# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020161
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## Stability for semivectorial bilevel programs

 1 School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, China 2 School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China 3 School of Mathematics Science, Chongqing Normal University, Chongqing, 401331, China

*Corresponding author

Received  May 2020 Revised  August 2020 Early access November 2020

Fund Project: This work was supported by NSFC (No.11901068, 11701057); China Postdoctoral Science Foundation (2020M673167); Natural Science Foundation of Chongqing (cstc2019jcyj-msxmX0456); the Education Committee Project Foundation of Bayu Young Scholarthe Education Committee Project Foundation of Bayu Young Scholar; Scientific and Technological Research Program of Chongqing Municipal Education Commission (KJQN201800810)

This paper studies the stability for bilevel program where the lower-level program is a multiobjective programming problem. As we know, the weakly efficient solution mapping for parametric multiobjective program is not generally lower semicontinuous. We first obtain this semicontinuity under a suitable assumption. Then, a new condition for the lower semicontinuity of the efficient solution mapping of this problem is also obtained. Finally, we get the continuities of the value functions and the solution set mapping for the upper-level problem based on the semicontinuities of solution mappings for the lower-level parametric multiobjective program.

Citation: Gaoxi Li, Liping Tang, Yingquan Huang, Xinmin Yang. Stability for semivectorial bilevel programs. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020161
##### References:
 [1] M. J. Alves and C. H. Antunes, A differential evolution algorithm to semivectorial bilevel problems, International Workshop on Machine Learning, Optimization, and Big Data. Springer, Cham, (2017), 172-185. doi: 10.1007/978-3-319-72926-8_15.  Google Scholar [2] M. J. Alves and C. H. Antunes, A semivectorial bilevel programming approach to optimize electricity dynamic time-of-use retail pricing, Computers and Operations Research, 92 (2018), 130-144.  doi: 10.1016/j.cor.2017.12.014.  Google Scholar [3] J. F. Bard, Practical Bilevel Optimization: Algorithms and Applications, Kluwer Academic Publishers, Dordrecht, 1998. doi: 10.1007/978-1-4757-2836-1.  Google Scholar [4] H. Bonnel and J. Morgan, Semivectorial bilevel optimization problem: Penalty approach, Journal of Optimization Theory and Applications, 131 (2006), 365-382.  doi: 10.1007/s10957-006-9150-4.  Google Scholar [5] H. 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Morgan, Topological existence and stability for stackelberg problems, Journal of Optimization Theory and Applications, 84 (1995), 145-169.  doi: 10.1007/BF02191740.  Google Scholar [16] Y. Lv and Z. Wan, Linear bilevel multiobjective optimization problem: penalty approach, Journal of Industrial and Management Optimization, 15 (2019), 1213-1223.  doi: 10.3934/jimo.2018092.  Google Scholar [17] Z. Y. Peng, J. W. Peng, X. J. Long and J. C. Yao, On the stability of solutions for semi-infinite vector optimization problems, Journal of Global Optimization, 70 (2018), 55-69.  doi: 10.1007/s10898-017-0553-6.  Google Scholar [18] T. Tanino, Stability and sensitivity analysis in multiobjective nonlinear programming, Annals of Operations Research, 27 (1990), 97-114.  doi: 10.1007/BF02055192.  Google Scholar [19] T. Tanino and Y. Sawaragi, Stability of nondominated solutions in multicriteria decision-making, Journal of Optimization Theory and Applications, 30 (1980), 229-253.  doi: 10.1007/BF00934497.  Google Scholar [20] G. Wang, X. Wang, Z. Wan and Y. Lv, A globally convergent algorithm for a class of bilevel nonlinear programming problem, Applied Mathematics and Computation, 188 (2007), 166-172.  doi: 10.1016/j.amc.2006.09.130.  Google Scholar [21] Y.-B. Xiao, T. N. Van and J.-C. Yao, Locally Lipschitz vector optimization problems: second-order constraint qualifications, regularity condition and KKT necessary optimality conditions, Positivity, 24 (2020), 313-337.  doi: 10.1007/s11117-019-00679-z.  Google Scholar [22] J. J. Ye, D. Zhu and Q. Zhu, Exact penalization and necessary optimality conditions for generalized bilevel programming problems, SIAM Journal on Optimization, 7 (1997), 481-507.  doi: 10.1137/S1052623493257344.  Google Scholar [23] J. J. Ye, Nondifferentiable multiplier rules for optimization and bilevel optimization problems, SIAM Journal on Optimization, 15 (2004), 252-274.  doi: 10.1137/S1052623403424193.  Google Scholar [24] J. Yu, Essential weak efficient solution in multiobjective optimization problems, Journal of Mathematical Analysis and Applications, 166 (1992), 230-235.  doi: 10.1016/0022-247X(92)90338-E.  Google Scholar [25] J. Zhao, The lower semicontinuity of optimal solution sets, Journal of Mathematical Analysis and Applications, 207 (1997), 240-254.  doi: 10.1006/jmaa.1997.5288.  Google Scholar [26] Y. Zheng, D. Fang and Z. Wan, A solution approach to the weak linear bilevel programming problems, Optimization, 65 (2016), 1437-1449.  doi: 10.1080/02331934.2016.1154553.  Google Scholar [27] Y. Zheng and Z. Wan, A solution method for semivectorial bilevel programming problem via penalty method, Journal of Applied Mathematics and Computing, 37 (2011), 207-219.  doi: 10.1007/s12190-010-0430-7.  Google Scholar [28] 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.  Google Scholar

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##### References:
 [1] M. J. Alves and C. H. Antunes, A differential evolution algorithm to semivectorial bilevel problems, International Workshop on Machine Learning, Optimization, and Big Data. Springer, Cham, (2017), 172-185. doi: 10.1007/978-3-319-72926-8_15.  Google Scholar [2] M. J. Alves and C. H. Antunes, A semivectorial bilevel programming approach to optimize electricity dynamic time-of-use retail pricing, Computers and Operations Research, 92 (2018), 130-144.  doi: 10.1016/j.cor.2017.12.014.  Google Scholar [3] J. F. Bard, Practical Bilevel Optimization: Algorithms and Applications, Kluwer Academic Publishers, Dordrecht, 1998. doi: 10.1007/978-1-4757-2836-1.  Google Scholar [4] H. Bonnel and J. Morgan, Semivectorial bilevel optimization problem: Penalty approach, Journal of Optimization Theory and Applications, 131 (2006), 365-382.  doi: 10.1007/s10957-006-9150-4.  Google Scholar [5] H. Bonnel, Optimality conditions for the semivectorial bilevel optimization problem, Pacific Journal of Optimization, 2 (2006), 447-467.   Google Scholar [6] H. Bonnel, L. Todjihound$\acute{e}$ and C. Udrit$\acute{e}$., Semivectorial bilevel optimization on riemannian manifolds, Journal of Optimization Theory and Applications, 167 (2015), 464-486.  doi: 10.1007/s10957-015-0789-6.  Google Scholar [7] S. Dempe, Foundations of Bilevel Programming, Kluwer Academic Publishers, Dordrecht, 2002. doi: 10.1007/b101970.  Google Scholar [8] S. Dempe and P. Mehlitz, Semivectorial bilevel programming versus scalar bilevel programming, Optimization, 69 (2020), 657-679.  doi: 10.1080/02331934.2019.1625900.  Google Scholar [9] S. Dempe, N. Gadhi and A. B. Zemkoho., New optimality conditions for the semivectorial bilevel optimization problem, Journal of Optimization Theory and Applications, 157 (2013), 54-74.  doi: 10.1007/s10957-012-0161-z.  Google Scholar [10] G. Eichfelder, Multiobjective bilevel optimization, Mathematical Programming, 123 (2010), 419-449.  doi: 10.1007/s10107-008-0259-0.  Google Scholar [11] W. W Hogan, Point-to-set maps in mathematical programming, SIAM Review, 15 (1973), 591-603.  doi: 10.1137/1015073.  Google Scholar [12] G. Li, Z. Wan and X. Zhao, Optimality conditions for bilevel optimization problem with both levels problems being multiobjective, Pacific Journal of Optimization, 13 (2017), 421-441.   Google Scholar [13] G. Li and Z. Wan, On bilevel programs with a convex lower-level problem violating slater's constraint qualification, Journal of Optimization Theory and Applications, 179 (2018), 820-837.  doi: 10.1007/s10957-018-1392-4.  Google Scholar [14] B. Liu, Z. Wan, J. Chen and G. Wang., Optimality conditions for pessimistic semivectorial bilevel programming problems, Journal of Inequalities and Applications, 2014 (2014), 1-26.  doi: 10.1186/1029-242X-2014-41.  Google Scholar [15] M. B. Lignola and J. Morgan, Topological existence and stability for stackelberg problems, Journal of Optimization Theory and Applications, 84 (1995), 145-169.  doi: 10.1007/BF02191740.  Google Scholar [16] Y. Lv and Z. Wan, Linear bilevel multiobjective optimization problem: penalty approach, Journal of Industrial and Management Optimization, 15 (2019), 1213-1223.  doi: 10.3934/jimo.2018092.  Google Scholar [17] Z. Y. Peng, J. W. Peng, X. J. Long and J. C. Yao, On the stability of solutions for semi-infinite vector optimization problems, Journal of Global Optimization, 70 (2018), 55-69.  doi: 10.1007/s10898-017-0553-6.  Google Scholar [18] T. Tanino, Stability and sensitivity analysis in multiobjective nonlinear programming, Annals of Operations Research, 27 (1990), 97-114.  doi: 10.1007/BF02055192.  Google Scholar [19] T. Tanino and Y. Sawaragi, Stability of nondominated solutions in multicriteria decision-making, Journal of Optimization Theory and Applications, 30 (1980), 229-253.  doi: 10.1007/BF00934497.  Google Scholar [20] G. Wang, X. Wang, Z. Wan and Y. Lv, A globally convergent algorithm for a class of bilevel nonlinear programming problem, Applied Mathematics and Computation, 188 (2007), 166-172.  doi: 10.1016/j.amc.2006.09.130.  Google Scholar [21] Y.-B. Xiao, T. N. Van and J.-C. Yao, Locally Lipschitz vector optimization problems: second-order constraint qualifications, regularity condition and KKT necessary optimality conditions, Positivity, 24 (2020), 313-337.  doi: 10.1007/s11117-019-00679-z.  Google Scholar [22] J. J. Ye, D. Zhu and Q. Zhu, Exact penalization and necessary optimality conditions for generalized bilevel programming problems, SIAM Journal on Optimization, 7 (1997), 481-507.  doi: 10.1137/S1052623493257344.  Google Scholar [23] J. J. Ye, Nondifferentiable multiplier rules for optimization and bilevel optimization problems, SIAM Journal on Optimization, 15 (2004), 252-274.  doi: 10.1137/S1052623403424193.  Google Scholar [24] J. Yu, Essential weak efficient solution in multiobjective optimization problems, Journal of Mathematical Analysis and Applications, 166 (1992), 230-235.  doi: 10.1016/0022-247X(92)90338-E.  Google Scholar [25] J. Zhao, The lower semicontinuity of optimal solution sets, Journal of Mathematical Analysis and Applications, 207 (1997), 240-254.  doi: 10.1006/jmaa.1997.5288.  Google Scholar [26] Y. Zheng, D. Fang and Z. Wan, A solution approach to the weak linear bilevel programming problems, Optimization, 65 (2016), 1437-1449.  doi: 10.1080/02331934.2016.1154553.  Google Scholar [27] Y. Zheng and Z. Wan, A solution method for semivectorial bilevel programming problem via penalty method, Journal of Applied Mathematics and Computing, 37 (2011), 207-219.  doi: 10.1007/s12190-010-0430-7.  Google Scholar [28] 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.  Google Scholar
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