Abstract
In this article, we look beyond convexity and introduce the four new classes of functions, namely, approximate pseudoconvex functions of type I and type II and approximate quasiconvex functions of type I and type II. Suitable examples illustrating the non emptiness of the newly defined classes and distinguishing them from the existing classical notions of pseudoconvexity and quasiconvexity are provided. These newly defined concepts are then employed to establish sufficient optimality conditions for the quasi efficient solutions of a vector optimization problem.
Similar content being viewed by others
References
Bector C.R., Chandra S., Dutta J.: Principles of Optimization Theory. Narosa Publishing House, India (2005)
Bhatia D., Jain P.: Generalized (F, ρ) convexity and duality for nonsmooth multiobjective programs. Optimization. 31, 153–164 (1994)
Chinchuluun A., Pardalos P.M.: A survey of recent developments in multiobjective optimization. Ann. Oper. Res. 154(1), 29–50 (2007)
Chinchuluun, A., Pardalos, P.M., Migdalas, A., Pitsoulis, L. (eds): Pareto Optimality, Game Theory and Equilibria. Springer, Berlin (2008)
Clarke F.H.: Optimization and Nonsmooth Analysis. Willey-Inteerscience, New York (1983)
Dutta J., Vetrivel V.: On approximate minima in vector optimization. Numer. Funct. Anal. Optim. 22, 845–859 (2001)
Egudo R.: Efficiency and generalized convex duality for multi-objective programs. J. Math. Anal. Appl. 138, 84–94 (1989)
Govil M.G., Mehra A.: \({\varepsilon}\)-optimality for multiobjective programming on a Banach space. Eur. J. Oper. Res. 157, 106–112 (2004)
Gupta A., Mehra A., Bhatia D.: Approximate convexity in vector optimization. Bull. Aust. Math. Soc. 74, 207–218 (2006)
Hanson M.A., Mond B.: Necessary and sufficient conditions in constrained optimization. Math. Program. 37, 51–58 (1987)
Jeykumar V., Mond B.: On generalized convex mathematical programming. J. Aust. Math. Soc. Ser. B. 34, 43–53 (1992)
Liang Z.-A., Huang H.X., Pardalos P.M.: Efficiency conditions and duality for a class of multiobjective fractional programming problem. J. Glob. Optim. 27(4), 447–471 (2003)
Liu J.C.: \({\varepsilon}\)-duality theorem of nondifferentiable nonconvex multiobjective programming. J. Optim. Theory Appl. 69, 153–167 (1991)
Loridan P.: \({\varepsilon}\)-solutions in vector minimization problems. J. Optim. Theory Appl. 43, 265–276 (1984)
Ngai H.V., Luc D., Thera M.: Approximate convex functions. J. Nonlinear Convex Anal. 1, 155–176 (2000)
Pardalos P.M., Yuan D., Liu X., Chinchuluun A.: Optimality conditions and duality for multiobjective programming involving (c, α, ρ, d)type-I functions. In: Konnov, I.V., Luc, D.T., Rubinov, A.M. (eds) Generalized Convexity and Related Topics, pp. 73–87. Springer, Berlin (2007)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bhatia, D., Gupta, A. & Arora, P. Optimality via generalized approximate convexity and quasiefficiency. Optim Lett 7, 127–135 (2013). https://doi.org/10.1007/s11590-011-0402-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-011-0402-3