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Approximate solutions in nonsmooth and nonconvex cone constrained vector optimization

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Abstract

In this paper, we employ some advanced tools of variational analysis to provide new necessary optimality conditions for approximate (weak) Pareto solutions of a nonconvex and nonsmooth cone constrained vector optimization problem. The obtained necessary conditions are exhibited in a fuzzy form and a Fritz-John type. Sufficient optimality conditions for approximate (weak) Pareto solutions of the multiobjective problem are established by using assumptions of (strictly) approximately generalized convexity. Moreover, we address an approximate dual vector problem for the cone constrained vector optimization problem and examine converse and strong dualities for approximate (weak) Pareto solutions.

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References

  • Bot, R. I., Hodrea, I. B., & Wanka, G. (2008). \(\epsilon \)-optimality conditions for composed convex optimization problems. Journal of Approximation Theory, 153(1), 108–121.

    Article  Google Scholar 

  • Chuong, T. D. (2019a). Optimality and duality in nonsmooth conic vector optimization. Journal of Optimization Theory and Applications, 183, 471–489.

    Article  Google Scholar 

  • Chuong, T. D. (2019b). Optimality and duality in nonsmooth composite vector optimization and applications. Annals of Operations Research. https://doi.org/10.1007/s10479-019-03349-1.

  • Chuong, T. D. (2020a). Robust optimality and duality in multiobjective optimization problems under data uncertainty. SIAM Journal on Optimization, 30, 1501–1526.

    Article  Google Scholar 

  • Chuong, T. D. (2020b). Second-order cone programming relaxations for a class of multiobjective convex polynomial problems. Annals of Operations Research. https://doi.org/10.1007/s10479-020-03577-w.

  • Chuong, T. D., & Kim, D. S. (2014). Optimality conditions and duality in nonsmooth multiobjective optimization problems. Annals of Operations Research, 217, 117–136.

    Article  Google Scholar 

  • Chuong, T. D., & Kim, D. S. (2016). Approximate solutions of multiobjective optimization problems. Positivity, 20(1), 187–207.

    Article  Google Scholar 

  • Durea, M., Dutta, J., & Tammer, Chr. (2010). Lagrange multipliers for \(\varepsilon \)-Pareto solutions in vector optimization with nonsolid cones in Banach spaces. Journal of Optimization Theory and Applications, 145(1), 196–211.

    Article  Google Scholar 

  • Ehrgott, M. (2005). Multicriteria optimization. Berlin: Springer.

    Google Scholar 

  • Engau, A., & Wiecek, M. M. (2007). Cone characterizations of approximate solutions in real vector optimization. Journal of Optimization Theory and Applications, 134(3), 499–513.

    Article  Google Scholar 

  • Fabian, M. (1989). Subdifferentiability and trustworthiness in the light of a new variational principle of Borwein and Preiss. Acta Universitatis Carolinae, Mathematica et Physica, 30, 51–56.

    Google Scholar 

  • Ghaznavi, M., Akbarib, F., & Khorram, E. (2019). Optimality conditions via a unified direction approach for (approximate) efficiency in multiobjective optimization. Optimization Methods and Software. https://doi.org/10.1080/10556788.2019.1571589.

  • Ghaznavi-Ghosoni, B. A., Khorram, E., & Soleimani-Damaneh, M. (2013). Scalarization for characterization of approximate strong/weak/proper efficiency in multi-objective optimization. Optimization, 62(6), 703–720.

    Article  Google Scholar 

  • Goh, C. J., & Yang, X. Q. (1999). Vector equilibrium problem and vector optimization. European Journal of Operational Research, 116, 615–628.

    Article  Google Scholar 

  • Govil, M. G., & Mehra, A. (2004). \(\varepsilon \)-optimality for multiobjective programming on a Banach space. European Journal of Operational Research, 157(1), 106–112.

    Article  Google Scholar 

  • Gupta, D., & Mehra, A. (2012). A new notion of quasi efficiency in vector optimization. Pacific Journal of Optimization, 8(2), 217–230.

    Google Scholar 

  • Gutiérrez, C., Jiménez, B., & Novo, V. (2010). Optimality conditions via scalarization for a new \(\epsilon \)-efficiency concept in vector optimization problems. European Journal of Operational Research, 201(1), 11–22.

    Article  Google Scholar 

  • Gutiérrez, C., Huerga, L., Jiménez, B., & Novo, V. (2018). Approximate solutions of vector optimization problems via improvement sets in real linear spaces. Journal of Global Optimization, 70(4), 875–901.

    Article  Google Scholar 

  • Gutiérrez, C., Jiménez, B., & Novo, V. (2006). A unified approach and optimality conditions for approximate solutions of vector optimization problems. SIAM Journal on Optimization, 17(3), 688–710.

    Article  Google Scholar 

  • Gutiérrez, C., Jiménez, B., & Novo, V. (2015). Optimality conditions for quasi-solutions of vector optimization problems. Journal of Optimization Theory and Applications, 167(3), 796–820.

    Article  Google Scholar 

  • Huang, X. X. (2002). Optimality conditions and approximate optimality conditions in locally Lipschitz vector optimization. Optimization, 51(2), 309–321.

    Article  Google Scholar 

  • Jiao, L., & Lee, J. H. (2018). Approximate optimality and approximate duality for quasi approximate solutions in robust convex semidefinite programs. Journal of Optimization Theory and Applications, 176(1), 74–93.

    Article  Google Scholar 

  • Kim, M. H., & Lee, G. M. (2013). On efficient applications of G-Karush–Kuhn–Tucker necessary optimality theorems to multiobjective programming problems. Journal of Global Optimization, 55(1), 5–11.

    Article  Google Scholar 

  • Jahn, J. (2004). Vector optimization, theory, applications, and extensions. Berlin: Springer.

    Google Scholar 

  • Lee, J. H., & Jiao, L. (2019). Finding efficient solutions in robust multiple objective optimization with SOS-convex polynomial data. Annals of Operations Research. https://doi.org/10.1007/s10479-019-03216-z.

  • Liu, J. C. (1991). \(\varepsilon \)-duality theorem of nondifferentiable nonconvex multiobjective programming. Journal of Optimization Theory and Applications, 69(1), 153–167.

    Article  Google Scholar 

  • Liu, J. C. (1996). \(\varepsilon \)-Pareto optimality for nondifferentiable multiobjective programming via penalty function. Journal of Mathematical Analysis and Applications, 198(1), 248–261.

    Article  Google Scholar 

  • Liu, C., & Lee, H. (2012). Lagrange multiplier rules for approximate solutions in vector optimization. Journal of Industrial and Management Optimization, 8(3), 749–764.

    Article  Google Scholar 

  • Loridan, P. (1984). \(\varepsilon \)-solutions in vector minimization problems. Journal of Optimization Theory and Applications, 43, 265–276.

    Article  Google Scholar 

  • Luc, D. T. (1989). Theory of vector optimization (Vol. 319). Lecture notes in economics and mathematical systems. Berlin: Springer.

    Google Scholar 

  • Luc, D. T., Tan, N. X., & Tinh, P. N. (1998). Convex vector functions and their subdifferential. Acta Mathematica Vietnamica, 23(1), 107–127.

    Google Scholar 

  • Mordukhovich, B. S. (2006). Variational analysis and generalized differentiation. I: Basic theory. Berlin: Springer.

    Book  Google Scholar 

  • Mordukhovich, B. S. (2018). Variational analysis and applications. Berlin: Springer.

    Book  Google Scholar 

  • Mordukhovich, B. S., Nam, N. M., & Yen, N. D. (2006). Fréchet subdifferential calculus and optimality conditions in nondifferentiable programming. Optimization, 55, 685–708.

    Article  Google Scholar 

  • Mordukhovich, B. S., Treiman, J. S., & Zhu, Q. J. (2003). An extended extremal principle with applications to set-valued optimization. SIAM Journal on Optimization, 14, 359–379.

    Article  Google Scholar 

  • Rockafellar, R. T. (1970). Convex analysis. Princeton, NJ: Princeton University Press.

    Book  Google Scholar 

  • Rockafellar, R. T., & Wets, R. J.-B. (1998). Variational analysis. Berlin: Springer.

    Book  Google Scholar 

  • Sawaragi, Y., Nakayama, H., & Tanino, T. (1985). Theory of multiobjective optimization. Mathematics in science and engineering, 176. Orlando, FL: Academic Press.

    Google Scholar 

  • Son, T. Q., & Kim, D. S. (2013). \(\varepsilon \)-mixed type duality for nonconvex multiobjective programs with an infinite number of constraints. Journal of Global Optimization, 57(2), 447–465.

    Article  Google Scholar 

Download references

Acknowledgements

The author would like to thank the editor and referees for the valuable comments and suggestions, which help improved the presentation of the paper.

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Authors and Affiliations

  1. Optimization and Applications Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Thai Doan Chuong

  2. Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Thai Doan Chuong

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  1. Thai Doan Chuong
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Correspondence to Thai Doan Chuong.

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This work was supported by the National Foundation for Science and Technology Development of Vietnam (NAFOSTED) under Grant No. 101.01-2020.09.

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Chuong, T.D. Approximate solutions in nonsmooth and nonconvex cone constrained vector optimization. Ann Oper Res 311, 997–1015 (2022). https://doi.org/10.1007/s10479-020-03740-3

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  • DOI: https://doi.org/10.1007/s10479-020-03740-3

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