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Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 103864, 7 pages http://dx.doi.org/10.1155/2013/103864 Research Article Semistrict πΊ-Preinvexity and Optimality in Nonlinear Programming Z. Y. Peng,1,2 Z. Lin,1 and X. B. Li1 1 2 College of Science, Chongqing JiaoTong University, Chongqing 400074, China Department of Mathematics, Inner Mongolia University, Hohhot 010021, China Correspondence should be addressed to Z. Y. Peng; [email protected] Received 26 October 2012; Accepted 10 April 2013 Academic Editor: Natig M. Atakishiyev Copyright © 2013 Z. Y. Peng et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A class of semistrictly πΊ-preinvex functions and optimality in nonlinear programming are further discussed. Firstly, the relationships between semistrictly πΊ-preinvex functions and πΊ-preinvex functions are further discussed. Then, two interesting properties of semistrictly πΊ-preinvexity are given. Finally, two optimality results for nonlinear programming problems are obtained under the assumption of semistrict πΊ-preinvexity. The obtained results are new and different from the corresponding ones in the literature. Some examples are given to illustrate our results. 1. Introduction It is well known that convexity and generalized convexity have been playing a central role in mathematical programming, economics, engineering, and optimization theory. The research convexity and generalized convexity are one of the most important aspects in mathematical programming and optimization theory in [1–4]. Various kinds of generalized convexity have been introduced by many authors (see, e.g., [5–21] and the references therein). In 1981, Hanson [5] introduced the concept of invexity which is extension of differentiable convex functions and proved the sufficiency of KuhnTucker condition. Later, Weir and Mond [6] considered functions (not necessarily differentiable) for which there exists a vector function π : π π × π π → π π such that, for all π₯, π¦ ∈ π π , π ∈ [0, 1], one hasthe following: π (π¦ + ππ (π₯, π¦)) ≤ ππ (π₯) + (1 − π) π (π¦) , (1) which has been named as preinvex functions with respect to vector-valued function π. In 2001, Yang and Li [8] obtained some properties of preinvex function. At the same time, Yang and Li [9] introduced the concept of semistrictly preinvex functions and investigated the relationships between semistrictly preinvex functions and preinvex functions. It is worth mentioning that many properties and applications in mathematical programming for invex functions and preinvex functions are discussed by many authors (see, e.g., [6–11, 21] and the references therein). On the other hand, Avriel et al. [12] introduced a class of πΊ-convex functions which is another generalization of convex functions and obtained some relations with other generalization of convex functions. In [13], Antczak introduced the concept of a class of πΊ-invex fuctions, which is a generalization of πΊ-convex functions and invex functions. Recently, Antczak [14] introduced a class of πΊ-preinvex functions, which is a generalization of πΊ-invex [13], preinvex functions [8] and derived some optimality results for constrained optimization problems under πΊ-preinvexity. Very recently, Luo and Wu introduced a new class of functions called semistrictly πΊ-preinvex functions in [15], which include semistrictly preinvex functions [9] as a special case. They investigated the relationships between semistrictly πΊpreinvex functions and πΊ-preinvex functions and gave a criterion for semistrict πΊ-preinvexity. Moreover, they also proposed three open questions (just as they said: “an interesting topic for our future research is to”: (1) investigate πΊ-preinvex functions and semicontinuity; (2) explore some properties of semistrictly πΊ-preinvex functions; (3) research into some applications in optimization problems under semistrictly πΊpreinvexity [15]). 2 Abstract and Applied Analysis However, as far as we know, there are few papers dealing with the properties and applications of the semistrictly πΊpreinvex functions [16]. The questions above in [15] have not been solved, and one condition in [15] is not mild in researching of relationships between πΊ-preinvexity and semistrict πΊpreinvexity. So, in this paper, we further discuss semistrictly πΊ-preinvex functions. The rest of the paper is organized as follows. Firstly, we investigate πΊ-preinvex functions and semicontinuity and obtain a criterion of πΊ-preinvexity under semicontinuity. Then, we give new relationships between πΊpreinvexity and semistrictly πΊ-preinvexity, which are different from the recent ones in the literature [15]. Finally, we get two optimality results under semistrict πΊ-preinvexity for nonlinear programming. The obtained results in this paper improve and extend the existing ones in the literature (e.g., [8, 9, 11, 15, 16]). Definition 6 (see [15]). Let πΎ be a nonempty invex (with respect to π) subset π π . A function π : πΎ → π is said to be semistrictly πΊ-preinvex at π’ on πΎ if there exists a continuous real-valued increasing function πΊ : πΌπ (πΎ) → π such that for all π₯ ∈ πΎ (π(π₯) =ΜΈ π(π’)) and π ∈ (0, 1), 2. Preliminaries and Definitions Remark 8. Every semistrictly preinvex function [8, 9] is semistrictly πΊ-preinvex with respect to the same function π, where πΊ : πΌπ (πΎ) → π is defined by πΊ(π₯) = π₯. π Throughout this paper, let πΎ be a nonempty subset of π . Let π : πΎ → π be a real-valued function and π : πΎ × πΎ → π π a vector-valued function. Let πΌπ (πΎ) be the range of π, that is, the image of πΎ under π. Now we recall some definitions. Definition 1 (see [5, 6]). A set πΎ is said to be invex if there exists a vector-valued function π : πΎ × πΎ → π π (π =ΜΈ 0) such that π₯, π¦ ∈ πΎ, π ∈ [0, 1] σ³¨⇒ π¦ + ππ (π₯, π¦) ∈ πΎ. (2) Definition 2 (see [6, 8]). Let πΎ ⊆ π π be an invex set with respect to π : π π ×π π → π π and let π : πΎ → π be a mapping. One says that π is preinvex if π (π¦ + ππ (π₯, π¦)) ≤ ππ (π₯) + (1 − π) π (π¦) , ∀π₯, π¦ ∈ πΎ, π ∈ [0, 1] . (3) Remark 3. Any convex function is a preinvex function with π(π₯, π¦) = π₯ − π¦. π Definition 4 (see [10]). Let πΎ ⊆ π be an invex set with respect to π : π π × π π → π π . Let π : πΎ → π . One says that π is prequasi-invex if π (π¦ + ππ (π₯, π¦)) ≤ max {π (π₯) , π (π¦)} , ∀π₯, π¦ ∈ πΎ, π ∈ [0, 1] . (4) Definition 5 (see [14]). Let πΎ be a nonempty invex (with respect to π) subset π π . A function π : πΎ → π is said to be πΊ-preinvex at π’ on πΎ if there exists a continuous real-valued increasing function πΊ : πΌπ (πΎ) → π such that for all π₯ ∈ πΎ and π ∈ [0, 1] (π ∈ (0, 1)), π (π’ + ππ (π₯, π’)) ≤ πΊ−1 (ππΊ (π (π₯)) + (1 − π) πΊ (π (π’))) . (5) If (5) is satisfied for any π’ ∈ πΎ then π is πΊ-preinvex on πΎ, with respect to π. π (π’ + ππ (π₯, π’)) < πΊ−1 (ππΊ (π (π₯)) + (1 − π) πΊ (π (π’))) . (6) If (6) is satisfied for any π’ ∈ πΎ, then π is semistrictly πΊ-preinvex on πΎ with respect to π. Remark 7. In order to define an analogous class of semistrictly πΊ-preincave functions with respect to π, the direction of the inequality in Definition 6 should be changed to the opposite one. In order to prove our main result, we need Condition C as follows. Condition C (see [9]). The vector-valued function π : π × π → π is said to satisfy Condition C if for any π₯, π¦ ∈ π, and π ∈ [0, 1], π (π¦, π¦ + ππ (π₯, π¦)) = −ππ (π₯, π¦) , π (π₯, π¦ + ππ (π₯, π¦)) = (1 − π) π (π₯, π¦) . (7) Example 9. Let π₯ − π¦, { { { { { { π₯ − π¦, { { { π (π₯, π¦) = { 5 − − π¦, { { { 2 { { {5 { { − π¦, {2 if π₯ ≥ 0, π¦ ≥ 0, if π₯ < 0, π¦ < 0, if π₯ > 0, π¦ ≤ 0, (8) if π₯ ≤ 0, π¦ > 0. It can be verified that π satisfies the Condition C. 3. Relationships with Semistrictly πΊ-Preinvexity In [15], Luo and Wu obtained a sufficient condition of πΊpreinvex functions under the condition of intermediate-point πΊ-preinvexity. Now we investigate πΊ-preinvex functions and semicontinuity without the condition of intermediate-point πΊ-preinvexity. Theorem 10. Let πΎ be a nonempty invex set with respect to π, where π satisfies Condition C. Let π : πΎ → π be lower semicontinuous and semistrictly πΊ-preinvex for the same π on πΎ, and let π(π¦ + π(π₯, π¦)) ≤ π(π₯) (for all π₯ ∈ πΎ). Then π is πΊ-preinvex function on πΎ. Abstract and Applied Analysis 3 Proof. Let π₯, π¦ ∈ πΎ. From the assumption of π(π¦ + π(π₯, π¦)) ≤ π(π₯), when π = 0, 1, we can know that π (π¦ + ππ (π₯, π¦)) ≤ πΊ−1 (ππΊ (π (π₯)) + (1 − π) πΊ (π (π¦))) . (9) According to (13) and the semistrictly πΊ-preinvexity of π, we have the following: π (π§πΌ ) = π (π§π½ + (1 − Then, there are two cases to be considered. (i) If π(π₯) =ΜΈ π(π¦), then by the semistrict πΊ-preinvexity of π, we have the following: π (π¦ + ππ (π₯, π¦)) < πΊ−1 (ππΊ (π (π₯)) + (1 − π) πΊ (π (π¦))) , ∀π ∈ (0, 1) . (10) (ii) If π(π₯) = π(π¦), to show that π is a πΊ-preinvex function, we need to show that π (π¦ + ππ (π₯, π¦)) ≤ πΊ−1 (ππΊ (π (π₯)) + (1 − π) πΊ (π (π¦))) = π (π₯) . (11) By contradiction, suppose that there exists an πΌ ∈ (0, 1) such that π (π¦ + πΌπ (π₯, π¦)) > π (π₯) . (12) Let π§πΌ = π¦ + πΌπ(π₯, π¦). Since π is lower semicontinuous, there exists π½ : πΌ < π½ < 1, such that π (π§π½ ) = π (π¦ + π½π (π₯, π¦)) > π (π₯) = π (π¦) . (13) From Condition C, π§π½ = π§πΌ + π½−πΌ π (π₯, π§πΌ ) . 1−πΌ (14) By the semistrict πΊ-preinvexity of π and (12), we have the following: π (π§π½ ) < πΊ−1 [(1 − πΌ πΌ ) πΊ (π (π¦)) + πΊ (π (π§π½ ))] π½ π½ < πΊ−1 [(1 − πΌ πΌ ) πΊ (π (π§π½ )) + πΊ (π (π§π½ ))] π½ π½ (17) = π (π§π½ ) which contradicts (15). This completes the proof. Remark 11. In Theorem 10, we investigate πΊ-preinvex functions and lower semicontinuity, and establish a new sufficient condition for πΊ-preinvexity without the condition of intermediate-point πΊ-preinvexity. Therefore, we also answer the open question (1) which proposed in [15] (“(1) investigate πΊ-preinvex functions and semicontinuity” [15]). Now, we give a new sufficient condition for semistrictly πΊ-preinvexity. Theorem 12. Let πΎ be a nonempty invex set with respect to π, where π satisfies Condition C. Let π : πΎ → π be a πΊ-preinvex function for the same π on πΎ. Suppose that for any π₯, π¦ ∈ πΎ with π(π₯) =ΜΈ π(π¦), there exists an πΌ ∈ (0, 1) such that π (π¦ + πΌπ (π₯, π¦)) < πΊ−1 (πΌπΊ (π (π₯)) + (1 − πΌ) πΊ (π (π¦))) . (18) Then, π is semistrictly πΊ-preinvex function on πΎ. Proof. For any π₯, π¦ ∈ πΎ with π(π₯) =ΜΈ π(π¦) and π ∈ (0, 1), by assumption, we have the following: π (π¦ + ππ (π₯, π¦)) ≤ πΊ−1 (ππΊ (π (π₯)) + (1 − π) πΊ (π (π¦))) . (19) (i) Let π ≤ πΌ. From Condition C, we can obtain the following: π½−πΌ = π (π§πΌ + π (π₯, π§πΌ )) 1−πΌ π¦+ π½−πΌ π½−πΌ <πΊ [ πΊ (π (π₯)) + (1 − ) πΊ (π (π§πΌ ))] 1−πΌ 1−πΌ −1 π½−πΌ π½−πΌ <πΊ [ πΊ (π (π§πΌ )) + (1 − ) πΊ (π (π§πΌ ))] 1−πΌ 1−πΌ −1 = π (π§πΌ ) . π π (π¦ + πΌπ (π₯, π¦) , π¦) πΌ =π¦+ π π (π¦ + πΌπ (π₯, π¦) , π¦ + πΌπ (π₯, π¦) − πΌπ (π₯, π¦)) πΌ =π¦+ π π (π¦ + πΌπ (π₯, π¦) , π¦ + πΌπ (π₯, π¦) πΌ (15) On the other hand, from Condition C, one can obtain the following: π§πΌ = π§π½ + (1 − πΌ ) π (π¦, π§π½ )) π½ πΌ ) π (π¦, π§π½ ) . π½ (16) +π (π¦, π¦ + πΌπ (π₯, π¦))) =π¦− π π (π¦, π¦ + πΌπ (π₯, π¦)) πΌ = π¦ + ππ (π₯, π¦) . (20) 4 Abstract and Applied Analysis Using (18) and the πΊ-preinvexity of π, we have the following: π (π¦ + ππ (π₯, π¦)) = π [π¦ + + π π (π¦ + πΌπ (π₯, π¦) , π¦)] πΌ 1−π (πΌπΊ (π (π₯)) + (1 − πΌ) πΊ (π (π¦)))] 1−πΌ (24) (21) and (24) imply that π is a semistrictly πΊ-preinvex function on πΎ. π + (1 − ) πΊ (π (π¦))] πΌ Remark 13. Theorem 12 extends and improves Theorem 1 in [15]. In Theorem 1 of [15], a uniform πΌ ∈ (0, 1) is needed; while in assumption (18) of Theorem 12, this condition has been weakened, where a uniform πΌ ∈ (0, 1) is not necessary. Moreover, our proof is also different from the corresponding result of [15]. π < πΊ [ πΊ (πΊ−1 (πΌπΊ (π (π₯)) + (1 − πΌ) πΊ (π (π¦)))) πΌ −1 π ) πΊ (π (π¦))] πΌ π = πΊ−1 [ (πΌπΊ (π (π₯)) + (1 − πΌ) πΊ (π (π¦))) πΌ + (1 − 1−π ) πΊ (π (π₯)) 1−πΌ = πΊ−1 ( ππΊ ( π (π₯)) + (1 − π) πΊ (π (π¦))) , π ≤ πΊ−1 [ πΊ ( π (π¦ + πΌπ (π₯, π¦)) πΌ + (1 − = πΊ−1 [(1 − The following example illustrates that assumption (18) in Theorem 12 is essential. π ) πΊ (π (π¦))] πΌ Example 14. Let 7 3 π (π₯) = ln ( |π₯| + ) , 2 4 = πΊ−1 (ππΊ ( π (π₯)) + (1 − π) πΊ (π (π¦))) . (21) (ii) Let πΌ < π; that is, 0< 1−π < 1. 1−πΌ (22) From Condition C, we have the following: π¦ + πΌπ (π₯, π¦) + (1 − = π¦ + [πΌ + (1 − 1−π ) π (π₯, π¦ + πΌπ (π₯, π¦)) 1−πΌ 1−π ) (1 − πΌ)] π (π₯, π¦) 1−πΌ (23) = π¦ + ππ (π₯, π¦) . π₯ − π¦, { { { {π₯ − π¦, π (π₯, π¦) = { { −π₯ − π¦, { { {−π₯ − π¦, if if if if π₯ ≥ 0, π₯ ≤ 0, π₯ > 0, π₯ < 0, π¦ ≥ 0; π¦ ≤ 0; π¦ < 0; π¦ > 0. It is obvious that π is a πΊ-preinvex function with respect to π, where πΊ(π‘) = ππ‘ , πΎ = π . However, from Definition 6, we can verify that π is a semistrictly πΊ-preinvex function. The reason is that the assumption (18) is violated. Indeed, there exist π₯ = 1/2, π¦ = 4 (with π(π₯) =ΜΈ π(π¦)), for any π ∈ (0, 1), we have the following: π (π¦ + ππ (π₯, π¦)) 7 σ΅¨σ΅¨ 7 3 σ΅¨σ΅¨ 7 = π (4 − π) = ln ( σ΅¨σ΅¨σ΅¨σ΅¨4 − πσ΅¨σ΅¨σ΅¨σ΅¨ + ) 2 2σ΅¨ 2 σ΅¨ 4 According to (18) and the πΊ-preinvexity of π, we get the following: = ln ( π (π¦ + ππ (π₯, π¦)) 7 3 σ΅¨ σ΅¨ = πΊ−1 ( (π |π₯| + (1 − π) σ΅¨σ΅¨σ΅¨π¦σ΅¨σ΅¨σ΅¨) + ) 2 4 = π [π¦ + πΌπ (π₯, π¦) + (1 − ≤ πΊ−1 [(1 − + 1−π ) πΊ (π (π₯)) 1−πΌ 1−π πΊ (π (π¦ + πΌπ (π₯, π¦))) ] 1−πΌ < πΊ−1 [(1 − + 1−π ) π (π₯, π¦ + πΌπ (π₯, π¦))] 1−πΌ 1−π ) πΊ (π (π₯)) 1−πΌ 1−π πΊ (πΊ−1 (πΌπΊ (π (π₯)) + (1 − πΌ) πΊ (π (π¦)))) ] 1−πΌ (25) 31 21 − π) 4 4 (26) = πΊ−1 (ππΊ (π (π₯)) + (1 − π) πΊ (π (π¦))) . Therefore, (18) is essential. Lemma 15 (see [16]). Let πΎ be a nonempty invex set with respect to π. Suppose that π is a semistrictly πΊ1 -preinvex function with respect to π and πΊ2 is a continuous and strictly increasing function on πΌπ (π). If π(π‘) = πΊ2 πΊ1−1 is convex on the image under πΊ1 of the range of π, then π is also semistrictly πΊ2 -preinvex function with respect to the same π on πΎ. Theorem 16. Let K be an invex set with respect to π and π : πΎ → π be a semistrictly πΊ-preinvex function on πΎ. If πΊ is Abstract and Applied Analysis 5 concave on πΌπ (π), then π is a preinvex function with respect to the same π on πΎ. Proof. Let π¦ and π§ be two points in πΌπ (π). Because πΊ is concave on πΌπ (π), the following inequality πΊ (ππ¦ + (1 − π) π§) ≥ ππΊ (π¦) + (1 − π) πΊ (π§) (27) holds for any π ∈ [0, 1]. Let πΊ(π¦) = π₯ and πΊ(π§) = π’. Then, for each pair of points π₯ and π’ in image πΊ of πΌπ (π), that is, πΊ−1 (π₯) = π¦ and πΊ−1 (π’) = π§, we have the following: πΊ (ππΊ−1 (π₯) + (1 − π) πΊ−1 (π’)) ≥ ππΊ (πΊ−1 (π₯)) + (1 − π) πΊ (πΊ−1 (π’)) (28) = ππ₯ + (1 − π) π’. It follows from (28) and the increasing property of πΊ−1 that πΊ−1 πΊ (ππΊ−1 (π₯) + (1 − π) πΊ−1 (π’)) ≥ πΊ−1 (ππΊ (πΊ−1 (π₯)) + (1 − π) πΊ (πΊ−1 (π’))) (29) (30) This means that πΊ−1 is convex. Letting πΊ1 = πΊ, πΊ2 (π‘) = π‘, then π(π‘) = πΊ2 πΊ1−1 (π‘) is convex. Hence, by virtue of Lemma 15, π is a semistrictly πΊ2 -preinvex function with respect to π. Because πΊ2 (π‘) = π‘ is identity function, π is a preinvex function with respect to the same π on πΎ. From Theorem 16 and Definitions 2-4, we can obtain the following Corollary easily. Corollary 17. Let K be an invex set with respect to π and π : πΎ → π is a semistrictly πΊ-preinvex function on πΎ. If πΊ is concave on πΌπ (π) then π be a prequasi-invex function with respect to the same π on πΎ. 4. Semistrictly πΊ-Preinvexity and Optimality In order to solve the open question (3) proposed in [15] (see, the part of Introduction), in this section, we consider nonlinear programming problems with constraint and obtain two optimality results under semistrict πΊ-preinvexity. We consider the following nonlinear programming Problem (P) with inequality constraint: (P) (31) where π : πΎ → π , ππ : πΎ → π , π ∈ π½, and πΎ is a nonempty subset of π π . We denote the set of all feasible solutions in (P) by the following: π· = {π₯ ∈ πΎ : ππ (π₯) ≤ 0, π ∈ π½} . (33) Let π§ (π§ =ΜΈ π₯∗ ) be an interior point of π·. By assumption, π· is an invex set with respect to π. It follows from the definition of invex set π· that there exists π¦ ∈ π· such that for some π ∈ (0, 1), (34) By assumption, π is semistrictly πΊ-preincave with respect to π on π·. Then, we have the following: > πΊ−1 (ππΊ (π (π¦)) + (1 − π) πΊ (π (π₯∗ ))) ππΊ−1 (π₯) + (1 − π) πΊ−1 (π’) ≥ πΊ−1 (ππ₯ + (1 − π) π’) . π ∈ π½ = 1, . . . , π, π (π₯∗ ) > π (π₯) . π (π§) = π (π₯∗ + ππ (π¦, π₯∗ )) Thus, ππ (π₯) ≤ 0, Proof. If problem (P) has no solution the theorem is trivially true. Let π₯ be an optimal solution in problem (P). By assumption, π is a nonconstant on π·. Then, there exists a feasible point π₯∗ ∈ π· such that π§ = π₯∗ + ππ (π¦, π₯∗ ) . = πΊ−1 (ππ₯ + (1 − π) π’) . min π (π₯) Theorem 18. Suppose the set of all feasible solutions π· of problem (P) is an invex set with respect to π, and π· at least contains two points with nonempty interior. Let π be a nonconstant semistrictly πΊ-preincave function with respect to π on π·. Then no interior of π· is an optimal solution of (P), or equivalently, any optimal solution π₯ in problem (P), if exists, must be a boundary point of π·. (32) > πΊ−1 (ππΊ (π (π₯)) + (1 − π) πΊ (π (π₯))) = π (π₯) , (35) where π§ is an interior point of π·. From the inequality above, we conclude that no interior of π· is an optimal solution of (P), that is, any optimal solution π₯ in problem (P), if exists, must be a boundary point of π·. This completes the proof. Theorem 19. Let π₯ ∈ π· be local optimal in problem (P). Moreover, we assume that π is semistrictly πΊ-preinvex with respect to π at π₯ on π· and the constraint functions ππ , π ∈ π½, are πΊpreinvex with respect to π at π₯ on π·. Then, π₯ is a global optimal solution in problem (P). Proof. Assume that π₯ ∈ πΎ is a local optimal solution in (P). Hence, there is an neighborhood ππ (π₯) around π₯ such that π (π₯) ≤ π (π₯) , ∀π₯ ∈ πΎ ∩ ππ (π₯) . (36) Suppose to the contrary, π₯ is not a global minimum in (P), there exists an π₯∗ ∈ π· such that π (π₯∗ ) < π (π₯) . (37) By assumption, the constraint functions ππ , π ∈ π½, are πΊ-preinvex with respect to the same π at π₯ on π·. By using Definition 5 together with π₯∗ , π₯ ∈ π·, then for all π ∈ π½ and any π ∈ [0, 1], we have the following: ππ (π₯ + ππ (π₯∗ , π₯)) ≤ πΊπ−1 (ππΊπ (ππ (π₯∗ )) + (1 − π) πΊπ (ππ (π₯))) ≤ πΊπ−1 (ππΊπ (0) + (1 − π) πΊπ (0)) = 0. (38) 6 Abstract and Applied Analysis Thus, for all π ∈ π½ and any π ∈ [0, 1], ∗ π₯ + ππ (π₯ , π₯) ∈ π·. (39) By assumption, π is semistrictly πΊ-preinvex with respect to the same π at π₯ on π·. Therefore, by Definition 6, we have the following: π (π₯ + ππ (π₯∗ , π₯)) < πΊ−1 (ππΊ (π (π₯∗ )) + (1 − π) πΊ (π (π₯))) , ∀π ∈ (0, 1) . (40) From (37) and (40), we have the following: π (π₯ + ππ (π₯∗ , π₯)) < πΊ−1 (ππΊ (π (π₯)) + (1 − π) πΊ (π (π₯))) = π (π₯) , (41) ∀π ∈ (0, 1) . For a sufficiently small π > 0, it follows that π₯ + ππ (π₯∗ , π₯) ∈ πΎ ∩ ππ (π₯) , (42) which contradicts that π₯ is local optimal in problem (P). This completes the proof. Now, we give an example to illustrate Theorem 19. Example 20. Let π : πΎ → π be defined as follows: ln (5 − 4 |π₯|) , if |π₯| ≤ 1, π (π₯) = { 0, if |π₯| ≥ 1, (43) and let ππ (π₯) = 3π₯ − 1, { 3 { { − π¦, { { 4 { { { { { { { 1, { { { { { π₯ − π¦, { { { { { −6π₯ − π¦, { { { { { 1 − π¦, { { { { { π₯ − π¦, { { { { { π¦ − π₯ − 2, { { { { { { { {π¦ − π₯, π (π₯, π¦) ={ { { { π¦ − π₯, { { { { { 1 { { −5 − π¦, { { { 2 { { { { 3 { { −4π₯ + π¦, { { 2 { { { { 3 { { 3π₯ + π¦, { { { 2 { { { 15 { { −π¦ − π₯2 − π₯ − , { { 2 { { { { π¦ − 2π₯, { { { {π¦ − 2π₯ − 6, π ∈ π½ = 1, . . . , π, if π₯ > 1, 0 ≤ π¦ < 1, or π₯ < −1, −1 < π¦ ≤ 0; if π¦ > 1, 0 ≤ π₯ < 1; if π¦ < −1, −1 < π₯ ≤ 0; if 0 ≤ π₯ < 1, 0 < π¦ < 1; if 0 ≤ π₯ < 1, π¦ = 1; σ΅¨ σ΅¨ if |π₯| ≥ 1, σ΅¨σ΅¨σ΅¨π¦σ΅¨σ΅¨σ΅¨ ≥ 1; if π₯ > 1, −1 < π¦ < 0; if π₯ > 1, −1 < π¦ < 0, or π¦ > 1, −1 < π₯ < 0; if π¦ < −1, 0 < π₯ < 1; if |π₯| < 1, −1 ≤ π¦ ≤ 0; if π₯ = 1, 0 < π¦ < 1; if π₯ = −1, −1 < π¦ < 0; if − 1 < π₯ < 0, 0 < π¦ ≤ 1; if π₯ = 1, −1 < π¦ ≤ 0; if π₯ = −1, 0 ≤ π¦ < 1. (44) From Definitions 5-6, we can get that π is semistrictly πΊpreinvex with respect to π and the constraint functions ππ , π ∈ π½, are πΊ-preinvex with respect to π, respectively, where πΊ(π‘) = ππ‘ , πΎ = π . Obviously, π₯ = −1 is a local optimal solution of (P). Then, all the conditions in Theorem 19 are satisfied. By virtue of Theorem 19, π₯ = −1 is a global optimal solution in problem (P). Acknowledgments The authors would like to express their thanks to Professor X. M. Yang and the anonymous referees for their valuable comments and suggestions which helped to improve the paper. This work was supported by the Natural Science Foundation of China (nos. 11271389, 11201509, and 71271226), the Natural Science Foundation Project of ChongQing (nos. CSTC, 2012jjA00016, and 2011AC6104), and the Research Grant of Chongqing Key Laboratory of Operations and System Engineering. References [1] O. L. Mangasarian, Nonlinear Programming, McGraw-Hill, New York, NY, USA, 1969. [2] M. S. Bazaraa, H. D. Sherali, and C. M. Shetty, Nonlinear Programming: Theory and Algorithms, John Wiley & Sons, New York, NY, USA, 1979. [3] S. Schaible and W. T. Ziemba, Generalized Concavity in Optimization and Economics, Academic Press Inc., London, UK, 1981. [4] S. K. Mishra and G. Giorgi, Invexity and Optimization, vol. 88 of Nonconvex Optimization and Its Applications, Springer, Berlin, Germany, 2008. [5] M. A. Hanson, “On sufficiency of the Kuhn-Tucker conditions,” Journal of Mathematical Analysis and Applications, vol. 80, no. 2, pp. 545–550, 1981. [6] T. Weir and B. Mond, “Pre-invex functions in multiple objective optimization,” Journal of Mathematical Analysis and Applications, vol. 136, no. 1, pp. 29–38, 1988. [7] T. Weir and V. Jeyakumar, “A class of nonconvex functions and mathematical programming,” Bulletin of the Australian Mathematical Society, vol. 38, no. 2, pp. 177–189, 1988. [8] X. M. Yang and D. Li, “On properties of preinvex functions,” Journal of Mathematical Analysis and Applications, vol. 256, no. 1, pp. 229–241, 2001. [9] X. M. Yang and D. Li, “Semistrictly preinvex functions,” Journal of Mathematical Analysis and Applications, vol. 258, no. 1, pp. 287–308, 2001. [10] X. M. Yang, X. Q. Yang, and K. L. Teo, “Characterizations and applications of prequasi-invex functions,” Journal of Optimization Theory and Applications, vol. 110, no. 3, pp. 645–668, 2001. [11] J. W. Peng and X. M. Yang, “Two properties of strictly preinvex functions,” Operations Research Transactions, vol. 9, no. 1, pp. 37–42, 2005. [12] M. Avriel, W. E. Diewert, S. Schaible, and I. Zang, Generalized Concavity, Plenum Press, New York, NY, USA, 1975. [13] T. Antczak, “On πΊ-invex multiobjective programming. I. Optimality,” Journal of Global Optimization, vol. 43, no. 1, pp. 97–109, 2009. Abstract and Applied Analysis [14] T. Antczak, “πΊ-pre-invex functions in mathematical programming,” Journal of Computational and Applied Mathematics, vol. 217, no. 1, pp. 212–226, 2008. [15] H. Z. Luo and H. X. Wu, “On the relationships between πΊ-preinvex functions and semistrictly πΊ-preinvex functions,” Journal of Computational and Applied Mathematics, vol. 222, no. 2, pp. 372–380, 2008. [16] Z. Y. Peng, “Semistrict G-preinvexity and its application,” Journal of Inequalities and Applications, vol. 2012, article 198, 2012. [17] X. J. Long, “Optimality conditions and duality for nondifferentiable multiobjective fractional programming problems with (πΆ, πΌ, π, π)-convexity,” Journal of Optimization Theory and Applications, vol. 148, no. 1, pp. 197–208, 2011. [18] X.-J. Long and N.-J. Huang, “Lipschitz π΅-preinvex functions and nonsmooth multiobjective programming,” Pacific Journal of Optimization, vol. 7, no. 1, pp. 97–107, 2011. [19] S. K. Mishra, Topics in Nonconvex Optimization: Theory and Applications, vol. 50 of Springer Optimization and Its Applications, Springer, New York, NY, USA, 2011. [20] S. K. Mishra and N. G. Rueda, “Generalized invexity-type conditions in constrained optimization,” Journal of Systems Science & Complexity, vol. 24, no. 2, pp. 394–400, 2011. [21] X. Liu and D. H. 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