Abstract
We consider two generalized Minty vector variational-like inequalities and investigate the relations between their solutions and vector optimization problems for non-differentiable α-invex functions.
Similar content being viewed by others
References
Al-Homidan S., Ansari Q.H.: Generalized minty vector variational-like inequalities and vector optimization problems. J. Optim. Theory Appl. 144, 1–11 (2010)
Ansari Q.H., Lee G.M.: Nonsmooth vector optimization problems and minty vector variational inequalities. J. Optim. Theory Appl. 145, 1–16 (2010)
Ansari, Q.H., Rezaie, M., Zafarani. J.: Generalized vector variational-like inequalities and vector optimization. J. Global Optim. (2012, in press)
Chen, B., Huang, N.J.: Vector variational-like inequalities and vector optimization problems in Asplund spaces. Optim. Lett. (2012, in press)
Chinchuluun A., Pardalos P.M.: A survey of recent developments in multiobjective optimization. Ann. Oper. Res. 154, 29–50 (2007)
Chinchuluun A., Pardalos P.M., Migdalas A., Pitsoulis L.: Pareto Optimality, Game Theory and Equilibria. Springer, New York (2008)
Ceng L.C., Yao J.C.: On generalized variational-like inequalities with generalized monotone multivalued mappings. Appl. Math. Lett. 22, 428–434 (2009)
Fang Y.P., Hu R.: A non-smooth version of minty variational principle. Optimization 58, 401–412 (2009)
Fakhar M., Zafarani J.: Generalized vector equilibrium problems for pseudomonotone bifunctions. J. Optim. Theory Appl. 126, 109–124 (2005)
Giannessi F.: Theorems on 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, Dordrecht (1997)
Garzon G.R., Gomez R.O., Lizana A.R.: Generalized invex monotonicity. Eur. J. Oper. Res. 144, 501–512 (2003)
Jabarootian T., Zafarani J.: Generalized invariant monotonicity and invexity of non-differentiable functions. J. Global Optim. 36, 537–564 (2006)
Jabarootian T., Zafarani J.: Generalized vector variational-like inequalities. J. Optim. Theory Appl. 136, 15–30 (2008)
Long, X.J., Peng, J.W., Wu, S.Y.: Generalized vector variational-like inequalities and nonsmooth vector optimization problems. Optimization (2012, in press)
Mordukhovich B.S.: Variational Analysis and Generalized Differential I, Basic theory, Grundlehren Ser., (Fundamental Principles of Mathematical Sciences). Springer, New York (2006)
Mohan S.R., Neogy S.K.: On invex sets and preinvex functions. J. Math. Anal. Appl. 189, 901–908 (1995)
Oveisiha M., Zafarani J.: Vector optimization problem and generalized convexity. J. Global Optim. 52, 29–43 (2012)
Polyak B.T.: Existence theorems and converegence of minimizing sequences in extremum problems with restrictions. Dokl. Akad. Nauk. USSR. 166, 287–290 (1966)
Rockafellar R.T., Wets R.J-B.: Variational Analysis. Springer, New York (1998)
Rezaie M., Zafarani J.: Vector optimization and variational-like inequalities. J. Global Optim. 43, 47–66 (2009)
Schirotzek W.: Nonsmooth Analysis. Springer, New York (2007)
Soleimani-damaneh M.: Characterizations and applications of generalized invexity and monotonicity in Asplund spaces. Top. 18(1), 1–22 (2010)
Weir T., Mond B.: Preinvex functions in multiple-objective optimization. J. Math. Anal. Appl. 136, 29–38 (1988)
Yang X.M., Yang X.Q.: Vector variational-like inequality with pseudoinvexity. Optimization 55, 157–170 (2006)
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 monotonicity. Eur. J. Oper. Res. 164, 115–119 (2005)
Zopounidis C., Pardalos P.M.: Handbook of Multicriteria Analysis Series: Applied Optimization. Springer, New York (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Oveisiha, M., Zafarani, J. Generalized Minty vector variational-like inequalities and vector optimization problems in Asplund spaces. Optim Lett 7, 709–721 (2013). https://doi.org/10.1007/s11590-012-0454-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-012-0454-z