THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 10, Number 2/2009, pp. 000-000 ON ZALMAI’S SEMIPARAMETRIC DUALITY MODEL FOR MULTIOBJECTIVE FRACTIONAL PROGRAMMING WITH -SET FUNCTIONS n ) Ioan M. STANCU-MINASIAN*, Vasile PREDA**, Miruna BELDIMAN*, Andreea Mădălina STANCU* * Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, Calea 13 Septembrie nr. 13, 050711, Bucharest 5, Romania, E-mail: stancum@csm.ro ** University of Bucharest, Faculty of Mathematics and Computer Sciences, Str. Academiei 14, 010014, Bucharest, Romania New duality results for a semiparametric duality model are given for a fractional programming problem involving n-set functions. Key words: multiobjective programming, n-set function, duality, generalized convexity. 1. INTRODUCTION AND PRELIMINARIES We consider the frame of optimization theory for n-set [2,5,8]. For formulating and proving various duality results, we use the class of generalized convex n-set functions called (F,b,φ,ρ, θ)-univex functions, which were defined in Zalmai [11]. Until now, F was assumed to be a sublinear function in the third argument. In our approach, we suppose that F is a convex function in the third argument, as in Preda et al. [7,8] and Bătătorescu et al. [1]. Let ( , , XA μ be a finite atomless measure space with ( ) 1 , , L XA μ separable, and let d be the pseudometric on A n defined by ( ) ( ) 1/2 2 1 , : n k k k dRS R S μ = ⎡ ⎤ = Δ ⎢ ⎥ ⎣ ⎦ ∑ where 1 1 ( ,···, ), ( ,···, ) n n R R R S S S A = = ⁿ ∈ ) and Δ stands for symmetric difference. Thus, is a pseudometric space. ( , A d ⁿ For ( ) 1 , , h L XA μ ∈ and T with indicator (characteristic) function A ∈ ( , , T L XA ) χ μ ∞ ∈ , the integral hd μ ∫ is denoted by , T h χ . Definition 1.1. [4] A function is said to be differentiable at S * ∈A if there exist : f A → \ ( ) ( * 1 Df S L XA ) , , μ ∈ , called the derivative of f at S * , and such that : V A A f × → \ * * * () ( ) D( ), (, ) S f S * f S FS fS V SS χ χ = + − + for each S ∈ A, where is , that is, * (, ) f V SS * ( (, )) odSS 0 * * ( , ) 0 (, ) lim 0 (, ) . f dSS V SS dSS → =