THE WEIGHTING METHOD AND MULTIOBJECTIVE PROGRAMMING UNDER NEW CONCEPTS OF GENERALIZED (, )-INVEXITY

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1 U.P.B. Sci. Bull., Series A, Vol. 80, Iss. 2, 2018 ISSN THE WEIGHTING METHOD AND MULTIOBJECTIVE PROGRAMMING UNDER NEW CONCEPTS OF GENERALIZED (, )-INVEXITY Tadeusz ANTCZAK 1, Manuel ARANA-JIMÉNEZ 2 In the paper, the weighting ethod is used for solving the considered nonconvex vector optiization proble. The equivalence between a wea Pareto solution of the original vector optiization proble and an optial solution of its corresponding unconstrained scalar optiization proble is established under (, )-invexity. Further, the definition of a differentiable KT-(, )-invex vector optiization proble is introduced and the sufficient optiality conditions are established for such nonconvex differentiable ultiobjective prograing probles. In order to prove several Mond-Weir duality results for a new class of nonconvex differentiable vector optiization probles, the concept of WD-(, )-invexity is introduced. It turns out that the results presented in the paper are proved also for such nonconvex vector optiization probles in which not all functions constituting the have the fundaental property of any generalized convexity notions, earlier introduced in the literature. Keywords: ultiobjective prograing; weighting ethod; KT-(, )-invexity; WD-(, )-invexity; Pareto optiality; Mond-Weir duality. 1. Introduction In recent years, attepts are ade by several authors to define various classes of nonconvex optiization probles and to study their optiality criteria and duality results. In [25], Martin proposed two weaer notions than invexity introduced by Hanson [14], called KT-invexity and WD-invexity. He proved that every Kuhn-Tucer point is a iniizer of a scalar optiization proble with inequality constraints if and only if this proble is KT-invex. Also, he established that Wolfe wea duality holds if and only if the prial optiization proble is WD-invex. In recent years, several generalizations of Martin s definitions have been introduced to optiization theory in order to weaen the assuption of convexity in establishing optiality and duality results for new classes of nonconvex differentiable optiization probles (see, for exaple, [1], [2], [3], [4], [5], [6], [13], [16], [17], [18], [19], [20], [21], [22]). Recently, Caristi et al. [8] and later Ferrara and Stefanescu [10] established sufficient optiality conditions and duality results for differentiable scalar and vector optiization probles under (, )-invexity hypotheses. In the paper, we use the weighting ethod for solving differentiable vector optiization probles involving (, )-invex functions. Thus, we relate a wea Pareto solution of such nonconvex sooth vector optiization proble to an optial solution of its corresponding weighting scalar optiization proble constructed in this ethod. Further, by taing the otivation fro Martin [14] and Caristi et al. [8], we generalize the definitions of KT-invexity, WD-invexity and (, )-invexity to the case of differentiable ultiobjective prograing probles with inequality constraints. Naely, we introduce the concepts of KT-(, )-invexity and WD-(, )-invexity for nonconvex differentiable 1 University of Lodz, Poland, e-ail: antcza@ath.uni.lodz.pl 2 University of Cádiz, Spain, e-ail: anuel.arana@uca.es

2 4 Tadeusz Antcza, Manuel Arana-Jiénez vector optiization probles. We use the introduced notion of KT-(, )-invexity to establish sufficient optiality conditions and WD-(, )-invexity to establish several Mond-Weir duality results for a new class of nonconvex differentiable vector optiization probles. We also prove the equivalence between a wealy efficient solution of the KT- (, )-invex vector optiization proble and a iniizer in its associated scalar optiization proble constructed in the weighting ethod. It turns out that the sufficient optiality conditions and Mond-Weir duality results are established for such nonconvex vector optiization probles for which other generalized convexity notions existing in the literature ay avoid. 2. Scalarization ethod for a new class of nonconvex differentiable vector optiization probles For any vectors x = (x 1,, x n ) T, y = (y 1,, y n ) T in R n, we define: x < y if and only if x i < y i for all i = 1,2,...,n; x y if and only if x i y i for all i = 1,2,...,n; x y if and only if x y and x y; x = y if and only if x i = y i for all i = 1,2,...,n; We consider the following unconstrained vector optiization proble: f(x) = (f 1 (x),, f (x)) in s.t. x X, (UVOP) where f: X R is a differentiable function on a nonepty open convex set X Rⁿ. Definition 2.1. A feasible point x is said to be a wea Pareto (wealy efficient, wea iniu) solution for (UVOP) if and only if there exists no x X such that f(x) < f(x ). Definition 2.2. A feasible point x is said to be a Pareto (efficient) solution for (UVOP) if and only if there exists no x X such that f(x) f(x ). Definition 2.3. [10] The function f: X R is said to be (, f )-invex at x X on X if there exist a function X X R n+1 R, where (x, x, ) is convex on R n+1, (x, x, (0, a)) 0 for all x X and any a R +, ρ = (ρ 1,, ρ ) R, such that, the following inequalities f i (x) f i (x ) (x, x, ( f i (x ), ρ fi )), i = 1,,, (1) hold for all x X. If inequalities (1) are satisfied at any point x X, then f is said to be a (, f )-invex function on X. Definition 2.4. A feasible point x X is said to be a vector critical point of the proble (UVOP) if there exists a vector R with 0 such that f(x ) = 0. Scalar stationary points are those whose vector gradients are zero. For vector optiization probles, vector critical points are those such that there exists a non-negative linear cobination of the gradient vectors of each coponent objective function, valued at this point, equal to zero. Craven [9] established the following result for the proble (UVOP): Theore 2.1. Let x X be a wealy efficient solution to the proble (UVOP). Then there exists vector R with 0 such that f(x ) = 0, in other words, x is a critical point to the proble (UVOP). Now, we give a condition under which any critical point of the unconstrained vector optiization proble (UVOP) is its wealy efficient solution.

3 The weighting ethod and ultiobjective prograing under new concepts of generalized (, )-invexity5 Theore 2.2. Let x X be a vector critical point of the unconstrained vector optiization proble (UVOP), that is, there exists 0 such that f(x ) = 0. Further, assue that the objective function f is (, f )-invex at x X on X, where i=1 i ρ fi 0. Then x is a wealy efficient solution of the proble (UVOP). Proof Since x is a vector critical point of the unconstrained vector optiization proble (UVOP), there exists 0 such that f(x ) = 0. By assuption, f is a (, f )- invex function at x on X. Then, by Definition 2.3, inequalities (1) are satisfied. Multiplying (1) by i, i=1,,, we have Let us denote α i = if i (x) if i (x ) i (x, x, ( f i (x ), ρ fi )), i = 1,,. (2) i i=1 t=1 t. Note that 0 α i 0, but at least one α i > 0 and, oreover, i=1 α i = 1. Then, (2) yields i=1 α if i (x) i=1 α if i (x ) i=1 α i (x, x, ( f i (x ), ρ fi )). (3) By definition, (x, x, ) is convex on R n+1. Since 0 α i 0 and, oreover, i=1 α i = 1, by the definition of a convex function, we have (x, x, ( i=1 α i f i (x ), i=1 α iρ fi )) i=1 α i (x, x, ( f i (x ), ρ fi )). (4) By (3) and (4), it follows that i=1 α if i (x) i=1 α if i (x ) (x, x, ( i=1 α i f i (x ), i=1 α iρ fi )). (5) By f(x ) = 0, (5) gives i=1 α if i (x) i=1 α if i (x ) (x, x, 1 t=1 t (0, i=1 iρ fi )). (6) By definition, (x, x, (0, a)) 0 for all x X and any a R +. Hence, hypothesis i=1 i ρ fi 0 and (6) yield that the following inequality i=1 α if i (x) i=1 α if i (x ) holds for all x X. Since α 0, the above inequality iplies that x is a wealy efficient solution of the proble (UVOP). One of the ethods used for solving vector optiization probles, the weighting ethod, relates their wealy efficient solutions to optial solutions of corresponding scalar probles. In this approach, the following scalar unconstrained optiization proble, the so-called weighting scalar optiization proble, is constructed for the considered ultiobjective prograing proble i=1 i f i (x) in s.t. x X, (P( )) where R. The following result is well-nown in the literature (see, [15]): Theore 2.3. A iniizer of the weighting scalar optiization proble (P( )) is a wealy efficient solution of the vector optiization proble (UVOP). If all weighting coefficients are positive, that is i > 0, i = 1,...,, then an optial solution of the proble (P( )) is an efficient solution of the proble (UVOP). Now, we prove the converse result for a new class of nonconvex vector optiization probles. Theore 2.4. Let the objective function f in the proble (UVOP) be (, f )- invex on X. Further, assue that x is a wealy efficient solution in proble (UVOP) and the necessary optiality conditions are satisfied at x with Lagrange ultiplier R,

4 6 Tadeusz Antcza, Manuel Arana-Jiénez 0 with i=1 i ρ fi 0. Then a wealy efficient solution x of (UVOP) solves a weighting scalar optiization proble. Proof The proof is siilar to the proof of Theore 2.2 and, hence, it is oitted. Corollary 2.1. Let the objective function f in the proble (UVOP) be (, f )- invex on X with ρ fi 0, i = 1,. Then every wealy efficient solution of the proble (UVOP) solves a weighting scalar optiization proble. 3. KT-(, )-invexity and optiality In any practical applications, a vector optiization proble has the set of all feasible solutions given by a nuber of inequality constraints. Therefore, we consider the following constrained vector optiization proble with inequality constraints: f(x) = (f 1 (x),, f (x)) in s.t. g j (x) 0, j = 1,,, x X, (VP) where f i : X R, i I = {1,...,} and g j : X R, j J = {1,...}, are differentiable functions defined on a nonepty open convex set X R n. Further, let D {x X: g j (x) 0, j = 1,, } be the set of all feasible solutions of the considered vector optiization proble (VP) and J(x ) {j J: g j (x ) = 0}. In this section, for the considered ultiobjective prograing proble (VP), we define a new concept of generalized convexity which is a generalization of a class of (, )- invex functions earlier defined by Caristi et al. [8] and the class of differentiable KT-invex vector optiization probles introduced by Osuna-Góez et al. [17]. Definition 3.1. Let u D be given. If there exist a function : D D R n+1 R, where (x, u, ) is convex on R n+1, (x, u, (0, a)) 0 for all x D and any a R +, ρ = (ρ f1,, ρ f, ρ g,, ρ g ) R +, such that x, u D f i (x) f i (u) (x, u, ( f i (u), ρ fi )), i I, g(x) 0] [ g(u) 0 (x, u, ( g j (u), ρ gj )) 0, j J(u), then the vector optiization proble (VP) is said to be a vector KT-(, )-invex optiization proble at u D on D (with respect to, f and g). If (7) is satisfied at any point u D, then the vector optiization proble (VP) is said to be a vector KT-(, )- invex optiization proble on D. Definition 3.2. Let u D be given. If there exist a function : D D R n+1 R, where (x, u, ) is convex on R n+1, (x, u, (0, a)) 0 for all x D and any a R +, ρ = (ρ f1,, ρ f, ρ g,, ρ g ) R +, such that x, u D, x u f i (x) f i (u) > (x, u, ( f i (u), ρ fi )), i I, g(x) 0 ] [ g(u) 0 (x, u, ( g j (u), ρ gj )) 0, j J(u), then the vector optiization proble (VP) is said to be a vector strict KT-(, )-invex optiization proble at u D on D (with respect to, f and g). If the above relation is satisfied at any point u D, then the vector optiization proble (VP) is said to be a vector strict KT-(, )-invex optiization proble on D. (7) (8)

5 The weighting ethod and ultiobjective prograing under new concepts of generalized (, )-invexity7 In this section, we prove the sufficient optiality conditions for (wea) Pareto optiality in the considered ultiobjective prograing proble (VP) under assuption that (VP) is vector (strict) KT-(, )-invex. It is well nown (see, for exaple, [11], [15]) that, under a suitable constraint qualification, if x D is a (wea) Pareto optial solution in the considered ultiobjective prograing proble (VP), then the following necessary optiality conditions, nown as Karush-Kuhn-Tucer conditions, are satisfied: Theore 3.1. (Karush-Kuhn-Tucer necessary optiality conditions). Let x D be a wea Pareto solution of the proble (VP) and a suitable constraint qualification be satisfied at x. Then, there exist R and μ R such that i=1 i f i (x ) + j=1 μ j g j (x ) = 0, (9) μ jg j (x ) = 0, j J, (10) 0, μ 0. (11) Definition 3.3. The point (x,, μ ) D R R is said to be a vector Karush- Kuhn-Tucer point of the considered vector optiization proble (VP), if the conditions (9)-(10) are satisfied at x with Lagrange ultipliers and μ. Theore 3.2. Let the considered ultiobjective prograing proble (VP) be a vector KT-(, )-invex optiization proble on D with respect to, f and g. Then, every vector Karush-Kuhn-Tucer point (x,, μ ) D R R of the proble (VP) is its wealy efficient solution if i=1 i ρ fi + j=1 μ j gj 0. Proof Let the considered vector optiization proble (VP) be a vector KT-(, )- invex optiization proble on D. Further, we assue that (x,, μ ) D R R is a vector Karush-Kuhn-Tucer point of the proble (VP). Suppose, contrary to the result, that x is not a wealy efficient solution of the proble (VP). Then, by definition, there exists a feasible solution x of the proble (VP) such that f(x ) < f(x ). Since the proble (VP) is a vector KT-(, )-invex optiization proble on D, by Definition 3.1 and the Karush- Kuhn-Tucer necessary optiality condition (11), we get i (x, x, ( f i (x ), ρ fi )) 0, i I, (12) i (x, x, ( f i (x ), ρ fi )) < 0 for at least one i I, (13) Let us denote α i = i=1, i =1,,, β j = μ j (x, x, ( g j (x ), ρ gj )) 0, j J(x ). (14) i i=1 i+ j=1 μ j μ j i=1 i=1 i+ j=1 μ j, j = 1,,. Note j=1 = that 0 α i 0, but at least one α i > 0, 0 β j 0, and, oreover, i=1 α i + β j 1. By (12)-(14), it follows that i=1 α i (x, x, ( f i (x ), ρ fi )) + j=1 β j (x, x, ( g j (x ), ρ gj )) < 0. (15) By Definition 3.1, we have that (x, x, ) is convex on R n+1. Thus, by (15) and convexity of (x, x, ), we obtain (x, x, ( i=1 α i f i (x ) + j=1 β j g j (x ), i=1 α i ρ fi + j=1 β jρ gj )) i=1 α i (x, x, ( f i (x ), ρ fi )) + j=1 β j (x, x, ( g j (x ), ρ gj )). (16)

6 8 Tadeusz Antcza, Manuel Arana-Jiénez Cobining (15) and (16), we get that the following inequality (x, x, ( i=1 α i f i (x ) + j=1 β j g j (x ), i=1 α i ρ fi + j=1 β jρ gj )) < 0 holds. By the Karush-Kuhn-Tucer necessary optiality condition (9), we have (x, x, (0, i=1 α i ρ fi + j=1 β jρ gj )) < 0. (17) By assuption, it follows that i=1 α i ρ fi + j=1 β j gj 0. As it follows fro Definition 3.1, (x, x, (0, a)) 0 for any a R +. This iplies that the inequality (x, x, (0, i=1 α i ρ fi + j=1 β jρ gj )) 0 holds, contradicting (17). This copletes the proof of this theore. Theore 3.3. Let the considered vector optiization proble (VP) be vector KT- (, )-invex on D, a suitable constraint qualification be satisfied at any wealy efficient solution x of the proble (VP) and the Karush-Kuhn-Tucer necessary optiality conditions be at x satisfied with Lagrange ultipliers R and μ R. If i=1 i ρ fi + j=1 μ j gj 0, then every wealy efficient solution of the original vector optiization proble (VP) solves a weighting scalar optiization proble. In order to illustrate the results established in this section, we consider the exaple of a ultiobjective prograing proble with KT-(, )-invex functions. Exaple 3.1. Consider the following nonconvex ultiobjective prograing proble f(x) = (ln((x 1 1) 2 + 1), ln((x 2 1) 2 + 1)) (VP1) g(x) = 1 x 1 x 2 0. Note that D = {(x 1, x 2 ) R 2 : x 1 x 2 1} and x = (1,1) is such a feasible solution at which the Karush-Kuhn-Tucer necessary optiality conditions are satisfied. It can be shown, by Definition 3.1, that (VP1) is KT-(, )-invex at x on D, where (x, x, (θ, ρ)) = ϑ 1 ln((x 1 1) 2 + 1) + ϑ 2 ln((x 2 1) 2 + 1) + (2 ρ 1)(ln((x 1 1) 2 + 1) + ln((x 2 1) 2 + 1)), and is equal to ρ f1 = 0, ρ f2 = 0 and ρ g = 1, respectively. Note that all hypotheses of Theore 3.3 are satisfied, then x is Pareto optial to the considered ultiobjective prograing proble. It is not difficult to show that the constraint function g is not invex on D with respect to any function D D R 2. This follows fro the fact that a stationary point of the constraint function g is not its global iniizer (see [7]). Since not all functions constituting the considered vector optiization proble are invex with respect to the sae function (what is ore, soe of the are not invex with respect to any ), then the sufficient optiality conditions given in [17] are not applicable in this case. Further, the objective function f and the constraint function g are not (, )-invex at x on D with respect to and defined above and, therefore, also the sufficient conditions given in [10] are not applicable in this case. Thus, the optiality conditions established in the paper are applicable for a larger class of nonconvex vector optiization probles than the sufficient optiality conditions established under other generalized convexity notions, even those ones entioned above.

7 The weighting ethod and ultiobjective prograing under new concepts of generalized (, )-invexity9 4. WD-(, )-invexity and duality In this section, for the considered ultiobjective prograing proble (VP), consider the following dual proble in the sense of Mond-Weir: f(y) in s. t. i=1 i f i (y) + j=1 μ j g j (y) = 0, μ j g j (y) = 0, j = 1,,, R, 0, μ R (VD), μ 0, x X. Let be the set of all feasible solutions in proble (VD). Further, denote Y = {y X : (y,, ) }. In order to prove several duality results between the considered vector optiization proble (VP) and its vector dual proble in the sense of Mond-Weir (VD), we now introduce the definition of WD-(, )-invexity on a nonepty subset of S containing the set D Y. Let S be a nonepty subset of X such that D Y S and u S be an arbitrary point. Definition 4.1. Let u S be given. If there exist a function S S R n+1 R, where (x, u, ) is convex on R n+1, (x, u, (0, a)) 0 for all x S and any a R +, ρ = (ρ f1,, ρ f, ρ g,, ρ g ) R +, such that x S f i (x) f i (u) (x, u, ( f i (u), ρ fi )), i I, u S ] [ g(x) 0 g j (u) (x, u, ( g j (u), ρ gj )) 0, j J, then the vector optiization proble (VP) is said to be a vector WD-(, )-invex optiization proble at u S on S (with respect to, f and g). If (18) is satisfied at any point u S, then the vector optiization proble (VP) is said to be a vector WD-(, )- invex optiization proble on S. Definition 4.2. Let u S be given. If there exist a function S S R n+1 R, where (x, u, ) is convex on R n+1, (x, u, (0, a)) 0 for all x S and any a R +, ρ = (ρ f1,, ρ f, ρ g,, ρ g ) R +, such that x, u S, x u f i (x) f i (u) > (x, u, ( f i (u), ρ fi )), i I, ] [ g(x) 0 g j (u) (x, u, ( g j (u), ρ gj )) 0, j J, then the vector optiization proble (VP) is said to be a vector strict WD-(, )-invex optiization proble at u S on S (with respect to, f and g). If (19) is satisfied at any point u S, then the vector optiization proble (VP) is said to be a vector strict WD- (, )-invex optiization proble on S. Theore 4.1. (Wea duality). Let x and (y,, ) be any feasible solutions of the vector optiization proble (VP) and its vector Mond-Weir dual proble (VD), respectively. Further, assue that proble (VP) is WD-(, )-invex on D Y with respect to, f and g. If i=1 i ρ fi + j=1 j gj 0, then f(x) f(y). Proof Suppose, contrary to the result, that f(x) < f(y). By the feasibility of (y,, ) to the proble (VD), it follows that i=1 i f i (x) < i=1 i f i (y). (20) By assuption, the vector optiization proble (VP) is WD-(, )-invex on D Y with respect to, f and g. Therefore, by Definition 4.1, the inequality (18) (19)

8 10 Tadeusz Antcza, Manuel Arana-Jiénez i=1 i f i (x) i=1 i f i (y) i=1 i (x, y, ( f i (y), ρ fi )) (21) holds. Hence, (20) and (21) yield i=1 i (x, y, ( f i (y), ρ fi )) < 0. (22) Using Definition 4.1 again together with (y,, ), we get j=1 μ j g j (u) μ j (x, u, ( g j (u), ρ gj )) Thus, the second constraint of proble (VD) iplies j=1. j=1 μ j (x, u, ( g j (u), ρ gj )) 0. (23) Cobining (22) and (23), we obtain i=1 i (x, y, ( f i (y), ρ fi )) + j=1 μ j (x, u, ( g j (u), ρ gj )) < 0. (24) Let us denote α i = i i=1 i=1 i + j=1 μ j, i =1,,, β j = μ j i=1 i=1 i + j=1 μ j, j = 1,,. Note j=1 = that 0 α i 0, but at least one α i > 0, 0 β j 0, and, oreover, i=1 α i + β j 1. By (24), it follows that i=1 α i (x, y, ( f i (y), ρ fi )) + j=1 β j (x, y, ( g j (y), ρ gj )) < 0. (25) By Definition 4.1, we have that (x, y, ) is convex on R n+1. Hence, by (25) and convexity of (x, y, ), we get (x, y, ( i=1 α i f i (y) + j=1 β j g j (y), i=1 α i ρ fi + j=1 β j ρ gj )) i=1 α i (x, y, ( f i (y), ρ fi )) + j=1 β j (x, y, ( g j (y), ρ gj )). (26) Thus, (25) and (26) yield that the following inequality (x, y, ( i=1 α i f i (y) + j=1 β j g j (y), i=1 α i ρ fi + j=1 β j ρ gj )) < 0 holds. By the first constraint of (VD), the inequality above iplies (x, y, 1 i=1 i + j=1 μ j (0, i i=1 ρ fi + j=1 j ρ gj )) < 0. (27) By assuption, i=1 i ρ fi + j=1 j ρ gj 0. Then, since fro Definition 3.1, (x, x, (0, a)) 0 for any a R +, then, by assuption, the following inequality (x, y, 1 i=1 i + j=1 μ j (0, i i=1 ρ fi + j=1 j ρ gj )) 0 holds, contradicts (27). This copletes the proof of this theore. Theore 4.2. (Wea duality). Let x and (y,, ) be any feasible solutions of (VP) and (VD), respectively. Further, assue that proble (VP) is strict WD-(, )-invex on D Y with respect to, f and g. If i=1 i ρ fi + j=1 j gj 0, then f(x) f(y). Theore 4.3. (Strong duality). Let x D be a (wea) Pareto solution of the vector optiization proble (VP) and the suitable constraint qualification be satisfied at x. Then, there exist R and μ R such that (x,, μ ) is feasible for (VD) and the objective functions of (VP) and (VD) are equal to these points. If all hypotheses of the wea duality

9 The weighting ethod and ultiobjective prograing under new concepts of generalized (, )-invexity11 theore (Theore 4.2 or Theore 4.3) are satisfied, then (x,, μ ) is a (wealy) efficient solution of a axiu type for (VD) Proof By assuption, x D is a (wea) Pareto solution of (VP) and the suitable constraint qualification is satisfied at x. Then, there exist Lagrange ultipliers R and μ R such that the Karush-Kuhn-Tucer necessary optiality conditions (9)-(11) are satisfied at x. Then, the feasibility of (x,, μ ) in (VD) follows directly fro these necessary optiality conditions. Hence, the objective functions of probles (VP) and (VD) are equal at x and (x,, μ ) are equal at these points. Thus, (wea) efficiency of (x,, μ ) in (VD) follows directly fro the wea duality theore (Theore 4.2 or Theore 4.3). Theore 4.4. (Converse duality). Let (y,, μ ) be a (wealy) efficient of a axiu type to the vector Mond-Weir dual proble (VD) such that y D. Further, assue that the considered ultiobjective prograing proble (VP) is (strict) WD- (, )-invex at y on D Y with respect to, f and g. If i=1 i ρ fi + j=1 μ j gj 0, then y is a wea Pareto solution (Pareto solution) of the considered ultiobjective prograing proble (VP). Proof Proof of this theore follows directly fro the wea duality theore (Theore 4.2 or Theore 4.3). Theore 4.5. (Restricted converse duality): Let (y,, μ ) be feasible to Mond- Weir vector dual proble (VD). Further, assue that the considered ultiobjective prograing proble (VP) is (strict) WD-(, )-invex at y on D Y with respect to, f and g with i=1 i ρ fi + j=1 μ j gj 0. If there exists x D such that f(x ) = f(y ), then x is a (wea) Pareto solution of the proble (VP). 5. Conclusions In the paper, the scalarization ethod, that is, the weighting ethod, has been used for solving a new class of nonconvex differentiable vector optiization probles. It has been established that a wealy efficient solution of an unconstrained sooth vector optiization proble in which the objective function is (, )-invex is related to an optial solution of its corresponding weighting scalar optiization proble constructed in this ethod. Further, we have established the sae result in the case when the weighting ethod has been used for solving the KT-(, )-invex constrained vector optiization proble. Hence, the weighting ethod has been used for a larger class of nonconvex differentiable vector optiization probles, in coparison to other siilar results, previously established in the literature under other generalized convexity notions. Further, new classes of nonconvex ultiobjective prograing probles have been defined in the paper. By introducing the concepts of KT-(, )-invexity and WD- (, )-invexity, we have generalized notions of generalized convexity introduced by Martin [14] for scalar optiization probles to new classes of nonconvex differentiable vector optiization probles. The definition of a KT-(, )-invex vector optiization proble introduced in the paper unifies any classes of generalized convex optiization probles, earlier defined in optiization theory. Therefore, the sufficient optiality conditions established in the paper are applicable also for such nonconvex vector optiization probles for which other generalized convexity notions ay avoid in proving such a result.

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