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doi: 10.3934/jimo.2018140

## Necessary optimality condition for trilevel optimization problem

 1 School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China 2 School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China

* Corresponding author

Received  January 2017 Revised  September 2017 Published  September 2018

Fund Project: This work was supported by the Natural Science Foundation of China (11871383, 11401487), and the Basic and Advanced Research Project of Chongqing(cstc2016jcyjA0239)

This paper mainly studies the optimality conditions for a class of trilevel optimization problem, of which all levels are nonlinear programs. We firstly transform this problem into an auxiliary bilevel optimization problem by applying KKT approach to the lower-level problem. Then we obtain a necessary optimality condition via the differential calculus of Mordukhovich. Finally, a theorem for existence of optimal solution is derived via Weierstrass Theorem.

Citation: Gaoxi Li, Zhongping Wan, Jia-wei Chen, Xiaoke Zhao. Necessary optimality condition for trilevel optimization problem. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018140
##### References:
 [1] N. Alguacil, A. Delgadillo and J. M. Arroyo, A trilevel programming approach for electric grid defense planning, Computers and Operations Research, 41 (2014), 282-290. doi: 10.1016/j.cor.2013.06.009. [2] J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, A Wiley-Interscience Publication, New York, 1984. [3] B. Bank, J. Guddat, D. Klatte, B. Kummer and K. Tammer, Non-linear Parametric Optimization, Birkha"user Verlag, Basel-Boston, Mass., 1983. [4] J. F. Bard, An investigation of the linear three level programming problem, IEEE Transactions on Systems, Man and Cybernetics, 5 (1984), 711-717. doi: 10.1109/TSMC.1984.6313291. [5] B. Si and Z. Gao, Optimal model for passenger transport pricing under the condition of market competition, Journal of Transportation Systems Engineering and Information Technology, 1 (2007), 72-78. doi: 10.1016/S1570-6672(07)60009-9. [6] X. Chi, Z. Wan and Z. Hao, Second order sufficient conditions for a class of bilevel programs with lower level second-order cone programming problem, Journal of Industrial and Management Optimization, 11 (2015), 1111-1125. doi: 10.3934/jimo.2015.11.1111. [7] S. Dempe and J. Dutta, Is bilevel programming a special case of a mathematical program with complementarity constraints?, Mathematical programming, 131 (2012), 37-48. doi: 10.1007/s10107-010-0342-1. [8] S. Dempe, B. S. Mordukhovich and A. B. Zemkoho, Sensitivity analysis for two-level value functions with applications to bilevel programming, SIAM Journal on Optimization, 22 (2012), 1309-1343. doi: 10.1137/110845197. [9] S. Dempe and A. B. Zemkoho, The generalized mangasarian-fromowitz constraint qualification and optimality conditions for bilevel programs, Journal of Optimization Theory and Applications, 148 (2011), 46-68. doi: 10.1007/s10957-010-9744-8. [10] S. Dempe and A. B. Zemkoho, The bilevel programming problem: Reformulations, constraint qualifications and optimality conditions, Mathematical Programming, 138 (2013), 447-473. doi: 10.1007/s10107-011-0508-5. [11] L. Guo, G. H. Lin, J. J. Ye and J. Zhang, Sensitivity analysis of the value function for parametric mathematical programs with equilibrium constraints, SIAM Journal on Optimization, 24 (2014), 1206-1237. doi: 10.1137/130929783. [12] J. Han, J. Lu, Y. Hu and G. Zhang, Tri-level decision-making with multiple followers: Model, algorithm and case study, Information Sciences, 311 (2015), 182-204. doi: 10.1016/j.ins.2015.03.043. [13] C. Huang, D. Fang and Z. Wan, An interactive intuitionistic fuzzy method for multilevel linear programming problems, Wuhan University Journal of Natural Sciences, 20 (2015), 113-118. doi: 10.1007/s11859-015-1068-y. [14] G. Li, Z. Wan and X. Zhao, Optimality conditions for bilevel optimization problem with both levels programs being multiobjective, Pacific journal of optimiization, 13 (2017), 421-441. [15] O. L. Mangasarian, Nonlinear Programming SIAM Classics in Applied Methematic, volume 10, 1969. [16] B. S. Mordukhovich, Variational Analysis and Generalized Differentiation I: Basic Theory, Springer Science and Business Media, 2006. doi: 10.1007/3-540-31247-1. [17] R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, volume 317. Springer Science and Business Media, 2009. [18] Z. Wan, L. Mao and G. Wang, Estimation of distribution algorithm for a class of nonlinear bilevel programming problems, Information Sciences, 256 (2014), 184-196. doi: 10.1016/j.ins.2013.09.021. [19] Z. Wan, G. Wang and B. Sun, A hybrid intelligent algorithm by combining particle swarm optimization with chaos searching technique for solving nonlinear bilevel programming problems, Swarm and Evolutionary Computation, 8 (2013), 26-32. doi: 10.1016/j.swevo.2012.08.001. [20] D. White, Penalty function approach to linear trilevel programming, Journal of Optimization Theory and Applications, 93 (1997), 183-197. doi: 10.1023/A:1022610103712. [21] H. Xu and B. Li, Dynamic cloud pricing for revenue maximization, IEEE Transactions on Cloud Computing, 1 (2013), 158-171. [22] J. J. Ye, Necessary optimality conditions for multiobjective bilevel programs, Mathematics of Operations Research, 36 (2011), 165-184. doi: 10.1287/moor.1100.0480. [23] G. Zhang, J. Lu and Y. Gao, Multi-level Decision Making, Springer-Verlag Berlin Heidelberg, 2015. [24] G. Zhang, J. Lu, J. Montero and Y. Zeng, Model, solution concept, and kth-best algorithm for linear trilevel programming, Information Sciences, 180 (2010), 481-492. doi: 10.1016/j.ins.2009.10.013. [25] Z. Zhang, G. Zhang, J. Lu and C. Guo, A fuzzy tri-level decision making algorithm and its application in supply chain, The 8th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT2013), Milan, Italy, 2013,154-160. doi: 10.2991/eusflat.2013.22. [26] Y. Zheng, J. Liu and Z. Wan, Interactive fuzzy decision making method for solving bilevel programming problem, Applied Mathematical Modelling, 38 (2014), 3136-3141. doi: 10.1016/j.apm.2013.11.008. [27] Y. Zheng, Z. Wan, S. Jia and G. Wang, A new method for strong-weak linear bilevel programming problem, Journal of Industrial and Management Optimization, 11 (2015), 529-547. doi: 10.3934/jimo.2015.11.529.

show all references

##### References:
 [1] N. Alguacil, A. Delgadillo and J. M. Arroyo, A trilevel programming approach for electric grid defense planning, Computers and Operations Research, 41 (2014), 282-290. doi: 10.1016/j.cor.2013.06.009. [2] J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, A Wiley-Interscience Publication, New York, 1984. [3] B. Bank, J. Guddat, D. Klatte, B. Kummer and K. Tammer, Non-linear Parametric Optimization, Birkha"user Verlag, Basel-Boston, Mass., 1983. [4] J. F. Bard, An investigation of the linear three level programming problem, IEEE Transactions on Systems, Man and Cybernetics, 5 (1984), 711-717. doi: 10.1109/TSMC.1984.6313291. [5] B. Si and Z. Gao, Optimal model for passenger transport pricing under the condition of market competition, Journal of Transportation Systems Engineering and Information Technology, 1 (2007), 72-78. doi: 10.1016/S1570-6672(07)60009-9. [6] X. Chi, Z. Wan and Z. Hao, Second order sufficient conditions for a class of bilevel programs with lower level second-order cone programming problem, Journal of Industrial and Management Optimization, 11 (2015), 1111-1125. doi: 10.3934/jimo.2015.11.1111. [7] S. Dempe and J. Dutta, Is bilevel programming a special case of a mathematical program with complementarity constraints?, Mathematical programming, 131 (2012), 37-48. doi: 10.1007/s10107-010-0342-1. [8] S. Dempe, B. S. Mordukhovich and A. B. Zemkoho, Sensitivity analysis for two-level value functions with applications to bilevel programming, SIAM Journal on Optimization, 22 (2012), 1309-1343. doi: 10.1137/110845197. [9] S. Dempe and A. B. Zemkoho, The generalized mangasarian-fromowitz constraint qualification and optimality conditions for bilevel programs, Journal of Optimization Theory and Applications, 148 (2011), 46-68. doi: 10.1007/s10957-010-9744-8. [10] S. Dempe and A. B. Zemkoho, The bilevel programming problem: Reformulations, constraint qualifications and optimality conditions, Mathematical Programming, 138 (2013), 447-473. doi: 10.1007/s10107-011-0508-5. [11] L. Guo, G. H. Lin, J. J. Ye and J. Zhang, Sensitivity analysis of the value function for parametric mathematical programs with equilibrium constraints, SIAM Journal on Optimization, 24 (2014), 1206-1237. doi: 10.1137/130929783. [12] J. Han, J. Lu, Y. Hu and G. Zhang, Tri-level decision-making with multiple followers: Model, algorithm and case study, Information Sciences, 311 (2015), 182-204. doi: 10.1016/j.ins.2015.03.043. [13] C. Huang, D. Fang and Z. Wan, An interactive intuitionistic fuzzy method for multilevel linear programming problems, Wuhan University Journal of Natural Sciences, 20 (2015), 113-118. doi: 10.1007/s11859-015-1068-y. [14] G. Li, Z. Wan and X. Zhao, Optimality conditions for bilevel optimization problem with both levels programs being multiobjective, Pacific journal of optimiization, 13 (2017), 421-441. [15] O. L. Mangasarian, Nonlinear Programming SIAM Classics in Applied Methematic, volume 10, 1969. [16] B. S. Mordukhovich, Variational Analysis and Generalized Differentiation I: Basic Theory, Springer Science and Business Media, 2006. doi: 10.1007/3-540-31247-1. [17] R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, volume 317. Springer Science and Business Media, 2009. [18] Z. Wan, L. Mao and G. Wang, Estimation of distribution algorithm for a class of nonlinear bilevel programming problems, Information Sciences, 256 (2014), 184-196. doi: 10.1016/j.ins.2013.09.021. [19] Z. Wan, G. Wang and B. Sun, A hybrid intelligent algorithm by combining particle swarm optimization with chaos searching technique for solving nonlinear bilevel programming problems, Swarm and Evolutionary Computation, 8 (2013), 26-32. doi: 10.1016/j.swevo.2012.08.001. [20] D. White, Penalty function approach to linear trilevel programming, Journal of Optimization Theory and Applications, 93 (1997), 183-197. doi: 10.1023/A:1022610103712. [21] H. Xu and B. Li, Dynamic cloud pricing for revenue maximization, IEEE Transactions on Cloud Computing, 1 (2013), 158-171. [22] J. J. Ye, Necessary optimality conditions for multiobjective bilevel programs, Mathematics of Operations Research, 36 (2011), 165-184. doi: 10.1287/moor.1100.0480. [23] G. Zhang, J. Lu and Y. Gao, Multi-level Decision Making, Springer-Verlag Berlin Heidelberg, 2015. [24] G. Zhang, J. Lu, J. Montero and Y. Zeng, Model, solution concept, and kth-best algorithm for linear trilevel programming, Information Sciences, 180 (2010), 481-492. doi: 10.1016/j.ins.2009.10.013. [25] Z. Zhang, G. Zhang, J. Lu and C. Guo, A fuzzy tri-level decision making algorithm and its application in supply chain, The 8th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT2013), Milan, Italy, 2013,154-160. doi: 10.2991/eusflat.2013.22. [26] Y. Zheng, J. Liu and Z. Wan, Interactive fuzzy decision making method for solving bilevel programming problem, Applied Mathematical Modelling, 38 (2014), 3136-3141. doi: 10.1016/j.apm.2013.11.008. [27] Y. Zheng, Z. Wan, S. Jia and G. Wang, A new method for strong-weak linear bilevel programming problem, Journal of Industrial and Management Optimization, 11 (2015), 529-547. doi: 10.3934/jimo.2015.11.529.
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