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Generalized higher-order cone-convex functions and higher-order duality in vector optimization

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Abstract

In this paper, we introduce a new class of higher-order cone-convex, \((K_1, K_2)\)-pseudoconvex and quasiconvex functions which encapsulates several already known functions. Higher-order sufficient optimality conditions have been established for a vector optimization problem over cones by using these functions, under weaker conditions on multipliers as compared to other papers in this domain. Wolfe type and Mond–Weir type higher-order duals are formulated and corresponding duality results are established. A number of previously studied problems appear as special cases of our primal-dual models. In case of nonlinear programming problem, our higher-order duals reduce to the corresponding higher-order duals given by Mangasarian (J Math Anal Appl 51:607–620, 1975) and Mond and Zhang (Generalized convexity, generalized monotonicity: recent results. Kluwer, Dordrecht, pp 357–372, 1998).

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References

  • Aghezzaf, B. (2003). Second order mixed type duality in multiobjective programming problems. Journal of Mathematical Analysis and Applications, 285, 97–106.

    Article  Google Scholar 

  • Ahmad, I., & Husain, Z. (2006). Second order \((f;\alpha; \rho; d)\)-convexity and duality in multiobjective programming. Information Sciences, 176, 3094–3103.

    Article  Google Scholar 

  • Askar, S., & Tiwari, A. (2009). First-order optimality conditions and duality results for multi-objective optimisation problems. Annals of Operations Research, 172, 277–289.

    Article  Google Scholar 

  • Bhatia, M. (2012). Higher order duality in vector optimization over cones. Optimization Letters, 6(1), 17–30.

    Article  Google Scholar 

  • Cambini, R. (1996). Some new classes of generalized concave vector-valued functions. Optimization, 36, 11–24.

    Article  Google Scholar 

  • Chinchuluun, A., & Pardalos, P. (2007). A survey of recent developments in multiobjective optimization. Annals of Operations Research, 154, 29–50.

    Article  Google Scholar 

  • Chinchuluun, A., Yuan, D., & Pardalos, P. (2007). Optimality conditions and duality for nondifferentiable multiobjective fractional programming with generalized convexity. Annals of Operations Research, 154, 133–147.

    Article  Google Scholar 

  • Craven, B. (1977). Lagrangean conditions and quasiduality. Bulletin of the Australian Mathematical Society, 16, 325–339.

    Article  Google Scholar 

  • Flores-Bazan, F., Hadjisavvas, N., & Vera, C. (2007). An optimal alternative theorem and applications to mathematical programming. Journal of Global Optimzation, 37, 229–243.

    Article  Google Scholar 

  • Gutierrez, C., Jimenez, B., Novo, V., & Ruiz-Garzon, G. (2015). Efficiency through variational-like inequalities with lipschitz functions. Applied Mathematics and Computation, 259, 438–449.

    Article  Google Scholar 

  • Hu, Y. (1990). Theory and application of multiobjective optimization in china a survey. Annals of Operations Research, 24(1), 45–51.

  • Jahn, J. (2004). Vector optimization. Heidelberg: Springer.

    Book  Google Scholar 

  • Luc, D. (1989). Theory of vector optimization. Berlin: Springer.

    Book  Google Scholar 

  • Mangasarian, O. (1975). Second and higher-order duality in nonlinear programming. Journal of Mathematical Analysis and Applications, 51, 607–620.

    Article  Google Scholar 

  • Mishra, S., & Rueda, N. (2000). Higher order generalized invexity and duality in mathematical programming. Journal of Mathematical Analysis and Applications, 247, 173–182.

    Article  Google Scholar 

  • Mond, B., & Weir, T. (1981–1983). Generalized convexity and higher order duality. Journal of Mathematical Science, 16(18), 74–94.

  • Mond, B., & Zhang, J. (1998). Higher order invexity and duality in mathematical programming. In J. P. Crouzeix (Ed.), Generalized convexity, generalized monotonicity: Recent results (pp. 357–372). Dordrecht: Kluwer.

    Chapter  Google Scholar 

  • Suneja, S., Aggarwal, S., & Davar, S. (2002). Multiobjective symmetric duality involving cones. European Journal of Operational Research, 141, 471–479.

    Article  Google Scholar 

  • Suneja, S., Louhan, P., & Grover, M. (2013). Higher-order cone-pseudoconvex, quasiconvex and other related functions in vector optimization. Optimization Letters, 7, 647–664.

    Article  Google Scholar 

  • Suneja, S., Sharma, S., & Vani, V. (2008). Second-order duality in vector optimization over cones. Journal of Applied Mathematics & Informatics, 26, 251–261.

    Google Scholar 

  • Wolfe, P. (1961). A duality theorem for nonlinear programming. Quarterly of Applied Mathematics, 19, 239–244.

    Article  Google Scholar 

  • Xu, Z. (1996). Mixed type duality in multiobjective programming problems. Journal of Mathematical Analysis and Applications, 198, 621–635.

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mathematics, University of Delhi, Delhi, 110007, India

    S. K. Suneja

  2. Department of Mathematics, Miranda House, University of Delhi, Delhi, 110007, India

    Sunila Sharma

  3. Department of Mathematics, Atma Ram Sanatan Dharma College, University of Delhi, Delhi, 110021, India

    Priyanka Yadav

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  1. S. K. Suneja
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  2. Sunila Sharma
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  3. Priyanka Yadav
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Correspondence to Priyanka Yadav.

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Suneja, S.K., Sharma, S. & Yadav, P. Generalized higher-order cone-convex functions and higher-order duality in vector optimization. Ann Oper Res 269, 709–725 (2018). https://doi.org/10.1007/s10479-017-2470-y

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  • DOI: https://doi.org/10.1007/s10479-017-2470-y

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