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Some Characterizations of Approximate Solutions for Robust Semi-infinite Optimization Problems

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Abstract

This paper deals with robust \(\varepsilon \)-quasi Pareto efficient solutions of an uncertain semi-infinite multiobjective optimization problem. By using robust optimization and a modified \(\varepsilon \)-constraint scalarization methodology, we first present the relationship between robust \(\varepsilon \)-quasi solutions of the uncertain optimization problem and that of its corresponding scalar optimization problem. Then, we obtain necessary optimality conditions for robust \(\varepsilon \)-quasi Pareto efficient solutions of the uncertain optimization problem in terms of a new robust-type subdifferential constraint qualification. We also deduce sufficient optimality conditions for robust \(\varepsilon \)-quasi Pareto efficient solutions of the uncertain optimization problem under assumptions of generalized convexity. Besides, we introduce a Mixed-type robust \(\varepsilon \)-multiobjective dual problem (including Wolfe type and Mond-Weir type dual problems as special cases) of the uncertain optimization problem, and explore robust \(\varepsilon \)-quasi weak, \(\varepsilon \)-quasi strong, and \(\varepsilon \)-quasi converse duality properties. Furthermore, we introduce an \(\varepsilon \)-quasi saddle point for the uncertain optimization problem and investigate the relationships between the \(\varepsilon \)-quasi saddle point and the robust \(\varepsilon \)-quasi Pareto efficient solution for the uncertain optimization problem.

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References

  1. Ben-Tal, A., Nemirovski, A.: Robust optimization-methodology and applications. Math. Program. Ser. B 92, 453–480 (2002)

    Article  MathSciNet  Google Scholar 

  2. Chankong, V., Haimes, Y.Y.: Multiobjective Decision Making: Theory and Methodology. North-Holland, New York (1983)

    MATH  Google Scholar 

  3. 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)

    Article  MathSciNet  Google Scholar 

  4. Chen, J.W., Li, J., Li, X.B., Lv, Y., Yao, J.C.: Radius of robust feasibility of system of convex inequalities with uncertain data. J. Optim. Theory Appl. 184, 384–399 (2020)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  6. Clarke, F.H.: Optimization and Nonsmooth Analysis. Willey, New York (1983)

    MATH  Google Scholar 

  7. Fakhar, M., Mahyarinia, M.R., Zafarani, J.: On approximate solutions for nonsmooth robust multiobjective optimization problems. Optimization 68, 1653–1683 (2019)

    Article  MathSciNet  Google Scholar 

  8. Fang, D.H., Li, C., Ng, K.F.: Constraint qualifications for extended Farkass lemmas and Lagrangian dualities in convex infinite programming. SIAM J. Optim. 20, 1311–1332 (2009)

    Article  MathSciNet  Google Scholar 

  9. Fang, D.H., Li, C., Ng, K.F.: Constraint qualifications for optimality conditions and total Lagrange dualities in convex infinite programming. Nonlinear Anal. 73, 1143–1159 (2010)

    Article  MathSciNet  Google Scholar 

  10. Fang, D.H., Li, C., Yao, J.C.: Stable Lagrange dualities for robust conical programming. J. Nonlinear Convex Anal. 16, 2141–2158 (2015)

    MathSciNet  MATH  Google Scholar 

  11. Fang, D.H., Zhao, X.P.: Local and global optimality conditions for DC infinite optimization problems. Taiwanese J. Math. 18, 817–834 (2014)

    Article  MathSciNet  Google Scholar 

  12. Fliege, J., Werner, R.: Robust multiobjective optimization and applications in portfolio optimization. Eur. J. Oper. Res. 234, 422–433 (2014)

    Article  MathSciNet  Google Scholar 

  13. Gabrel, V., Murat, C., Thiele, A.: Recent advances in robust optimization: an overview. Eur. J. Oper. Res. 235, 471–483 (2014)

    Article  MathSciNet  Google Scholar 

  14. Goberna, M., López, M.A.: Linear Semi-Infinite Optimization. Wiley, Chichester (1998)

    MATH  Google Scholar 

  15. Guo, T.T., Yu, G.L.: Optimality conditions of the approximate quasi-weak robust efficiency for uncertain multi-objective convex optimization. Pac. J. Optim. 15, 623–638 (2019)

    MathSciNet  MATH  Google Scholar 

  16. Ide, J., Schöbel, A.: Robustness for uncertain multiobjective optimization: a survey and analysis of different concepts. OR Spectrum. 38, 235–271 (2016)

    Article  Google Scholar 

  17. Khantree, C., Wangkeeree, R.: On quasi approximate solutions for nonsmooth robust semi-infinite optimization problems. Carpathian J. Math. 35, 417–426 (2019)

    Article  MathSciNet  Google Scholar 

  18. Kim, D.S., Son, T.Q.: An approach to \(\varepsilon \)-duality theorems for nonconvex semi-infinite multiobjective optimization problems. Taiwan J. Math. 22, 1261–1287 (2018)

    MathSciNet  MATH  Google Scholar 

  19. Lee, G.M., Son, P.T.: On nonsmooth optimality theorems for robust optimization problems. Bull. Korean Math. Soc. 51, 287–301 (2014)

    Article  MathSciNet  Google Scholar 

  20. Long, X.J., Liu, J., Huang, N.J.: Characterizing the solution set for nonconvex semi-infinite programs involving tangential subdifferentials. Numer. Funct. Anal. Optim. 42, 279–297 (2021)

    Article  MathSciNet  Google Scholar 

  21. Long, X.J., Peng, Z.Y., Wang, X.: Stable Farkas lemmas and duality for nonconvex composite semi-infinite programming problems. Pac. J. Optim. 15, 295–315 (2019)

    MathSciNet  MATH  Google Scholar 

  22. Long, X.J., Tang, L.P., Peng, J.W.: Optimality conditions for semi-infinite programming problems under relaxed quasiconvexity assumptions. Pac. J. Optim. 15, 519–528 (2019)

    MathSciNet  MATH  Google Scholar 

  23. 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)

    Article  MathSciNet  Google Scholar 

  24. Ou, X.Q., Chen, J.W., Deng, G.J., Lv, Y.B.: Necessary optimality conditions of approximated robust solutions of multi-criteria decision making problem with uncertainty data. J. Nonlinear Convex Anal. 21, 909–922 (2020)

    MathSciNet  MATH  Google Scholar 

  25. Peng, Z.Y., Peng, J.W., Long, X.J., Yao, J.C.: On the stability of solutions for semi-infinite vector optimization problems. J. Glob. Optim. 70, 55–69 (2018)

    Article  MathSciNet  Google Scholar 

  26. Peng, Z.Y., Wang, X., Yang, X.M.: Connectedness of approximate efficient solutions for generalized semi-infinite vector optimization problems. Set-Valued Var. Anal. 27, 103–118 (2019)

    Article  MathSciNet  Google Scholar 

  27. Piao, G.R., Jiao, L.G., Kim, D.S.: Optimality conditions in nonconvex semi-infinite multiobjective optimization problems. J. Nonlinear Convex Anal. 17, 167–175 (2016)

    MathSciNet  MATH  Google Scholar 

  28. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Book  Google Scholar 

  29. Shitkovskaya, T., Kim, D.S.: \(\varepsilon \)-Efficient solutions in semi-infinite multiobjective optimization. RAIRO-Oper. Res. 52, 1397–1410 (2018)

    Article  MathSciNet  Google Scholar 

  30. Son, T.Q., Kim, D.S.: \(\varepsilon \)-Mixed type duality for nonconvex multiobjective programs with an infinite number of constraints. J. Glob. Optim. 57, 447–465 (2013)

    Article  MathSciNet  Google Scholar 

  31. Son, T.Q., Kim, D.S.: A new approach to characterize the solution set of a pseudoconvex programming problem. J. Comput. Appl. Math. 261, 333–340 (2014)

    Article  MathSciNet  Google Scholar 

  32. 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)

    Article  MathSciNet  Google Scholar 

  33. Sun, X.K., Li, X.B., Long, X.J., Peng, Z.Y.: On robust approximate optimal solutions for uncertain convex optimization and applications to multi-objective optimization. Pac. J. Optim. 13, 621–643 (2017)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  35. Sun, X.K., Teo, K.L., Tang, L.P.: Dual approaches to characterize robust optimal solution sets for a class of uncertain optimization problems. J. Optim. Theory Appl. 182, 984–1000 (2019)

    Article  MathSciNet  Google Scholar 

  36. Sun, X.K., Teo, K.L., Zeng, J., Guo, X.L.: On approximate solutions and saddle point theorems for robust convex optimization. Optim. Lett. 14, 1711–1730 (2020)

  37. Sun, X.K., Teo, K.L., Zeng, J., Liu, L.Y.: Robust approximate optimal solutions for nonlinear semi-infinite programming with uncertainty. Optimization 69, 2109–2129 (2020)

    Article  MathSciNet  Google Scholar 

  38. Wei, H.Z., Chen, C.R., Li, S.J.: Characterizations for optimality conditions of general robust optimization problems. J. Optim. Theory Appl. 177, 835–856 (2018)

    Article  MathSciNet  Google Scholar 

  39. Wei, H.Z., Chen, C.R., Li, S.J.: A unified characterization of multiobjective robustness via separation. J. Optim. Theory Appl. 179, 86–102 (2018)

    Article  MathSciNet  Google Scholar 

  40. Wei, H.Z., Chen, C.R., Li, S.J.: A unified approach through image space analysis to robustness in uncertain optimization problems. J. Optim. Theory Appl. 184, 466–493 (2020)

    Article  MathSciNet  Google Scholar 

  41. Woolnough, D., Jeyakumar, N., Li, G., Loy, C.T., Jeyakumar, V.: Robust optimization and data classification for characterization of Huntington disease onset via duality methods. J. Optim. Theory Appl. https://doi.org/10.1007/s10957-021-01835-w

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Acknowledgements

The authors would like to express their sincere thanks to the anonymous reviewers and the Associate Editor for many valuable comments and suggestions which have contributed to the final preparation of this paper.

Funding

This research was supported by the Natural Science Foundation of Chongqing (cstc2020jcyj-msxmX0016), the ARC Discovery Grant (DP190103361), the Science and Technology Research Program of Chongqing Municipal Education Commission (KJZDK201900801), the Education Committee Project Foundation of Chongqing for Bayu Young Scholar, and the Project of CTBU (ZDPTTD201908).

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Authors and Affiliations

  1. Chongqing Key Laboratory of Social Economy and Applied Statistics, College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, China

    Xiangkai Sun & Xian-Jun Long

  2. School of Mathematical Sciences, Sunway University, Bandar Sunway, 47500, Selangor Darul Ehsan, Malaysia

    Kok Lay Teo

  3. Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin, 300222, China

    Kok Lay Teo

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  1. Xiangkai Sun
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  2. Kok Lay Teo
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  3. Xian-Jun Long
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Correspondence to Xiangkai Sun.

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Communicated by Marco Antonio López-Cerdá

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Sun, X., Teo, K.L. & Long, XJ. Some Characterizations of Approximate Solutions for Robust Semi-infinite Optimization Problems. J Optim Theory Appl 191, 281–310 (2021). https://doi.org/10.1007/s10957-021-01938-4

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