ε-Quasi-Weakly Solution for Semi-infinite Vector Optimization Problems with Data Uncertainty

doi:10.1007/s40305-023-00489-x

Abstract

Abstract: This paper is concerned with an ε-quasi-weakly solution for a semi-infinite vector optimization problem with data uncertainty in constraints by using the Clarke subdifferential. Both necessary and sufficient optimality conditions for a semi-infinite vector optimization problem with data uncertainty in constraints are established. We also investigate a Mond–Weir-type dual problem with respect to the primal problem. An application to a fractional semi-infinite vector optimization problem with data uncertainty in constraints is provided. Some examples are given to illustrate our results.

Key words: Optimality condition, Duality theorem, Semi-infinite multiobjective/vector optimization, Generalized convexity, Data uncertainty

CLC Number:

Thanh-Hung Pham, Thanh-Sang Nguyen. ε-Quasi-Weakly Solution for Semi-infinite Vector Optimization Problems with Data Uncertainty[J]. Journal of the Operations Research Society of China, 2025, 13(2): 415-439.

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References

[1] Goberna, M.A., López, M.A.: Linear Semi-Infinite Optimization. Wiley, Chichester (1998)
[2] López, M., Still, G.: Semi-infinite programming. Eur. J. Oper. Res. 180, 491-518 (2007)
[3] Chuong, T.D., Kim, D.S.: Nonsmooth semi-infinite multiobjective optimization problems. J. Optim. Theory Appl. 160, 748-762 (2014)
[4] Chuong, T.D., Yao, J.-C.: Isolated and proper efficiencies in semi-infinite vector optimization problems. J. Optim. Theory Appl. 162, 447-462 (2014)
[5] Chuong, T.D.: Nondifferentiable fractional semi-infinite multiobjective optimization problems. Oper. Res. Lett. 44, 260-266 (2016)
[6] Correa, R., López, M.A., Pérez-Aros, P.: Necessary and sufficient optimality conditions in DC Semi-infinite programming. SIAM J. Optim. 31, 837-865 (2021)
[7] Goberna, M.A., Kanzi, N.: Optimality conditions in convex multiobjective SIP. Math. Program. Ser. A 164, 167-191 (2017)
[8] Goberna, M.A., López, M.A.: Recent contributions to linear semi-infinite optimization: an update. Ann. Oper. Res. 271, 237-278 (2018)
[9] Jiao, L.G., Dinh, B.V., Kim, D.S., Yoon, M.: Mixed type duality for a class of multiple objective optimization problems with an infinite number of constraints. J. Nonlinear Convex Anal. 21, 49-61 (2020)
[10] Khanh, P.Q., Tung, N.M.: On the Mangasarian-Fromovitz constraint qualification and Karush-Kuhn-Tucker conditions in nonsmooth semi-infinite multiobjective programming. Optim. Lett. 14, 2055-2072 (2020)
[11] Kanzi, N.: Constraint qualifications in semi-infinite systems and their applications in nonsmooth semi-infinite problems with mixed constraints. SIAM J. Optim. 24, 559-572 (2014)
[12] Kanzi, N., Nobakhtian, S.: Optimality conditions for nonsmooth semi-infinite multiobjective programming. Optim. Lett. 8, 1517-1528 (2014)
[13] Kanzi, N.: On strong KKT optimality conditions for multiobjective semi-infinite programming problems with Lipschitzian data. Optim. Lett. 9, 1121-1129 (2015)
[14] Mordukhovich, B.S., Pérez-Aros, P.: New extremal principles with applications to stochastic and semi-infinite programming. Math. Program. (2020). https://doi.org/10.1007/s10107-020-01548-4
[15] Tung, L.T.: Karush-Kuhn-Tucker optimality conditions and duality for multiobjective semi-infinite programming via tangential subdifferentials. Numer. Funct. Anal. Optim. 41, 659-684 (2020)
[16] Tung, L.T.: Strong Karush-Kuhn-Tucker optimality conditions for Borwein properly efficient solutions of multiobjective semi-infinite programming. Bull. Braz. Math. Soc. 52, 1-22 (2021)
[17] Ben-Tal, A., Ghaoui, L.E., Nemirovski, A.: Robust Optimization: Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2009)
[18] Bertsimas, D., Brown, D., Caramanis, C.: Theory and applications of robust optimization. SIAM Rev. 53(3), 464-501 (2011)
[19] Chen, J.W., Köbis, E., Yao, J.C.: Optimality conditions and duality for robust nonsmooth multiobjective optimization problems with constraints. J. Optim. Theory Appl. 181, 411-436 (2019)
[20] Chuong, T.D.: Optimality and duality for robust multiobjective optimization problems. Nonlinear Anal. 134, 127-143 (2016)
[21] Chuong, T.D.: Robust optimality and duality in multiobjective optimization problems under data uncertainty. SIAM J. Optim. 30, 1501-1526 (2020)
[22] Dinh, N., Goberna, M.A., Lopez, M.A., Volle, M.: A unifying approach to robust convex infinite optimization duality. J. Optim. Theory Appl. 174, 650-685 (2017)
[23] Dinh, N., Long, D.H., Yao, J.C.: Duality for robust linear infinite programming problems revisited. Vietnam J. Math. 46, 293-328 (2020)
[24] Fakhar, M., Mahyarinia, M.R., Zafarani, J.: On nonsmooth robust multiobjective optimization under generalized convexity with applications to portfolio optimization. Eur. J. Oper. Res. 265, 39-48 (2018)
[25] Goberna, M.A., Jeyakumar, V., Li, G., López, M.: Robust linear semi-infinite programming duality. Math. Program Ser. B 139, 185-203 (2013)
[26] Goberna, M.A., Jeyakumar, V., Li, G., Vicente-Pérez, J.: Robust solutions of multiobjective linear semi-infinite programs under constraint data uncertainty. SIAM J. Optim. 24, 1402-1419 (2014)
[27] Kerdkaew, J., Wangkeeree, R., Lee, G.M.: On optimality conditions for robust weak sharp solution in uncertain optimizations. Carpathian J. Math. 36, 443-452 (2020)
[28] Lee, J.H., Lee, G.M.: On optimality conditions and duality theorems for robust semi-infinite multiobjective optimization problems. Ann. Oper. Res. 269, 419-438 (2018)
[29] Mashkoorzadeh, F., Movahedian, N., Nobakhtian, S.: Robustness in nonsmooth nonconvex optimization problems. Positivity 25, 701-729 (2021)
[30] Loridan, P.: Necessary conditions for ε-optimality. Optimality and stability in mathematical programming. Math. Program. Study 19, 140-152 (1982)
[31] Loridan, P.: ε-solutions in vector minimization problems. J. Optim. Theory Appl. 43, 265-276 (1984)
[32] Chuong, T.D., Kim, D.S.: Approximate solutions of multiobjective optimization problems. Positivity 20, 187-207 (2016)
[33] Fakhara, M., Mahyarinia, M.R., Zafarani, J.: On approximate solutions for nonsmooth robust multiobjective optimization problems. Optimization 68, 1653-1683 (2019)
[34] Sun, X.K., Tang, L.P., Zeng, J.: Characterizations of approximate duality and saddle point theorems for nonsmooth robust vector optimization. Numer. Funct. Anal. Optim. 41, 462-482 (2020)
[35] Sun, X.K., Teo, K.L., Long, X.J.: Characterizations of robust ε-quasi optimal solutions for nonsmooth optimization problems with uncertain data. Optimization 70, 847-870 (2021)
[36] Long, X.J., Xiao, Y.B., Huang, N.J.: Optimality conditions of approximate solutions for nonsmooth semi-infinite programming problems. J. Oper. Res. Soc. China 6, 289-299 (2018)
[37] Kim, D.S., Son, T.Q.: An approach to ε-duality theorems for nonconvex semi-infinite multiobjective optimization problems. Taiwan. J. Math. 22, 1261-1287 (2018)
[38] 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 Math. Vietnam 45, 435-448 (2020)
[39] Shitkovskaya, T., Hong, Z., Kim, D.S., Piao, G.R.: Approximate necessary optimality in fractional semi-infinite multiobjective optimization. J. Nonlinear Convex Anal. 21, 195-204 (2020)
[40] Lee, J.H., Lee, G.M.: On ε-solutions for robust semi-infinite optimization problems. Positivity 23, 651-669 (2019)
[41] Khantree, C., Wangkeeree, R.: On quasi approximate solutions for nonsmooth robust semiinfinite optimization problems. Carpathian J. Math. 35, 417-426 (2019)
[42] Sun, X.K., Teo, K.L., Zheng, J., Liu, L.: Robust approximate optimal solutions for nonlinear semi-infinite programming with uncertainty. Optimization 69, 2109-2020 (2020)
[43] Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983)
[44] Rockafellar, R.T.: Convex Analysis. Princeton Landmarks in Mathematics. Princeton University Press, Princeton (1997)