On sufficiency and duality for nonsmooth multiobjective programming problems involving generalized V

Abstract In this paper, a class of nonsmooth multiobjective programming problems with inequality constraints is considered. We introduce the concepts of V – r -pseudo-invex, strictly V – r -pseudo-invex and V – r -quasi-invex functions, in which the involved functions are locally Lipschitz. Based upon these generalized V – r -invex functions, sufficient optimality conditions for a feasible point to be an efficient or a weakly efficient solution are derived. Appropriate duality theorems are proved for a Mond–Weir-type dual program of a nonsmooth multiobjective programming under the aforesaid functions.

[1]  Manuel Arana-Jiménez,et al.  Pseudoinvexity, optimality conditions and efficiency in multiobjective problems; duality , 2008 .

[2]  Tadeusz Antczak,et al.  Optimality and duality for nonsmooth multiobjective programming problems with V-r-invexity , 2009, J. Glob. Optim..

[3]  M. A. Hanson,et al.  Multiobjective duality with invexity , 1987 .

[4]  Rita Pini,et al.  A survey of recent[1985-1995]advances in generalized convexity with applications to duality theory and optimality conditions , 1997 .

[5]  P. Kanniappan,et al.  Necessary conditions for optimality of nondifferentiable convex multiobjective programming , 1983 .

[6]  Tadeusz Antczak.  (p, r)-Invexity in multiobjective programming , 2004, Eur. J. Oper. Res..

[7]  T. Antczak,et al.  Υ -preinvexity and Υ -invexity in mathematical programming , 2005 .

[8]  Manuel Arana-Jiménez,et al.  A characterization of pseudoinvexity for the efficiency in non-differentiable multiobjective problems. Duality , 2010 .

[9]  B. Mond,et al.  Pre-invex functions in multiple objective optimization , 1988 .

[10]  T. R. Gulati,et al.  Sufficiency and duality in nonsmooth multiobjective optimization involving generalized (F, alpha, pi, d)-Type I functions , 2006, Comput. Math. Appl..

[11]  Vaithilingam Jeyakumar,et al.  On generalised convex mathematical programming , 1992, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[12]  d–ρ–(η,θ)-invexity in multiobjective optimization , 2009 .

[13]  Vasile Preda,et al.  On efficiency and duality for multiobjective programs , 1992 .

[14]  M. A. Hanson.  On sufficiency of the Kuhn-Tucker conditions , 1981 .

[15]  Izhar Ahmad,et al.  Sufficiency and Duality in Multiobjective Programming under Generalized Type I Functions , 2007 .

[16]  T. Antczak.  The Notion of V-r-Invexity in Differentiable Multiobjective Programming , 2005 .

[17]  Adi Ben-Israel,et al.  What is invexity? , 1986, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[18]  M. A. Hanson,et al.  Optimality criteria in mathematical programming involving generalized invexity , 1988 .

[19]  I. Ahmad.  Sufficiency and Duality in Multiobjective Programming with Generalized (F, ρ)-Convexity , 2005 .

[20]  R. Kaul,et al.  Optimality criteria and duality in multiple-objective optimization involving generalized invexity , 1994 .

[21]  Panos M. Pardalos,et al.  A survey of recent developments in multiobjective optimization , 2007, Ann. Oper. Res..

[22]  Kaisa Miettinen,et al.  Nonlinear multiobjective optimization , 1998, International series in operations research and management science.