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A note on characterizing solution set of nonsmooth pseudoinvex optimization problem

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Abstract

In this paper, by making use of some examples, we first point out some results are not true in the earlier works of Mishra and Lai (On Characterization of Solution Sets of Nonsmooth Pseudoinvex Minimization Problems. IEEE Computer Society, pp. 739–741, 2009). Furthermore, we correct and modify corresponding results. In the end, we give an example to illustrate the main result. Our study modifies and improves some previously known results.

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References

  1. Mangasarian O.L.: A Simple characterization of solution sets of convex programs. Oper. Res. Lett. 7, 21–26 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  2. Burke J.V., Ferris M.C.: Characterization of solution sets of convex programs. Oper. Res. Lett. 10, 57–60 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  3. Jeyakumar V.: Infinite-dimensional convex programming with applications to constrained approximation. J. Optim. Theory Appl. 75, 469–586 (1992)

    Article  MathSciNet  Google Scholar 

  4. Jeyakumar V., Yang X.Q.: Convex composite multi-objective nonlinear programming. Math. Program. 59, 325–343 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  5. Jeyakumar V., Lee G.M., Dinh N.: Lagrange multiplier conditions characterizing the optimal solution Sets of cone-constrained convex programs. J. Optim. Theory Appl. 123, 83–103 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Jeyakumar V., Lee G.M., Dinh N.: Characterizations of solution sets of convex vector minimization problems. Eur. J. Oper. Res. 174, 1380–1395 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Wu Z.L., Wu S.Y.: Characterizations of the solution sets of convex programs and variational inequality problems. J. Optim. Theory Appl. 130, 339–358 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chew K.L., Choo E.V.: Pseudolinearity and efficiency. Math. Program. 28, 226–239 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  9. Papesak T.: On pseudolinear functions. Eur. J. Oper. Res. 50, 353–360 (1991)

    Article  Google Scholar 

  10. Komlosi S.: First and second-order characterizations of pseudolinear functions. Eur. J. Oper. Res. 67, 278–286 (1993)

    Article  MATH  Google Scholar 

  11. Jeyakumar V., Yang X.Q.: On characterizing the solution sets of pseudolinear programs. J. Optim. Theory Appl. 87, 747–755 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hanson M.A.: On sufficiency of the Kuhn–Tucker conditions. J. Math. Anal. Appl. 80, 545–550 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  13. Weir T., Mond B.: Preinvex functions in multi-objective optimization. J. Math. Anal. Appl. 136, 29–38 (1988)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  15. Yang X.M., Yang X.Q., Teo K.L.: Generalized invexity and generalized invariant monotonicity. J. Optim. Theory Appl. 117, 607–625 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jabarootian T., Zafarani J.: Generalized invariant monotonicity and invexity of non-differentiable functions. J. Global. Optim. 36, 537–564 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  17. Yang X.M.: On characterizing the solution sets of pseudoinvex extremum problems. J. Optim. Theory Appl. 140, 537–542 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Yang X.M., Yang X.Q., Teo K.L.: Characterizations and applications of prequasi-invex functions. J. Optim. Theory Appl. 110, 645–668 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mohan S.R., Neogy S.K.: On invex sets and preinvex functions. J. Math. Anal. Appl. 189, 901–908 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  20. Rezaie M., Zafarani J.: Vector optimization and variational-like inequalities. J. Global. Optim. 43, 47–66 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mishra, S.K., Lai, K.K.: On Characterization of Solution Sets of Nonsmooth Pseudoinvex Minimization Problems. IEEE Computer Society, pp. 739–741 (2009)

  22. Zhao, K.Q., Tang, L.P.: On characterizing solution set of non-differentiable η-pseudolinear extremum problems. Optimization. doi:10.1080/02331934.2010.505653 (2011)

  23. Yuan D., Chinchuluun A., Liu X., Pardalos P.M.: Generalized convexities and generalized gradients based on algebraic operations. J. Math. Anal. Appl. 321, 675–690 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Sach P.H., Kim D.S., Lee G.M.: Generalized convexity and nonsmooth problems of vector optimization. J. Global. Optim. 31, 383–403 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ansari, Q.H., Rezaei, M.: Generalized pseudolinearity. Optim. Lett. doi:10.1007/s11590-010-0238-2 (2010)

  26. Pardalos P.M., Rassias T.M., Khan A.A.: Nonlinear Analysis and Variational Problems. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  27. Floudas, C.A., Pardalos, P.M. (eds): Encyclopedia of Optimization, 2nd edn. Springer, Berlin (2009)

    MATH  Google Scholar 

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Authors and Affiliations

  1. College of Mathematics science, Chongqing Normal University, Chongqing, 400047, China

    K. Q. Zhao, X. Wan & X. M. Yang

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  1. K. Q. Zhao
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  2. X. Wan
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  3. X. M. Yang
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Correspondence to K. Q. Zhao.

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Zhao, K.Q., Wan, X. & Yang, X.M. A note on characterizing solution set of nonsmooth pseudoinvex optimization problem. Optim Lett 7, 117–126 (2013). https://doi.org/10.1007/s11590-011-0399-7

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  • DOI: https://doi.org/10.1007/s11590-011-0399-7

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