Abstract
In this paper, some properties of pseudoinvex functions are obtained. We study the equivalence between different solutions of the vector variational-like inequality problem. Some relations between vector variational-like inequalities and vector optimization problems for non-differentiable functions under generalized monotonicity are established.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ansari, Q.H., Yao, J.C.: On non-differentiable and nonconvex vector optimization problems. J. Optim. Theory Appl. 106, 475–488 (2000)
Clarke, F.H., Stern, R.J., Ledyaev, Y.S., Wolenski, P.R.: Nonsmooth Analysis and Control Theory. Springer, New York (1998)
Chen, G.Y.: Existence of solutions for a vector variational inequality. An extension of Hartman-Stampacchia theorem. J. Optim. Theory Appl. 74, 445–456 (1992)
Chen, G.Y., Yang, X.Q.: The vector complementarity problem and its equivalence with the weak minimal element in ordered spaces. J. Math. Anal. Appl. 153, 136–158 (1990)
Chiang, Y.: Semicontinuous mapping in t. v. s. with applications to mixed vector variational like inequalities. J. Global Optim. 32, 467–486 (2005)
Chinaie, M., Jabarootian, T., Rezaie, M., Zafarani, J.: Minty’s lemma and vector variational-like inequalities. J. Global Optim. 40, 463–473 (2008)
Daniilidis, A., Hadjisavvas, N.: Existence theorems for vector variational inequalities. Bull. Austral. Math. Soc. 54, 473–481 (1996)
Daniilidis, A., Hadjisavvas, N.: On generalized cyclically monotone operators and proper quasimonotonicity. Optimization 47, 123–135 (2000)
Fakhar, M., Zafarani, J.: Generalized vector equilibrium problems for pseudomonotone bifunctions. J. Optim. Theory Appl. 126, 109–124 (2005)
Fang, Y.P., Huang, N.J.: On minty vector prevariational inequalities and vector optimization problems. Manuscript (2005)
Giannessi, F.: Theorems of the alternative quadratic programs and complementarity problems. In: Cottle, R.W., Giannessi, F., Lions, J.L. (eds.) Variational Inequalities and Complementarity Problems, pp. 151–186. Wiley, Chichester (1980)
Giannessi, F.: On Minty Variational Principle. New Trends in Mathematical Programming. Kluwer Academic Publishers, Dordrecht, Netherlands (1997)
Giannessi, F. (ed.): Vector Variational Inequalities and Vector Equilibria. Kluwer Academic Publishers, Dordrecht (2000)
Giannessi, F., Maugeri, A.: Variational Inequalities and Network Equilibrium Problems. Plenum Press, New York (1995)
Giannessi, F., Maugeri, A.: Variational Analysis and Applications. Non-convex Optimization and Its Applications. Springer, New York (2005)
Jabarootian, T., Zafarani, J.: Generalized invariant monotonicity and invexity of nondifferentiable functions. J. Global. Optim. 36, 537–564 (2004)
Jabarootian, T., Zafarani, J.: Generalized vector variational-like inequalities. J. Optim. Theory Appl. 136, 15–30 (2008)
Konnov, I.V., Yao, J.C.: On the generalized vector variational inequality problem. J. Math. Anal. Appl. 206, 42–58 (1997)
Lee, G.M., Lee, K.B.: Vector variational inequalities for non-differentiable convex vector optimization problems. J. Global Optim. 32, 597–612 (2005)
Mishra, S.K., Wang, S.Y.: Vector variational-like inequalities and non-smooth vector optimization problems. Nonlinear Anal. 64, 1939–1945 (2005)
Mohan, S.R., Neogy, S.K.: On invex sets and priinvex functions. J. Math. Anal. Appl. 189, 901–908 (1995)
Santos, L.B., Medar, M.R., Lizana, A.R.: Some relations between variational-like inequalities and efficient solutions of certain non-smooth optimizations problems. Int. J. Math. Math. Sci. Art. ID 26808, 16 pp (2006)
Weir, T., Mond, B.: Preinvex functions in multiple-objective optimization. J. Math. Anal. Appl. 136, 29–38 (1988)
Yang, X.Q.: On vector variational inequalities: application to vector equilibria. J. Optim. Theory Appl. 95, 729–734 (1997)
Yang, X.M.: Generalized convexity in optimization. Ph.D. Dissertation (2002)
Yang, X.Q., Goh, C.J.: On vector variational inequality with application to vector traffic equilibria. J. Optim. Theory Appl. 95, 431–443 (1997)
Zhao, Y., Xia, Z.: Existence results for systems of vector variational-like inequalities. Non-linear Anal:Real World Appl. 8, 1370–1378 (2007)
Yang, X.M., Yang, X.Q., Teo, K.L.: Generalized invexity and generalized invariant monotonicity. J. Optim. Theory Appl. 117, 607–625 (2003)
Yang, X.M., Yang, X.Q., Teo, K.L.: Some remarks on the minty vector variational inequality. J. Optim. Theory Appl. 121, 193–201 (2004)
Yang, X.M., Yang, X.Q., Teo, K.L.: Criteria for generalized invex monotonicities. Eur. J. Oper. Res. 164, 115–119 (2005)
Yang, X.M., Yang, X.Q.: Vector variational-like inequality with pseudoinvexity. Optimization 55, 157–170 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
J. Zafarani was partially supported by the Center of Excellence for Mathematics (University of Isfahan).
Rights and permissions
About this article
Cite this article
Rezaie, M., Zafarani, J. Vector optimization and variational-like inequalities. J Glob Optim 43, 47–66 (2009). https://doi.org/10.1007/s10898-008-9290-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-008-9290-1
Keywords
- Vector variational-like inequalities
- Pseudoinvex functions
- Pseudomonotone mappings
- Vector optimization problems