# Nonsmooth Vector Optimization Problem Involving Second-Order Semipseudo, Semiquasi Cone-Convex Functions

### Abstract

Recently, Suneja et al. [26] introduced new classes of second-order cone-(\eta; \xi)-convex functions along with their generalizations and used them to prove second-order Karush–Kuhn–Tucker (KKT) type optimality conditions and duality results for the vector optimization problem involving first-order differentiable and second-order directionally differentiable functions. In this paper, we move one step ahead and study a nonsmooth vector optimization problem wherein the functions involved are first and second-order directionally differentiable. We introduce new classes of nonsmooth second-order cone-semipseudoconvex and nonsmooth second-order cone-semiquasiconvex functions in terms of second-order directional derivatives. Second-order KKT type sufficient optimality conditions and duality results for the same problem are proved using these functions.### References

S. Aggarwal, Optimality and duality in mathematical programming involving generalized convex functions, Ph.D. thesis, University of Delhi, Delhi, 1998.

B. Aghezzaf, Second order mixed type duality in multiobjective programming problems, J. Math. Anal.Appl., vol. 285, pp. 97-106, 2003.

I. Ahmad, and Z. Husain, Second order (F; α; ρ; d)-convexity and duality in multiobjective programming, Information Sciences, vol. 176, pp. 3094-3103, 2006 .

I. Ahmad, and S. Sharma, Second-Order Duality for Nondifferentiable Multiobjective Programming Problems, Numer. Funct. Anal. Optim., vol. 28, pp. 975–988, 2007.

Q. H. Ansari, and J. C. Yao, Recent Developments in Vector Optimization, Springer-Verlag, Berlin, Heidelberg, 2012.

A. Auslender, Penalty methods for computing points that satisfy second order necessary conditions, Mathematical Programming, vol. 17, pp. 229–238, 1979.

W. F. Demyanov, and A. B. Pevnyi, Expansion with Respect to a Parameter of the Extremal Values of Game Problems, USSR Computational Mathematics and Mathematical Physics, vol. 14, pp. 33-45, 1974.

R. Dubey, and V.N. Mishra, Symmetric duality results for second-order nondifferentiable multiobjective programming problem, RAIRO-Oper. Res., vol. 53, pp. 539–558, 2019.

R. Dubey, V.N. Mishra, and P. Tomar, Duality relations for second-order programming problem under (G, αf )-bonvexity assumptions, Asian-Eur. J. Math., vol. 13, no. 2, pp. 1–17, 2020.

R. Dubey, A. Kumar, R. Ali, and L.N. Mishra, New class of G-Wolfe-type symmetric duality model and duality relations under Gf -bonvexity over arbitrary cones, J Inequal Appl, vol. 30, https://doi.org/10.1186/s13660-019-2279-0, 2020.

M. Feng, and S. Li, On second-order optimality conditions for continuously Frchet differentiable vector optimization problems, Optimization, DOI: 10.1080/02331934.2018.1545122, 2018.

M. Feng, and S. Li, On second-order Fritz John type optimality conditions for a class of differentiable optimization problems, Applicable Analysis, DOI:10.1080/00036811.2019.1573989, 2019.

M. Feng, and S. Li, Second-Order Strong Karush/KuhnTucker Conditions for Proper Efficiencies in Multiobjective Optimization, Journal of Optimization Theory and Applications, https://doi.org/10.1007/s10957-019-01484-0, 2019.

F. Flores-Baz´an, N. Hadjisavvas, and C. Vera, An Optimal Alternative Theorem and Applications to Mathematical Programming, J Glob Optim, vol. 37, no. 2, pp. 229–243, 2007.

M. A. Hanson, Second order invexity and duality in mathematical programming, Opsearch, vol. 30, pp. 313-320, 1993.

I. Husain, N. G. Rueda, and Z. Jabeen, Fritz John second-order duality for nonlinear programming, Appl. Math. Lett., vol. 14, pp.513–518, 2001.

V. I. Ivanov, Second-order invex functions in nonlinear programming, Optimization, vol. 61, no. 5, pp. 489–503, 2012.

V. I. Ivanov, Second-order optimality conditions and Lagrange multiplier characterizations of the solution set in quasiconvex programming, Optimization, DOI: 10.1080/02331934.2019.1625351, 2019.

A. Jayswal, and S. Jha, Second order symmetric duality in fractional variational problems over cone constraints, Yugoslav Journal of Operations Research, vol. 28, no.1, pp. 39–57, 2018.

D. G. Luenberger, and Y. Ye, Linear and Nonlinear Programmming, Springer, New York, 2008.

O. L. Mangasarian, Second and higher-order duality in nonlinear programming, J.Math.Anal.Appl., vol.51, pp. 607-620, 1975.

S. K. Mishra, Second order generalized invexity and duality in mathematical programming, Optimization., vol. 42, no.1, pp. 51–69, 1997.

B. Mond, Second order duality for nonlinear programs, Opsearch, vol. 11, pp. 90-99, 1974.

J. Nocedal, and S. J. Wright, Numerical Optimization, Springer, New York, 2006.

S. K. Suneja, S. Sharma, and Vani, Second-order duality in vector optimization over cones, J.Appl.Math.Inform., vol. 26, pp. 251-261, 2008.

S. K. Suneja, S. Sharma, and M. Kapoor, Second-order optimality and duality in vector optimization over cones, Stat. Optim. Inf Comput., vol. 4,no. 2, pp. 163–173, 2016.

L. Tang, H. Yan, and X. Yang, Second order duality for multiobjective programming with cone constraints, Sci. China Math.,vol. 59, no. 7, pp. 1285–1306, 2016.

N. V. Tuyen, N. Q. Huy, and D. S. Kim, Strong second-order KarushKuhnTucker optimality conditions for vector optimization, Applicable Analysis, DOI: 10.1080/00036811.2018.1489956, 2018.

R. U. Verma, Multiobjective Fractional Programming Problems and Second Order Generalized Hybrid Invexity Frameworks, Stat. Optim. Inf Comput., vol. 2, no.4, pp. 280-304, 2014.

R. U. Verma, and G. J. Zalmai, Generalized second-order parametric optimality conditions in semiinfinite discrete minmax fractional programming and second order (F, β, ϕ, ρ, θ,m)-univexity, Stat. Optim. Inf Comput., vol. 4, no.1, pp. 15-29, 2016.

R. U. Verma, and T. Antczak, New Class of duality models in discrete minmax fractional programming based on second-order univexities, Stat. Optim. Inf Comput., vol. 5,no. 3, pp.262-277, 2017.

Y. B. Xiao, N. V. Tuyen, J. C. Yao, and C. F. Wen, Locally Lipschitz vector optimization problems: second-order constraint qualifications, regularity condition and KKT necessary optimality conditions, Positivity, https://doi.org/10.1007/s11117-019-00679-z, 2019.

X. M. Yang, X. Q. Yang, K. L. Teo, and S.H. Hou, Second order duality for nonlinear programming, Indian J Pure Appl Math, vol. 35, pp. 699-708, 2004.

*Statistics, Optimization & Information Computing*,

*9*(2), 383-398. https://doi.org/10.19139/soic-2310-5070-839

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