Mathematical Programming With (φ,ρ)-Subvexormal Functions

Optimality and duality conditions for multi-objective optimisation prob-lems (MOOP) have particularly grown and become one of the most interestingtopics in optimisation. The convexity concept plays an important factor inthese conditions. In order to relax convexity assumptions imposed on thefunctions in theorems on optimality and duality, various generalized convex-ity concepts have been proposed. Many authors have extended this conceptand diferent kinds of functions, with and without diferentiability assumption-s. A number of duality results corresponding to this sort of multi-objectiveproblems are also shown. Hanson and Mond [1] introduced the concept ofF-convexity and Jeyakumar [2] generalized Vial’s ρ-convexity [3] introducingthe concept of ρ-invexity. Preda [4] further extended F-convexity to (F, ρ)-convexity by combining the concepts of the F-convexity and ρ-convexity. Andthen Giuseppe Caristi1, Massimiliano Ferrara, and Anton Stefanescu [5] gen-eralized (F, ρ)-convexity to (, ρ)-invexity, which was used by several authorsto obtain relevant results.However, all of the concepts mentioned above were defined under the as-sumption of diferentiable. It is well known that, by substituting generalizedconvexity for convexity, many theoretical problems in diferentiable program-ming can be solved [6-15]. However, the corresponding conclusions cannot beobtained in nondiferentiable programming with the aid of generalized convex-ity because the existence of a derivative is required in the definitions of thediferent forms of generalized convexity mentioned. There exists a generaliza-tion of convexity to locally Lipschitz functions, with derivative replaced by theClarke generalized gradient. Zheng XJ and Cheng L [16] introduced (F, ρ, θ)-d-univexty by using Clarke generalized gradient. Consequently, in this paper,consider a nondiferentiable programming, where the objective functions and the constraint functions need not necessarily be either diferentiable or con-vex. we introduce a new class of generalized convex function called (, ρ)-subvexormal function, which is the extension of the concept of (, ρ)-invesityto the nonsmooth case.