Sufficiency and Duality for Nonsmooth Multiobjective Programming Problems Involving Generalized (Φ, ρ)-V-Type I Functions
Tóm tắt
In this paper, a class of nonsmooth multiobjective programming problems is considered. We introduce the new concepts of (Φ, ρ)-V-type I, (pseudo, quasi) (Φ, ρ)-V-type I and (quasi, pseudo) (Φ, ρ)-V-type I functions, in which the involved functions are locally Lipschitz. Based upon these generalized (Φ, ρ)-V-type I functions, the sufficient optimality conditions for weak efficiency, efficiency and proper efficiency are derived. Mond-Weir duality results are also established under the aforesaid functions.
Tài liệu tham khảo
Hanson, M.A.: On sufficiency of the kuhn-tucker conditions. J. Math. Anal. Appl. 80, 545–550 (1981)
Craven, B.D.: Invex functions and constrained local minima. Bull. Aust. Math. Soc. 24, 357–366 (1981)
Hanson, M.A., Mond, B.: Necessary and sufficient conditions in constrained optimization. Math. Program. 37, 51–58 (1987)
Preda, V.: On efficiency and duality for multiobjective program. J. Math. Anal. Appl. 166, 365–377 (1992)
Hanson, M.A., Mond, B.: Further generalizations of convexity in mathematical programming. J. Inform. Optim. Sci. 3, 25–32 (1982)
Vial, J.P.: Strong and weak convexity of sets and functions. Math. Oper. Res. 8, 231–259 (1983)
Caristi, G., Ferrara, M., Stefanescu, A.: Mathematical programming with (Φ, ρ)-invexity. In: Konnor, I.V., Luc, D.T., Rubinov, A.M. (eds.): Generalized Convexity and Related Topics. Lecture Notes in Economics and Mathematical Systems, vol. 583, pp. 167–176. Springer, Berlin-Heidelberg-New York (2006)
Pitea, A., Postolache, M.: Duality theorems for a new class of multitime multiobjective variational problems. J. Glob. Optim. 54, 47–58 (2012)
Pitea, A., Postolache, M.: Minimization of vectors of curvilinear functionals on the second order jet bundle. Necessary conditions. Optim. Lett. 6, 459–470 (2012)
Pitea, A., Postolache, M.: Minimization of vectors of curvilinear functionals on the second order jet bundle. Sufficient efficiency conditions. Optim. Lett. 6, 1657–1669 (2012)
Reiland, T.W.: Nonsmooth invexity. Bull. Aust. Math. Soc. 42, 437–446 (1990)
Kaul, R.N., Suneja, S.K., Lalitha, C.S.: Generalized nonsmooth invexity. J. Inform. Optim. Sci. 15, 1–17 (1994)
Lee, G.M.: Nonsmooth invexity in multiobjective programming. J. Inform. Optim. Sci. 15, 127–136 (1994)
Kuk, H., Tanino, T.: Optimality and duality in nonsmooth multiobjective optimization involving generalized type I functions. Comput. Math. Appl. 45, 1497–1506 (2003)
Ahmad, I., Gupta, S.K., Jayswal, A.: On sufficiency and duality for nonsmooth multiobjective programming problems involving generalized V -r-invex functions. J. Nonlinear. Anal. 74, 5920–5928 (2011)
Nahak, C., Mohapatra, R.N.: Nonsmooth ρ- (η, 𝜃)-invexity in multiobjective programming problems. Optim. Lett. 6, 253–260 (2012)
Antczak, T., Stasiak, A.: (Φ, ρ)-invexity in nonsmooth optimization. Num. Func. Anal. Optim 32, 1–25 (2011)
Antczak, T.: Proper efficiency conditions and duality results for nonsmooh vector optimization in Banach spaces under (Φ, ρ)-invexity. Nonlinear. Anal. 75, 3107–3121 (2012)
Clarke, F.H.: Nonsmooth Optimization. Wiley, New York (1983)
Mishra, S.K., Wang, S.Y., Lai, K.K.: Optimality and duality for a nonsmooth multiobjective optimization involving generalized type I functions. Math. Meth. Oper. Res. 67, 493–504 (2008)
Geoffrion, A.M.: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl. 22, 618–630 (1968)
Miettinen, K.M.: Nonlinear Multiobjective Optimization, Kluwer Academic, Boston MA (1999)