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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  October 2019 Early access  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 and Management Optimization, 2019, 15 (4) : 1653-1675. doi: 10.3934/jimo.2018116
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.

[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.

[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.

[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.

[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.

[6]

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

[7]

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

[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.

[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.

[10]

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

[11]

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

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[19]

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

[20]

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

[21]

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

[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.

[23]

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

[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.

[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.

[26]

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

[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.

[28]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[6]

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

[7]

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

[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.

[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.

[10]

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

[11]

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

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[19]

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

[20]

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

[21]

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

[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.

[23]

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

[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.

[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.

[26]

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

[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.

[28]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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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