C:/Documents and Settings/Profesor/Escritorio/Cubo/Cubo/Cubo/Cubo 2011/Cubo2011-13-01/Vol13 n\2721/Art N\2608 .dvi CUBO A Mathematical Journal Vol.13, No¯ 01, (125–136). March 2011 Multiple Objective Programming Involving Differentiable (Hp, r)-invex Functions Xiaoling Liu, Dehui Yuan, Shengyun Yang, Guoming Lai Department of Mathematics and Inform., Tech.,Hanshan Normal University, Chaozhou, Guangdong, 521041, China. email: ydhliu@gmail.com, ydhlxl@sohu.com and Chuanqing Xu Department of Mathematics, Beijing Armed Forced Eng. Institute, Beijing 100072, China. ABSTRACT In this paper, we introduce new types of generalized convex functions which include locally (Hp, r)-pre-invex functions and (Hp, r)-invex functions. Relationship between these two new classes of functions are established. We also present the conditions for optimality in differentiable mathematical programming problems where the functions considered are (Hp, r)-invex functions introduced in this paper. RESUMEN Este trabajo, establece nuevos tipos de funciones convexas generalizadas que incluyen localmente funciones (Hp, r) de pre-invex y funciones (Hp, r)-invex. La relacin entre 126 Xiaoling Liu, Dehui Yuan and Shengyun Yang CUBO 13, 1 (2011) estas dos nuevas clases de funciones estn establecidas. Tambin se presentan las condi- ciones de optimalidad en diferenciables problemas de programacin matemtica, donde las funciones consideradas en este artculo son funciones (Hp, r)-invex. Keywords: Differentiable mathematical programming. Mathematics Subject Classification: 90B50. 1 Introduction Convexity plays a central role in many aspects of mathematical programming (see[22, 5, 8]) includ- ing analysis of stability [9, 16], sufficient optimality conditions and duality [12, 15, 11]. Based on convexity assumptions, nonlinear programming problems can be solved efficiently. There have been many attempts to weaken the convexity assumptions in order to treat many practical problems. Therefore, many concepts of generalized convex functions have been introduced and applied to mathematical programming problems in the literature [1, 2, 19, 27, 28, 25]. One of these concepts, invexity, was introduced by Hanson in [11]. Hanson has shown that invexity has a common property in mathematical programming with convexity that Karush Kuhn Tucker conditions are sufficient for global optimality of nonlinear programming under the invexity assumptions. Ben-Israel and Mond [7] introduced the concept of pre-invex functions which is a special case of invexity. Following [11] and [7], many authors have introduced concepts of generalized invexity and pre-invexity including strictly pseudoinvex functions and quasiinvex functions [13], prepseudoinvex and prequasiinvex functions [21] and r-pre-pseudoinvex functions [1]. The relationships between some of these generalized invex functions were studied in [20, 21]. Recently, Antczak [3] introduced new definitions of p-invex sets and (p, r)-invex functions which can be seen as generalization of invex functions. He also discussed nonlinear programming problems involving the (p, r)-invexity-type functions in [2, 4]. On the other hand, Kaul et al. [14] introduced the classes of locally connected sets which generalizes the arcwise connected sets [6] and locally star-shaped sets [10]. Based on the new class of sets, they [14, 15] introduced a new class of functions called locally connected functions. They [15]also defined the directional derivative (with respect to a vector function) of a real valued function, and also defined locally P -connected functions in terms of its right differential. Motivated by [7, 14, 1, 2, 3, 4, 15, 18], Yuan et al. introduced definition of a new class of sets, locally Hp-invex sets, and definitions of classes of generalized convex functions called locally (Hp, r, α)-pre-invex functions. And we give the concept of locally differentiable (Hp, r)- invex functions, discuss the relationship between (Hp, r)-invexity and locally (Hp, r, 1)-pre-invexity. Based on the definition of locally differentiable (Hp, r)-invexity, we have managed to deal with nonlinear programming problems under some assumptions. The rest of the paper is organized as follows: In Section 2, we give some preliminary concepts CUBO 13, 1 (2011) Multiple Objective Programming Involving Differentiable (Hp, r)-invex Functions 127 regarding locally Hp-convex sets, locally (Hp, r, α)-preinvex function, and differentiable (Hp, r)- invex function, discuss the relationship between (Hp, r)-invexity and locally (Hp, r, 1)-pre-invexity. In Section 3, we present the conditions for optimality in differentiable mathematical programming problems in which the functions considered belong to the classes of functions introduced in section. In Section 4, we present the conditions of optimality for the following nonlinear differentiable fractional programming problems (MFP) with the same convexity assumption. 2 Differentiable (Hp, r)-invex Functions Let Rn be the n-dimensional Euclidean space, Rn+ = {x ∈ R n|x > 0} and Ṙn+ = {x ∈ R n|x > 0}. In this section, we give definitions of locally Hp-invex set. Definition 1. [3] Let a1, a2 > 0, λ ∈ (0, 1) and r ∈ R. Then the weighted r-mean of a1 and a2 is given by Mr(a1, a2; λ) := { (λar1 + (1 − λ)a r 2) 1 r for r 6= 0, aλ1 a 1−λ 2 for r = 0. Definition 2. [29] S ⊂ Rn is a locally Hp-invex set if and only if, for any x, u ∈ S, there exist a maximum positive number a(x, u) ≤ 1 and a vector function Hp : S × S × [0, 1] → R n, such that Hp(x, u; 0) = e u, Hp(x, u; λ) ∈ Ṙ n +, ln (Hp(x, u; λ)) ∈ S, ∀ 0 < λ < a(x, u) f or p ∈ R. and Hp(x, u; λ) is continuous on the interval (0, a(x, u)), where the logarithm and the exponentials appearing in the relation are understood to be taken componentwise. Definition 3. [29] A function f : S → R defined on a locally Hp-invex set S ⊂ R n is said to be locally (Hp, r)-pre-invex on S if, for any x, u ∈ S, there exists a maximum positive number a(x, u) ≤ 1 such that f (ln (Hp(x, u; λ))) ≤ ln ( Mr(e f (x), ef (u); λα) ) , ∀ 0 < λ < a(x, u) f or p ∈ R where the logarithm and the exponentials appearing on the left-hand side of the inequality are understood to be taken componentwise. If u is fixed, then f is said to be locally (Hp, r)-pre-invex at u. Correspondingly, if the direction of above inequality is changed to the opposite one, then f is said to (Hp, r)-pre-incave on S or at u. Now, we introduce the classes of differentiable (Hp, r)-invex functions. For convenience, we assume that S be a Hp-invex set, Hp is right differentiable at 0 with respect to variable λ for each given pair x, u ∈ S, and f : S → R is differentiable on S. The symbol H′p(x, u; 0+) , (H′p1(x, u; 0+), · · · , H ′ pn(x, u; 0+)) T denotes the right derivative of Hp at 0 with respect to variable λ for each given pair x, u ∈ S; ▽f (x) , (▽1f (x), . . . , ▽nf (x)) T denotes the differential of f at x, where ∂if (x) is partial differential of f with respect to the i-th componentwise; ▽f (u) e u denotes ( ▽1f (u) e u 1 , · · · , ▽nf (u) e un )T . 128 Xiaoling Liu, Dehui Yuan and Shengyun Yang CUBO 13, 1 (2011) Definition 4. Let S be a Hp-invex set, Hp is right differentiable at 0 with respect to variable λ for each given pair x, u ∈ S, and f : S → R is differentiable on S. If for all x ∈ S, one of the relations 1 r erf (x) > 1 r erf (u) [ 1 + r ▽f (u) eu T H′p(x, u; 0+) ] (>) f or r 6= 0, f (x) − f (u) > ▽f (u) e u T H′p(x, u; 0+) (>) f or r = 0, (2.1) holds, then f is said to be (Hp, r)-invex (strictly (Hp, r)-invex) at u ∈ S. If the inequalities (2.1) are satisfied at any point u ∈ S, then f is said to be (Hp, r)-invex (strictly (Hp, r)-invex) on S. Remark 5. Any function f satisfying (2.1) is called (Hp, r)-invex (strictly (Hp, r)-invex) on S. However, if Hp(x, u; λ) = Mp(e η(x,u)+u, eu; λ) and a(x, u) = 1 for all x, u ∈ S, we will say that f is (p, r)-invex (strictly (p, r)-invex) with respect to η on S; furthermore, f is r-invex (strictly (p, r)-invex) with respect to η on S in the case p = 0 and f is invex (strictly invex) with respect to η on S in the case p = 0 and r = 0. Remark 6. In order to define an analogous class of (strict) (Hp, r)-incave functions on the Hp-invex set, the direction of the inequality (2.1) in the definition of these functions should be changed to the opposite one. Theorem 7. Let S ⊂ Rn be a Hp-invex set, Hp is right differentiable at 0 with respect to variable λ for each given pair x, u ∈ S, and f : S → R is differentiable on S. If f is (Hp, r, α)-pre-invex ((Hp, r, α)-pre-incave) on S and α = 1, then f is (Hp, r)-invex ((Hp, r)-incave) on S. Proof. The theorem will be proved only in the case f is (Hp, r, α)-pre-invex, and we just prove that the theorem is true when r > 0 (the proof in the case when r < 0 is analogous to the one when r > 0; only the directions of the inequalities should be changed to the opposite ones). Firstly, we prove the theorem is true in the case when r > 0. Since f : S → R is a (Hp, r, α)- pre-invex function and α = 1, by Definition 3, we have erf (u) [ erf (ln(Hp(x,u;λ)))−rf (u) − 1 ] λ−1 6 erf (x) − erf (u). By letting λ → 0, we get the inequality erf (x) − erf (u) > rerf (u) ▽f (u) eu T H′p(x, u; 0+) that is 1 r erf (x) > 1 r erf (u) [ 1 + r ▽f (u) eu T H′p(x, u; 0+) ] Now, we prove the case r = 0. By Definition 3, we have [f (ln (Hp(x, u; λ))) − f (u)] λ −1 6 f (x) − f (u). By letting λ → 0, we get the inequality f (x) − f (u) > ▽f (u) eu T H′p(x, u; 0+) Therefore, we get the desired result. � CUBO 13, 1 (2011) Multiple Objective Programming Involving Differentiable (Hp, r)-invex Functions 129 3 Optimality for Multiple Objective Programming In this section, we present the conditions for optimality in differentiable mathematical programming problems in which the functions considered belong to the classes of functions introduced earlier in this paper. Consider the following form of optimization problem (VOP) min f (x) g(x) 6 0, x ∈ S, where S ⊂ Rn, f : S → Rq, g : S → Rm. Let us denote by E the set of feasible solutions of (VOP), i.e., the set of the form E := {x ∈ S|g(x) 6 0}. From now on, we assume that Hp is right differentiable at 0 with respect to variable λ for each given pair x, u ∈ S, fi : S → R(i = 1, · · · , q), gj : S → R(j = 1, · · · , m) are differentiable on S, f = (f1, · · · , fq), g = (g1, · · · , gm) and S is an Hp-invex (nonempty) set. Definition 8. x̄ ∈ E is said to be an efficient solution for problem (VOP), if there exists no x ∈ E such that f (x) 6 f (x̄); x̄ ∈ E is said to be a weak efficient solution for problem (VOP), if there exists no x ∈ E such that f (x) < f (x̄); Theorem 9. Let E be a Hp-invex set with the respect to the same Hp. Assume that x̄ ∈ S is feasible for problem (VOP), and there exists λ ∈ Rq, µ ∈ Rm such that ∑q i λi▽fi(x̄) + ∑m j=1 µj ▽gj (x̄) = 0, (3.1) ∑m j=1 µj gj (x̄) = 0, (3.2) λ = (λ1, · · · , λq) > 0, ∑q i λi = 1, µ = (µ1, · · · , µm) > 0. (3.3) If fi(i = 1, · · · , q) are strictly (Hp, r)-invex and gj (j = 1, · · · , m) are (Hp, r)-invex at x̄ on S, then x̄ is an efficient solution of problem (VOP). Proof. Here we prove only the cases when r > 0 or r = 0(the proof of the case when r < 0 is similar to the one when r > 0; the only changes arise from the form of inequalities defining the class of (Hp, r)-invex functions). Assume that x is an arbitrary feasible point for problem (VOP). On the contrary to suppose that x̄ is not an efficient solution of problem (VOP). Thus, there exists x ∈ E such that f (x) 6 f (x̄). We first consider the case when r > 0. Therefore, q ∑ i λi { erfi(x) − erfi(x̄) } 6 0. By hypothesis, f and gi(i = 1, · · · , m) are (Hp, r)-invex at x̄ on S; therefore, for all x ∈ S, the 130 Xiaoling Liu, Dehui Yuan and Shengyun Yang CUBO 13, 1 (2011) inequalities 1 r erfi(x) > 1 r erfi(x̄) [ 1 + r ▽fi(x̄) ex̄ T H′p(x, u; 0+) ] , i = 1, · · · , q, (3.4) 1 r ergj (x) > 1 r ergj (x̄) [ 1 + r ▽gj (x̄) ex̄ T H′p(x, x̄; 0+) ] , j = 1, · · · , m, (3.5) are true. Denote I(x̄) , {j|µj > 0, j = 1, · · · , m}. By (3.2), we have gj (x̄) = 0 if j ∈ I(x̄) and µj = 0 if gj (x̄) 6= 0, thus gj (x) 6 gj (x̄) for j ∈ I(x̄). Therefore, from (3.4) and (3.5), we have er(fi(x)−fi(x̄)) > 1 + r ▽fi(x̄) ex̄ T H′p(x, u; 0+) (3.6) r ▽gj (x̄) ex̄ T H′p(x, x̄; 0+) 6 0, j ∈ I(x̄). (3.7) By(3.6), (3.7) and (3.3), we deduce that ( ∑m i=1 λi▽fi(x̄) ex̄ + ∑ j∈I(x̄) µj ▽gj (x̄) ex̄ )T H′p(x, x̄; 0+) < 0. (3.8) Notice that ∑m j=1 µj ▽gj (x̄) = ∑ j∈I(x̄) µj ▽gj (x̄), by (3.8), we have ( ∑m i=1 λi▽fi(x̄) + ∑m j=1 µj ▽gj (x̄) ex̄ )T H′p(x, x̄; 0+) < 0. This, together with (3.1), follows a contradiction 0 < 0. Now, we prove the theorem is true in the case when r = 0. Since f (x) 6 f (x̄), then ∑q i λi (fi(x) − fi(x̄)) 6 0. By Definition 4, we have fi(x) − fi(x̄) > ▽fi(x̄) ex̄ T H′p(x, x̄; 0+), i = 1, . . . , q, gj (x) − gj (x̄) > ▽gj (x̄) ex̄ T H′p(x, x̄; 0+), j = 1, · · · , m, On the same line as the case when r > 0, we have the contradiction 0 < 0 again. Therefore, x̄ is an efficient solution for problem (VOP). � Theorem 10. Let E be a Hp-invex set with the respect to the same Hp. Assume that x̄ ∈ S is feasible for problem (VOP), and there exists λ ∈ Rq, µ ∈ Rm such that ∑q i λi▽fi(x̄) + ∑m j=1 µj ▽gj (x̄) = 0, (3.9) ∑m j=1 µj gj (x̄) = 0, (3.10) λ = (λ1, · · · , λq) > 0, ∑q i λi = 1, µ = (µ1, · · · , µm) > 0. (3.11) If fi(i = 1, · · · , q), gj (j = 1, · · · , m) are (Hp, r)-invex at x̄ on S, then x̄ is an efficient solution of problem (VOP). CUBO 13, 1 (2011) Multiple Objective Programming Involving Differentiable (Hp, r)-invex Functions 131 Proof. The theorem can be proved on simple lines as Theorem 9. � The assumption on functions in Theorem 9 ( or Theorem 10) could also be given in another form. It is enough to assume that the Lagrange function fi + ∑m j=1 µj gj (i = 1, . . . , m) are strictly (Hp, r)-invex (or strictly (Hp, r)-invex) . And so, the following two theorems are true. Their proofs are on the same line as Theorem 9, therefore we delete them here. Theorem 11. Assume that x̄ ∈ S is feasible for problem (VOP), and there exists λ ∈ Rq, µ ∈ Rm satisfying (3.1), (3.2) and (3.3). If fi + ∑m j=1 µj gj (i = 1, . . . , m) are strictly (Hp, r)-invex at x̄ on S, then x̄ is an efficient solution of problem (VOP). Theorem 12. Assume that x̄ ∈ S is feasible for problem (VOP), and there exists λ ∈ Rq, µ ∈ Rm satisfying (3.9), (3.10) and (3.11). If fi + ∑m j=1 µj gj (i = 1, . . . , m) are (Hp, r)-invex at x̄ on S, then x̄ is an efficient solution of problem (VOP). For weak efficient solution of problem (VOP), we have the following theorems. Theorem 13. Assume that x̄ ∈ S is feasible for problem (VOP), and there exists λ ∈ Rq, µ ∈ Rm satisfying (3.1), (3.2) and (3.3). If fi(i = 1, · · · , q), gj (j = 1, · · · , m) are (Hp, r)-invex at x̄ on S, then x̄ is a weak efficient solution of problem (VOP). Theorem 14. Assume that x̄ ∈ S is feasible for problem (VOP), and there exists λ ∈ Rq, µ ∈ Rm satisfying (3.1), (3.2) and (3.3). If fi + ∑m j=1 µj gj (i = 1, . . . , m) are (Hp, r)-invex at x̄ on S, then x̄ is a weak efficient solution of problem (VOP). 4 Multiple Objective Fractional Programming In this section, we present the conditions of optimality for the following nonlinear differentiable fractional programming problems (MFP). (MFP) min f (x) g(x) , ( f1(x) g1(x) , f2(x) g2(x) , · · · , fq(x) gq(x) )T s.t. h(x) = (h1(x), h2(x), · · · , hm(x)) 6 0 x ∈ S, where S ⊂ Rn, f : S → Rq, g : S → Rq, h : S → Rm, f = (f1, · · · , fq), g = (g1, · · · , gq), h = (h1, · · · , hm), are differentiable on S and S is an Hp-invex (nonempty) set. Moreover, for i = 1, . . . , q, gi(x) > 0 for all x ∈ S. Let us denote by E the set of feasible solutions of (MFP), i.e., the set of the form E := {x ∈ S|h(x) 6 0}. We assume that Hp is right differentiable at 0 with respect to variable λ for each given pair x, u ∈ S. Definition 15. x̄ ∈ E is said to be an efficient solution for problem (MFP), if there exists no x ∈ E such that f (x) g(x) 6 f (x̄) g(x̄) ; x̄ ∈ E is said to be a weak efficient solution for problem (MFP), if there exists no x ∈ E such that f (x) g(x) < f (x̄) g(x̄) ; 132 Xiaoling Liu, Dehui Yuan and Shengyun Yang CUBO 13, 1 (2011) Theorem 16. Assume that x̄ ∈ S is feasible for problem (MFP), and there exists λ ∈ Rq, u ∈ Rq, µ ∈ Rm such that ∑q i λi(▽fi(x̄) − ui▽gi(x̄)) + ∑m j=1 µj ▽hj (x̄) = 0, (4.1) ∑m j=1 µj hj (x̄) = 0, (4.2) λ = (λ1, · · · , λq) > 0, ∑q i λi = 1, µ = (µ1, · · · , µm) ≧ 0, (4.3) u = (u1, · · · , uq) ≧ 0, ui = fi(x̄)/gi(x̄), i = 1, . . . , q. (4.4) If fi −uigi(i = 1, · · · , q) are strictly (Hp, r)-invex at x̄ on S, and hj (j = 1, · · · , m) are (Hp, r)-invex at x̄ on S, then x̄ is an efficient solution of problem (MFP). Proof. Similar to Theorem 9, here we prove only the cases when r > 0 or r = 0. Assume that x is an arbitrary feasible point for problem (MFP). On the contrary to suppose that x̄ is not an efficient solution of problem (MFP). Thus, there exists x ∈ E such that f (x) g(x) 6 f (x̄) g(x̄) = u. We first consider the case when r > 0. Therefore, q ∑ i λi { er(fi(x)−uigi(x)) − 1 } 6 0. By the convexity of fi − uigi(i = 1, · · · , q) and hj (j = 1, · · · , m), we have er(fi(x)−uigi(x)) − 1 > r ( ▽fi(x̄) − ui▽gi(x̄) ex̄ )T H′p(x, u; 0+), i = 1, · · · , q, (4.5) 1 r erhj (x) > 1 r erhj (x̄) [ 1 + r ▽hj (x̄) ex̄ T H′p(x, x̄; 0+) ] , j = 1, · · · , m, (4.6) Thus, by (4.2), (4.3), (4.5) and (4.6), we deduce that ( ∑q i=1 λi(▽fi(x̄) − ui▽gi(x̄)) + ∑m j=1 µj ▽hj (x̄) ex̄ )T H′p(x, u; 0+) < 0 Therefore, we get a contradiction 0 < 0. Similarly, we can prove the theorem in the case when r = 0. � Theorem 17. Assume that x̄ ∈ S is feasible for problem (MFP), and there exists λ ∈ Rq, u ∈ Rq, µ ∈ Rm satisfying (4.1), (4.2), (4.3) and (4.4). If one of the following holds: 1) fi(i = 1, · · · , q) are strictly (Hp, r)-invex at x̄ on S, uigi(i = 1, · · · , q) are (Hp, r)-incave at x̄ on S, and hj (j = 1, · · · , m) are (Hp, r)-invex at x̄ on S; 2) uigi(i = 1, · · · , q) are strictly (Hp, r)-incave at x̄ on S, fi(i = 1, · · · , q) and hj (j = 1, · · · , m) are (Hp, r)-invex at x̄ on S; then x̄ is an efficient solution of problem (MFP). CUBO 13, 1 (2011) Multiple Objective Programming Involving Differentiable (Hp, r)-invex Functions 133 Proof. In the same way as Theorem 16, here we prove only 1) for the cases when r > 0, 2) can be proved similar to 1). Assume that x is an arbitrary feasible point for problem (MFP). On the contrary to suppose that x̄ is not an efficient solution of problem (MFP). Thus, there exists x ∈ E such that f (x) g(x) 6 f (x̄) g(x̄) = u. Therefore, q ∑ i λi { erfi(x) − eruigi(x) } 6 0. (4.7) 1) Since fi(i = 1, · · · , q) are strictly (Hp, r)-invex at x̄ on S, uigi(i = 1, · · · , q) are (Hp, r)- incave at x̄ on S, and hj (j = 1, · · · , m) are (Hp, r)-invex at x̄ on S, then er(fi(x)) − er(fi(x̄)) > r ( ▽fi(x̄) ex̄ )T H′p(x, u; 0+), i = 1, · · · , q, (4.8) er(uigi(x)) − er(uigi(x̄)) 6 r ( ui▽gi(x̄) ex̄ )T H′p(x, u; 0+), i = 1, · · · , q, (4.9) 1 r erhj (x) > 1 r erhj (x̄) [ 1 + r ▽hj (x̄) ex̄ T H′p(x, x̄; 0+) ] , j = 1, · · · , m, (4.10) Notice that erfi(x̄)) = er(uigi(x̄)), by (4.2)-(4.4), (4.7)-(4.10), we have ( ∑q i=1 λi(▽fi(x̄) − ui▽gi(x̄)) + ∑m j=1 µj ▽hj (x̄) ex̄ )T H′p(x, u; 0+) < 0. Therefore, we get a contradiction 0 < 0. � Theorem 18. Assume that x̄ ∈ S is feasible for problem (MFP), and there exists λ ∈ Rq, u ∈ Rq, µ ∈ Rm such that ∑q i λi(▽fi(x̄) − ui▽gi(x̄)) + ∑m j=1 µj ▽hj (x̄) = 0, (4.11) ∑m j=1 µj hj (x̄) = 0, (4.12) λ = (λ1, · · · , λq) > 0, ∑q i λi = 1, µ = (µ1, · · · , µm) ≧ 0, (4.13) u = (u1, · · · , uq) ≧ 0, ui = fi(x̄)/gi(x̄), i = 1, . . . , q. (4.14) If fi − uigi(i = 1, · · · , q) and hj (j = 1, · · · , m) are (Hp, r)-invex at x̄ on S, then x̄ is an efficient solution of problem (MFP). Proof. The theorem can be proved on simple lines as Theorem 16. Theorem 19. Assume that x̄ ∈ S is feasible for problem (MFP), and there exists λ ∈ Rq, u ∈ Rq, µ ∈ Rm satisfying (4.11), (4.12), (4.13) and (4.14). If uigi(i = 1, · · · , q) are (Hp, r)-incave at x̄ on S, fi(i = 1, · · · , q) and hj (j = 1, · · · , m) are (Hp, r)-invex at x̄ on S, then x̄ is an efficient solution of problem (MFP). 134 Xiaoling Liu, Dehui Yuan and Shengyun Yang CUBO 13, 1 (2011) Proof. The theorem can be proved on simple lines as Theorem 17. Theorem 20. Assume that x̄ ∈ S is feasible for problem (MFP), and there exists λ ∈ Rq, u ∈ Rq, µ ∈ Rm such that ∑q i λi(▽fi(x̄) − ui▽gi(x̄)) + ∑m j=1 µj ▽hj (x̄) = 0, ∑m j=1 µj hj (x̄) = 0, λ = (λ1, · · · , λq) > 0, ∑q i λi = 1, µ = (µ1, · · · , µm) ≧ 0, u = (u1, · · · , uq) ≧ 0, ui = fi(x̄)/gi(x̄), i = 1, . . . , q. If one of the following holds: 1) fi − uigi(i = 1, · · · , q) and hj (j = 1, · · · , m) are (Hp, r)-invex at x̄ on S; 2) uigi(i = 1, · · · , q) are (Hp, r)-incave at x̄ on S, fi(i = 1, · · · , q) and hj (j = 1, · · · , m) are (Hp, r)-invex at x̄ on S; then x̄ is a weak efficient solution of problem (MFP). Proof. The theorem can be proved on simple lines as Theorem 16. Received: September 2009. Revised: November 2009. 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