On vector variational-like inequality and vector optimization problem with $(G,\alpha)$-univexity

Document Type : Research Paper


Department of Mathematics, University of Jammu, Jammu-180006, India


In this paper, we introduce and study $(G,\alpha)$-univex functions by generalizing the $\alpha- $ univex functions and establishing relationships between vector variational-like inequality problems and vector optimization problems. Furthermore, we formulate equivalence among the vector critical points, weak efficient points of vector optimization problems and the solution of weak vector variational-like inequality problems under pseudo-$(G,\alpha)$-univexity assumptions. An example is also constructed to validate the main result.


[1] S. Al-Homidan and Q.H. Ansari, Generalized Minty vector variational-like inequalities with vector optimization
problems, J. Optim. Theory Appl. 144 (2010), no. 1, 1–11.
[2] T. Antezak, New optimality conditions and duality results of G-type in differentiable mathematical programming
problem, Nonlinear Anal. 66 (2007), 1617-1632.
[3] M.S. Bazaraa, H.D. Sherali and C.M. Shetty, Nonlinear programming: Theory and algorithms, Wiley-Interscience,
New York, 2006.
[4] C.R. Bector, S. Chandra, S. Gupta and S. K. Suneja, Univex sets, functions and univex nonlinear programming,
Lecture Notes in Economics and Mathematical Systems, Springer Verlag, Berlin, 1994.
[5] F. Giannessi, Theorems of alternative, quadratic programs and complementarity problems, in variational inequality
and complementarity problems (ed. by R. W. Cottle, F. Giannessi, and J. L. Lions), John Wiley and Sons, New
York, 1980.
[6] M.A. Hanson, On sufficiency of the Kuhn-Tucker conditions, J. Math. Anal. Appl. 80 (1981), 545–550.
[7] T. Jabarootian and J. Zafarani, Generalized vector variational-like inequalities, J. Optim. Theory Appl. 136
(2008), 15–30.
[8] A. Jayswal and S. Choudhury, On vector variational-like inequalities and vector optimization problems with (G, α)-
invexity, Appl. Math. Chinese Univ. 32 (2017), no. 3, 323–338.
[9] A. Jayswal and S. Singh, On vector variational-like inequalities involving right upper-Dini-derivative functions,
Afr. Mat. 29 (2018), no. 3, 383–398.
[10] M. Jennane, L.E. Fadil and E.M. Kalmoun, On local quasi efficient solutions for nonsmooth vector optimization
problems, Crotian Oper. Res. Rev. 11 (2020), 1–10.
[11] P. Gupta and S.K. Mishra, On minty variational principle for nonsmooth vector optimization problems with
generalized approximate convexity, J. Math. Prog. Oper. Res. 67 (2018), 1157–1167.
[12] J.K. Kim, T. Ram and A.K. Khanna, On η generalized operator variational-like inequalities, Commun. Optim.
Theory 2018 (2018), Article ID 14, 1-11.
[13] X.J. Long, J.W. Peng and S.Y. Wu, Generalized vector variational-like inequalities and nonsmooth vector optimization problems, Optim. 61 (2012), 1075–1086.
[14] S.K. Mishra, On multiple objective optimization with generalized univexity, J. Math. Anal. Appl. 224 (1998),
[15] S.K. Mishra, G. Giorgi, Optimality and duality with generalized semi-univexity, Opsearch 37 (2000), 340–350.
[16] S.K. Mishra and M.A. Noor, On vector variational-like inequality problems, J. Math. Anal. Appl. 311 (2005),
[17] S.K. Mishra, S.Y. Wang and K.K. Lai, Nondifferentiable multiobjective programming under generalized dunivexity, Eur. J. Oper. Res. 160 (2005), 218–226.
[18] S.K. Mishra, S.Y. Wang and K.K. Lai, On nonsmooth α-invex functions and vector variational-like inequality,
Optim. Lett. 2 (2008), 91–98.
[19] S.K. Mishra, S.Y. Wang and K.K. Lai, Role of α-pseudo-univex functions in vector variational-like inequality
problems, J. Syst. Sci. Complexity 20 (2007), 344–349.
[20] S.K. Mishra and S.Y. Wang, Vector variational-like inequalities and non-smooth vector optimization problems,
Nonlinear Anal. 64 (2006), 1939–1945.
[21] M.A. Noor, On generalized preinvex functions and monotonicities, J. Ineq. Pure and Appl. Math. 5 (2004), no.
4, Article ID 110.
[22] R. Osuna, A. Rufian and G. Ruiz, Invex functions and generalized convexity in multiobjective programming, J.
Optim. Theory Appl. 98, (1998) 651–661.[23] G. Ruiz-Garzon, R. Osuna-Gomez and A. Rufian-Lizana, Relationships between vector variational-like inequality
and optimization problems, Eur. J. Oper. Res. 157 (2004), 113–119.
[24] R. Li and G. Yu, A class of generalized invex functions and vector variational-like inequalities, J. Inequal. Appl.
70 (2017), 1–14.
[25] A.H. Siddiqi, Q.H. Ansari and R. Ahmad, On vector variational-like inequalities, Indian J. Pure Appl. Math. 28
(1997), no. 8, 1009–1016.
[26] X.Q. Yang, Generalized convex functions and vector variational inequalities, J. Optim. Theory Appl. 79 (1993),
Volume 13, Issue 2
July 2022
Pages 643-650
  • Receive Date: 09 December 2021
  • Revise Date: 26 March 2022
  • Accept Date: 21 April 2022