Optimality Conditions for Approximate Pareto Solutions of a Nonsmooth Vector Optimization Problem with an Infinite Number of Constraints


In this paper, we present some new necessary and sufficient optimality conditions in terms of Clarke subdifferentials for approximate Pareto solutions of a nonsmooth vector optimization problem which has an infinite number of constraints. As a consequence, we obtain optimality conditions for the particular cases of cone-constrained convex vector optimization problems and semidefinite vector optimization problems. Examples are given to illustrate the obtained results.

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  1. 1.

    Araya, Y.: Ekeland’s variational principle and its equivalent theorems in vector optimization. J. Math. Anal. Appl. 346, 9–16 (2008)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer (2000)

  3. 3.

    Blekherman, G., Parrilo, P.A., Thomas, R.R.: Semidefinite Optimization and Convex Algebraic Geometry. SIAM, Philadelphia (2013)

    Google Scholar 

  4. 4.

    Canovas, M.J., López, M.A., Mordukhovich, B.S., Parra, J.: Variational analysis in semi-infinite and infinite programming. I. Stability of linear inequality systems of feasible solutions. SIAM J. Optim. 20, 1504–1526 (2009)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Caristi, G., Ferrara, M., Stefanescu, A.: Semi-infinite multiobjective programming with generalized invexity. Math. Rep. 12, 217–233 (2010)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Chankong, V., Haimes, Y.Y.: Multiobjective Decision Making. North-Holland Publishing Co., New York (1983)

    Google Scholar 

  7. 7.

    Chuong, T.D., Huy, N.Q., Yao, J.-C.: Subdifferentials of marginal functions in semi-infinite programming. SIAM J. Optim. 20, 1462–1477 (2009)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Chuong, T.D., Huy, N.Q., Yao, J.-C.: Stability of semi-infinite vector optimization problems under functional perturbations. J. Glob. Optim. 45, 583–595 (2009)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Chuong, T.D., Huy, N.Q., Yao, J.-C.: Pseudo-lipschitz property of linear semi-infinite vector optimization problems. Eur. J. Oper. Res. 200, 639–644 (2010)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Chuong, T.D., Kim, D.S.: Nonsmooth semi-infinite multiobjective optimization problems. J. Optim. Theory Appl. 160, 748–762 (2014)

    MathSciNet  Article  Google Scholar 

  11. 11.

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

    MathSciNet  Article  Google Scholar 

  12. 12.

    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983)

    Google Scholar 

  13. 13.

    Dinh, N., Goberna, M.A., López, M. A., Son, T.Q.: New Farkas-type constraint qualifications in convex infinite programming. ESAIM Control Optim. Calc. Var. 13, 580–597 (2007)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Dinh, N., Mordukhovich, B.S., Nghia, T.T.A.: Qualification and optimality conditions for DC programs with infinite constraints. Acta. Math. Vietnam. 34, 123–153 (2009)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Gorberna, M.A., López, M. A.: Linear Semi-infinite Optimization. Wiley, Chichester (1998)

    Google Scholar 

  17. 17.

    Ha, T.X.D.: Variants of the Ekeland variational principle for a set-valued map involving the Clarke normal cone. J. Math. Anal. Appl. 316, 346–356 (2006)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Hettich, R., Kortanek, K.O.: Semi-infinite programming: theory, methods, and applications. SIAM Rev. 35, 380–429 (1993)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Hiriart-Urruty, J.B.: On optimality conditions in nondifferentiable programming. Math. Program. 14, 73–86 (1978)

    Article  Google Scholar 

  20. 20.

    Huy, N.Q., Kim, D.S., Tuyen, N.V.: Existence theorems in vector optimization with generalized order. J. Optim. Theory Appl. 174, 728–745 (2017)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Ioffe, A.D., Tikhomirov, V.M.: Theory of Extremal Problems. Stud. Math Appl. 6, North-Holland, Amsterdam (1979)

  22. 22.

    Jahn, J.: Existence theorems in vector optimization. J. Optim. Theory Appl. 50, 397–406 (1986)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Kanzi, N., Nobakhtian, S.: Optimality conditions for nonsmooth semi-infinite programming. Optimization 59, 717–727 (2010)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Kanzi, N.: Constraint qualifications in semi-infinite systems and their applications in nonsmooth semi-infinite programs with mixed constraints. SIAM J. Optim. 24, 559–572 (2014)

    MathSciNet  Article  Google Scholar 

  25. 25.

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

    MathSciNet  Article  Google Scholar 

  26. 26.

    Kim, D.S., Phạm, T.S., Tuyen, N.V.: On the existence of Pareto solutions for polynomial vector optimization problems. Math. Program. 177, 321–341 (2019)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Kim, D.S., Mordukhovich, B.S., Phạm, T.S., Tuyen, N.V.: Existence of efficient and properly efficient solutions to problems of constrained vector optimization. https://arxiv.org/abs/1805.00298

  28. 28.

    Lee, G.M., Kim, G.S., Dinh, N.: Optimalityconditionsforapproximatesolutionsofconvexsemi-infinitevectoroptimizationproblems. In: Ansari, Q.H., Yao, J.-C. (eds.) RecentDevelopmentsinVectorOptimization,VectorOptimization, vol. 1, pp 275–295. Springer, Berlin (2012)

  29. 29.

    Li, C., Zhao, X.P., Hu, Y.H.: Quasi-slaterandFarkas–Minkowskiqualificationsforsemi-infiniteprogrammingwithapplications. SIAMJ.Optim. 23, 2208–2230 (2013)

    Article  Google Scholar 

  30. 30.

    Long, X.J., Xiao, Y.B., Huang, N.J.: Optimalityconditionsofapproximatesolutionsfornonsmoothsemi-infiniteprogrammingproblems. J.Oper.Res.Soc.China 6, 289–299 (2018)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Loridan, P.: Necessaryconditionsforε-optimality. Math.Program.Study 19, 140–152 (1982)

    Article  Google Scholar 

  32. 32.

    Loridan, P.: 𝜖-solutionsinvectorminimizationproblems. J.Optim.TheoryAppl. 43, 265–276 (1984)

    Article  Google Scholar 

  33. 33.

    Luc, D.T.: TheoryofVectorOptimization.Springer (1989)

  34. 34.

    Mishra, S.K., Jaiswal, M., LeThi, H.A.: Nonsmoothsemi-infiniteprogrammingproblemusinglimitingsubdifferentials. J.Glob.Optim. 53, 285–296 (2012)

    Article  Google Scholar 

  35. 35.

    Reemtsen, R., Rückmann, J.J.: Semi-InfiniteProgrammingNonconvexOptimizationandItsApplications, vol. 25. KluwerAcademicPublishers, Boston (1998)

    Google Scholar 

  36. 36.

    Son, T.Q., Strodiot, J.J., Nguyen, V.H.: ε-Optimalityandε-Lagrangiandualityforanonconvexprogrammingproblemwithaninfinitenumberofconstraints. J.Optim.TheoryAppl. 141, 389–409 (2009)

    Article  Google Scholar 

  37. 37.

    Shitkovskaya, T., Kim, D.S.: ε-efficientsolutionsinsemi-infinitemultiobjectiveoptimization. RAIROOper.Res. 52, 1397–1410 (2018)

    Google Scholar 

  38. 38.

    Tuyen, N.V.: ConvergenceoftherelativeParetoefficientsets. TaiwaneseJ.Math. 20, 1149–1173 (2016)

    Article  Google Scholar 

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The authors would like to thank the anonymous referee and the handling Associate Editor for their valuable remarks and detailed suggestions that allowed us to improve the original version.


The research of Ta Quang Son was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01-2017.08. The research of Nguyen Van Tuyen was supported by the Ministry of Education and Training of Vietnam (grant number B2018-SP2-14). The research of Ching-Feng Wen was supported by the Taiwan MOST (grant number 107-2115-M-037-001) as well as the grant from Research Center for Nonlinear Analysis and Optimization, Kaohsiung Medical University, Taiwan.

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Son, T.Q., Van Tuyen, N. & Wen, C. 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). https://doi.org/10.1007/s40306-019-00358-x

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  • Approximate Pareto solutions
  • Optimality conditions
  • Clarke subdifferential
  • Semi-infinite vector optimization
  • Infinite vector optimization

Mathematics Subject Classification (2010)

  • 41A65
  • 65K10
  • 90C34
  • 90C29
  • 90C46