Advanced Search
Article Contents
Article Contents

Necessary optimality condition for trilevel optimization problem

  • * Corresponding author

    * Corresponding author 
This work was supported by the Natural Science Foundation of China (11871383, 11401487), and the Basic and Advanced Research Project of Chongqing(cstc2016jcyjA0239).
Abstract Full Text(HTML) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: 90C46, 90C30, 90C31.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] N. AlguacilA. 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. EkelandApplied 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. ChiZ. 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. DempeB. 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. GuoG. H. LinJ. 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. HanJ. LuY. 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. HuangD. 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. LiZ. 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. WanL. 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. WanG. 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. ZhangJ. LuJ. 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. ZhengJ. 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. ZhengZ. WanS. 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.
  • 加载中

Article Metrics

HTML views(2617) PDF downloads(568) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint