic mathematical programs with equilibrium constraints via approximation and smoothing implicit programming with penalization, Math. Program. Ser. B, 116 (2009), 343-368. doi: 10.1007/s10107-007-0119-3. Google Scholar

[18]

K. Massimiliano and R. Daris, Multi-Objective stochastic optimization programs for a non-life insurance company under solvency constraints, Risk, 3 (2015), 390-419. doi: 10.3390/risks3030390. Google Scholar

[19]

K. Miettinen, Nonlinear Multiobjective Optimization, Kluwer Academic Publishers, Boston, MA, 1999. Google Scholar

[20]

B. S. Mordukhovich, Equilibrium problems with equilibrium constraints via multiobjective optimization, Optimization Methods and Software, 19 (2004), 479-492. doi: 10.1080/1055678042000218966. Google Scholar

[21]

B. Mordukhovich, Variational Analysis and Generalized Differentiation I, Springer, Berlin, 2006. Google Scholar

[22]

B. S. Mordukhovich and J. V. Outrata, Coderivative analysis of quasi-variational inequalities with applications to stability and optimization, SIAM J. Optim., 18 (2007), 752-777. doi: 10.1137/060652889. Google Scholar

[23]

B. S. Mordukhovich, Multiojective optimization problems with equilibrium constraints, Mathematical Programming, 117 (2009), 331-354. doi: 10.1007/s10107-007-0172-y. Google Scholar

[24]

L. P. PangF. Y. MengS. Chen and D. Li, Optimality condition for multi-objective optimization problem constrained by parameterized variational inequalities, Set-Valued and Variational Analysis, 22 (2014), 285-298. doi: 10.1007/s11228-014-0277-4. Google Scholar

[25]

M. Patriksson and L. Wynter, Stochastic mathematical programs with equilibrium constraints, Operations Reaserch Letters, 25 (1999), 159-167. doi: 10.1016/S0167-6377(99)00052-8. Google Scholar

[26]

R. T. Rockafellar and R. J. B. Wets, Variational Analysis, Springer, Berlin, 1998 doi: 10.1007/978-3-642-02431-3. Google Scholar

[27]

E. RoghanianS. J. Sadjadi and M. B. Aryanezhad, A probabilistic bi-level linear multi-objective programming problem to supply chain planning, Applied Mathematics and Computation, 188 (2007), 786-800. doi: 10.1016/j.amc.2006.10.032. Google Scholar

[28]

A. Shapiro, Stochastic programming with equilibrium constraints, J. Optim. Theory Appl., 128 (2006), 223-243. doi: 10.1007/s10957-005-7566-x. Google Scholar

[29]

A. Shapiro and H. F. Xu, Stochastic mathematical programs with equilibrium constraints modeling and sample average approximation, Optimization, 57 (2008), 395-418. doi: 10.1080/02331930801954177. Google Scholar

[30]

R. Slowinski and J. Teghem, Stochastic Versus Fuzzy Approaches to Multiobjective Mathematical Programming Under Uncertainty, Kluwer Academic, Dordrecht, 1990. doi: 10.1007/978-94-009-2111-5. Google Scholar

[31]

H. F. Xu and J. J. Ye, Approximating stationary points of stochastic mathematical programs with equilibrium constraints via sample averaging, Set-Valued Analysis, 19 (2011), 283-309. doi: 10.1007/s11228-010-0160-x. Google Scholar

[32]

H. F. Xu and J. J. Ye, Necessary conditions for two-stage stochastic programs with equilibrium constraints, SIAM J. Optim, 20 (2010), 1685-1715. doi: 10.1137/090748974. Google Scholar

[33]

H. F. Xu, Uniform exponential convergence of sample average random functions under general sampling with applications in stochastic programming, Journal of Mathematical Analysis and Applications, 368 (2010), 692-710. doi: 10.1016/j.jmaa.2010.03.021. Google Scholar

[34]

J. J. Ye and Q. J. Zhu, Multiobjective optimization problems with variational inequality constraints, Mathematical Program, 96 (2003), 139-160. doi: 10.1007/s10107-002-0365-3. Google Scholar

[35]

J. ZhangL. W. Zhang and L. P. Pang, On the convergence of coderivative of SAA solution mapping for a parametric stochastic variational inequality, Set-Valued Ana, 20 (2012), 75-109. doi: 10.1007/s11228-011-0181-0. Google Scholar

Figure 1.  Convergence of SAA optimal values
Figure 2.  Convergence of SAA optimal solutions
Figure 3.  The boundary of set $\mathbb{E}[\phi(C,\cdot)]$
Figure 4.  Convergence of SAA optimal values
Figure 5.  Convergence of SAA optimal solutions
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