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Optimality and duality for vector optimization problem with non-convex feasible set

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Abstract

The Karush–Kuhn–Tucker (KKT) optimality conditions are necessary and sufficient for a convex programming problem under suitable constraint qualification. Recently, several papers (Dutta and Lalitha in Optim Lett 7(2):221–229, 2013; Lasserre in Optim Lett 4(1):1–5, 2010; Suneja et al. Am J Oper Res 3(6):536–541, 2013) have appeared wherein the convexity of constraint function has been replaced by convexity of the feasible set. Further, Ho (Optim Lett 11(1):41–46, 2017) studied nonlinear programming problem with non-convex feasible set. We have used this modified approach in the present paper to study vector optimization problem over cones. The KKT optimality conditions are proved by replacing the convexity of the objective function with convexity of strict level set, convexity of feasible set is replaced by a weaker condition and no condition is assumed on the constraint function. We have also formulated a Mond–Weir type dual and proved duality results in the modified setting. Our results directly extend the work of Ho (2017) Suneja et al. (2013) and Lasserre (2010).

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Notes

  1. 1.

    Slaters’s condition holds for S if there exists \(x_0 \in S, g_i (x_0) < 0\) for every \(i= 1,\ldots ,m\).

References

  1. 1.

    Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming. Wiley, New Jersey (2006)

  2. 2.

    Chinchuluun, A., Pardalos, P.: A survey of recent developments in multiobjective optimization. Ann. Oper. Res. 154, 29–50 (2007)

  3. 3.

    Chinchuluun, A., Migdalas, A., Pardalos, P.M., Pitsoulis, L.: Pareto Optimality. Game Theory and Equilibria. Springer, New York (2008)

  4. 4.

    Coladas, L., Li, Z., Wang, S.: Optimality conditions for multiobjective and nonsmooth minimization in abstract spaces. Bull. Aust. Math. Soc. 50(2), 205–218 (1994)

  5. 5.

    Dutta, J., Lalitha, C.S.: Optimality conditions in convex optimization revisited. Optim Lett. 7(2), 221–229 (2013)

  6. 6.

    Ehrgott, M.: Multicriteria Optimization. Springer, Berlin (2005)

  7. 7.

    Hiriart-Urruty, J.B., Lemarechal, C.: Convex analysis and minimization algorithms I. Springer, Berlin (1993)

  8. 8.

    Ho, Q.: Necessary and sufficient KKT optimality conditions in non-convex optimization. Optim Lett. 11(1), 41–46 (2017)

  9. 9.

    Hu, Y.: Theory and application of multiobjective optimization in China a survey. Ann. Oper. Res. 24(1), 45–51 (1990)

  10. 10.

    Khanh, P.Q., Quyen, H.T., Yao, J.C.: Optimality conditions under relaxed quasi-convexity assumptions using star and adjusted subdifferentials. Eur. J. Oper. Res. 212, 235–241 (2011)

  11. 11.

    Lasserre, J.B.: On representations of the feasible set in convex optimization. Optim Lett. 4(1), 1–5 (2010)

  12. 12.

    Mond, B., Weir, T.: Generalized concavity and duality, generalized concavity in optimization and economics. In: Schaible, S., Ziemba, W.T. (eds.), pp. 263–279. Academic Press, New York (1981)

  13. 13.

    Schy, A., Giesy, D.P.: Multicriteria optimization techniques for design aircraft control systems. In: Stadler, W. (ed.) Multieriteria Optimization in Engineering and in the Sciences. Plenum Press, New York (1988)

  14. 14.

    Stadler, W.: Multicriteria optimization in mechanics: a survey. Appl. Mech. Rev. 37, 277–286 (1984)

  15. 15.

    Stadler, W. (ed.): Multicriteria Optimization in Engineering and in the Sciences. Plenum Press, New York (1988)

  16. 16.

    Suneja, S.K., Aggarwal, S., Davar, S.: Multiobjective symmetric duality involving cones. Eur. J. Oper. Res. 141(3), 471–479 (2002)

  17. 17.

    Suneja, S., Sharma, S., Grover, M., Kapoor, M.: A different approach to cone-convex optimization. Am. J. Oper. Res. 3(6), 536–541 (2013)

  18. 18.

    Weir, T., Mond, B.: Generalized convexity and duality in multiple objective programming. Bull. Aust. Math. Soc. 39, 287–299 (1989)

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Acknowledgements

The authors are grateful to the referees and the editor for their valuable comments and helpful suggestions.

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Correspondence to Priyanka Yadav.

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Suneja, S.K., Sharma, S. & Yadav, P. Optimality and duality for vector optimization problem with non-convex feasible set. OPSEARCH 57, 1–12 (2020). https://doi.org/10.1007/s12597-019-00401-3

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Keywords

  • Vector optimization
  • Cones
  • Level sets
  • KKT conditions
  • Duality

Mathematics Subject Classification

  • 90C26
  • 90C29
  • 90C30
  • 90C46