Optimality conditions and duality for multiobjective semi-infinite programming with data uncertainty via Mordukhovich subdifferential

Authors

  • Thanh-Hung Pham Faculty of Pedagogy and Faculty of Social Sciences & Humanities, Kien Giang University, Kien Giang Province, Vietnam

DOI:

https://doi.org/10.2298/YJOR201017013P

Keywords:

Robust Optimality Condition, Semi-Infinite Programming, Constraint Qualification, Generalized Convexity, Mordukhovich Subdifferential

Abstract

Based on the notation of Mordukhovich subdifferential in [27], we propose some of new concepts of convexity to establish optimality conditions for quasi ε−solutions for nonlinear semi-infinite optimization problems with data uncertainty in constraints. Moreover, some examples are given to illustrate the obtained results.

This article has been corrected. Link to the correction 10.2298/YJOR211117025E

References

Ben-Tal, A., Ghaoui, L. E., Nemirovski, A., Robust Optimization, Princeton: Princeton University Press, 2009.

Bertsimas, D., Brown, D. B., Caramanis, C., “Theory and applications of robust optimization”, SIAM Review, 53 (3) (2011) 464–501.

Chen, J. W., Köbis, E., Yao, J. C., “Optimality conditions and duality for robust nonsmooth multiobjective optimization problems with constraints”, Journal of Optimization Theory and Applications, 181 (2) (2019) 411–436.

Chuong, T. D., Kim, D. S., “Nonsmooth semi-infinite multiobjective optimization problems”, Journal of Optimization Theory and Applications, 160 (3) (2014) 748–762.

Chuong, T. D., Yao, J. C., “Isolated and proper efficiencies in semi-infinite vector optimization problems”, Journal of Optimization Theory and Applications, 162 (2) (2014) 447–462.

Chuong, T. D., Kim, D. S., “Approximate solutions of multiobjective optimization problems”, Positivity, 20 (1) (2016) 187–207.

Chuong, T. D., “Optimality and duality for robust multiobjective optimization problems”, Nonlinear Analysis, 134 (2016) 127–143.

Fakhara, M., Mahyariniab, M. R., Zafarani, J., “On nonsmooth robust multiobjective optimization under generalized convexity with applications to portfolio optimization”, European Journal of Operational Research, 265 (1) (2018) 39–48.

Fakhara, M., Mahyariniab, M. R., Zafarani, J., “On approximate solutions for nonsmooth robust multiobjective optimization problems”, Optimization, 68 (9) (2019) 1653–1683.

Goberna, M. A., López, M. A., Linear Semi-Infinite Optimization, Chichester, Wiley, 1998.

Goberna, M. A., Jeyakumar, V., Li, G., López, M. A., “Robust linear semi-infinite programming duality under uncertainty”, Mathematical Programming, 139 (1) (2013) 185–203.

Goberna, M. A., López, M. A., “Recent contributions to linear semi-infinite optimization: an update”, Annals of Operations Research, 271 (1) (2018) 237–278.

Jiao, L., Lee, J. H., “Approximate optimality and approximate duality for quasi approximate solutions in robust convex semidefinite programs”, Journal of Optimization Theory and Applications, 176 (1) (2018) 74–93.

Jiao, L., Kim, D. S., Zhou, Y., “Quasi ε−solutions in a semi-infinite programming problem with locally Lipschitz data”, Optimization Letters, 15 (2) (2021) 1759–1772.

Joshi, B. C., Mishra, S. K., Kumar, P., “On semi-infinite Mathematical Programming Problems with Equilibrium Constraints using Generalized Convexity”, Journal of Operations Research Society of China, 8 (4) (2020) 619–636.

Joshi, B. C., “Optimality and duality for nonsmooth semi-infinite mathematical program with equilibrium constraints involving generalized invexity of order σ>0”, RAIRO Operations Research, 55 (4) (2021) 2221-2240.

Joshi, B. C., “On Generalized Approximate Convex Functions and Variational Inequalities”, RAIRO- Operations Research, 55 (5) (2020) 2999-3008.

Kanzi, N., “Constraint qualifications in semi-infinite systems and their applications in nonsmooth semi-infinite problems with mixed constraints”, SIAM Journal on Optimization, 24 (2) (2014) 559–572.

Kanzi, N., Nobakhtian, S., “Optimality conditions for nonsmooth semi-infinite multiobjective programming”, Optimization Letters, 8 (4) (2014) 1517–1528.

Khantree, C., Wangkeeree, R., “On quasi approximate solutions for nonsmooth robust semiinfinite optimization problems”, Carpathian Journal of Mathematics, 35 (3) (2019) 417–426.

Lee, J. H., Jiao, L., “On quasi ε−solution for robust convex optimization problems”, Optimization Letters, 11 (8) (2017) 1609-1622.

Lee, J. H., Lee, G. M., “On ε−solutions for robust semi-infinite optimization problems”, Positivity, 23 (3) (2019) 651–669.

Long, X. J., Xiao, Y. B., Huang, N. J., “Optimality conditions of approximate solutions for nonsmooth semi-infinite programming problems”, Journal of the Operations Research Society of China, 6 (2) (2018) 289–299.

López, M., Still, G., “Semi-infinite programming”, European Journal of Operational Research, 180 (1) (2007) 491–518.

Loridan, P., “Necessary conditions for ε−optimality”, Mathematical Programming Study, 19 (1982) 140–152.

Loridan, P., “ε−solutions in vector minimization problems”, Journal of Optimization Theory and Applications, 42 (1984) 265–276.

Mordukhovich, B. S., Variational Analysis and Generalized Differentiation. I: Basic Theory, Springer, Berlin, 2006.

Mordukhovich, B. S., Nghia, T. T. A., “Constraint qualifications and optimality conditions for nonconvex semi-infinite and infinite programs”, Mathematical Programming, 139 (1-2) (2013) 271–300.

Son, T. Q., Tuyen, N. V., Wen, C. F., “Optimality conditions for approximate Pareto solutions of a nonsmooth vector optimization problem with an infinite number of constraints”, Acta Mathematica Vietnamica, 45 (2) (2020) 435–448.

Sun, X. K., Ten, K. L., Zeng, J., Guo, X., “On approximate solutions and saddle point theorems for robust convex optimization”, Optimization Letters, 14 (7) (2020) 1711-1730.

Sun, X. K., Fu, H. Y., Zeng, J., “Robust approximate optimality conditions for uncertain nonsmooth optimization with infinite number of constraints”, Mathematics, 7 (1) (2019) 1-14.

Sun, X. K., Tang, L. P., Zeng, J., “Characterizations of approximate duality and saddle point theorems for nonsmooth robust vector optimization”, Numerical Functional Analysis and Optimization, 41 (4) (2020) 462–482.

Sun, X. K., Teo, K. L., Zeng, J., Liu, L., “Robust approximate optimal solutions for nonlinear semi-infinite programming with uncertainty”, Optimization, 69 (9) (2020) 1-21.

Downloads

Published

2021-11-01

Issue

Section

Research Articles