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Semilocally Prequasi-Invex Functions and Characterizations JOURNAL OF INDUSTRIAL AND Website: http://AIMsciences.org MANAGEMENT OPTIMIZATION Volume 3, Number 3, August 2007 pp. 503–517 SEMILOCALLY PREQUASI-INVEX FUNCTIONS AND CHARACTERIZATIONS Jin-bao Jian, Hua-qin Pan and Han-jun Zeng College of Mathematics and Information Science Guangxi University, 530004 Nanning, P.R. China. (Communicated by Duan Li) Abstract. In this paper, based on semilocal convexity, invexity and prequasi- invexity, a new kind of generalized convexity called (strictly/ semistrictly) semilocally prequasi-invex functions is presented, and some of their basic char- acterizations are discussed. Then, several necessary and sufficient conditions for the proposed generalized convexity are established. Finally, some impor- tant properties of the optimal solutions to the associated generalized convex optimization problems are discussed. 1. Introduction. Convexity and generalized convexity (see [1]∼ [11]) play an im- portant role in mathematical economics, engineering, management science, opti- mization and so on (see [12]∼[16]). Therefore, the research on convexity and gen- eralized convexity is one of the important aspects in mathematical programming. During the past years, various significant generalizations of convexity have been presented. In 1981, Craven (see [1]) brought forward a class of new convexity called invex functions, and discussed some of its characterizations. In 1986, Ben-Israel and Mond (see [2]) further obtained the general characterizations of invexity. In 1988, a significant generalization of invex functions, preinvex function, was given by Weir and Mond (see [3]) and by Weir and Jeyakumar (see [4]). In 1991, Pini (see [5]) discussed some of the basic properties of invexity and generalized invex- ity, and established some optimality results on invex programming. In 1997, Khan and Hanson (see [6]) also discussed some of the basic properties on ratio invexity, Mukherjee and Reddy (see [7]) also discussed the characterizations of quasiconvex function under the condition of semicontinuity and obtained some basic proper- ties. In 1999, Wang, Li and Craven (see [8]) gave some of the basic properties of global efficiency in multiobjective programming. In 2001, Yang, Yang and Teo (see [9]) introduced two new generalized convex functions —semistrictly and strictly prequasi-invex functions, and discussed their some properties under semicontinuity and other given conditions. 2000 Mathematics Subject Classification. 90C30, 90C53, 49M37, 65K10, 65K05. Key words and phrases. locally invex set, prequasi-invex functions, semilocally prequasi-invex functions, generalized convexity. Project supported by the National Natural Science Foundation of China (No. 10261001) and Guangxi Province Science Foundation (Nos.02361001, 0640001) as well as Guangxi University Key Program for Science and Technology Research (No. 2005ZD02). 503 504 JIN-BAO JIAN, HUA-QIN PAN AND HAN-JUN ZENG In this paper, motivated by Yang, Yang and Teo (see [9]), we present three kinds of new generalized convex functions: semilocally prequasi-invex, semistrictly semilocally prequasi-invex and strictly semilocally prequasi-invex functions, then discuss their basic properties. Under the conditions of semicontinuity and others, we obtain some relationships among the proposed generalized convex functions, and analyze the optimality conditions for the corresponding generalized convex programs. 2. Preliminaries and semilocal prequasi-invexity. In this section, we first re- view the related generalized convexity, then put forward our generalized convexity— (strict/semistrict) semilocal prequasi-invexity. Definition 2.1 (see [2]). A set K ⊆ Rn is said to be invex if there exits a vector function η : Rn × Rn → Rn such that ∀x, y ∈ K, λ ∈ [0, 1] ⇒ y + λη(x, y) ∈ K. Definition 2.2 (see [3]). Let K ⊆ Rn be an invex set with respect to η : Rn ×Rn → Rn, and let f : K → R. We say that f is preinvex if f(y + λη(x, y)) ≤ λf(x)+(1 − λ)f(y), ∀x, y ∈ K, λ ∈ [0, 1]. Definition 2.3 (see [5]). Let K ⊆ Rn be an invex set with respect to η : Rn ×Rn → Rn, and let f : K → R. We say that f is prequasi-invex if f(y + λη(x, y)) ≤ max{f(x),f(y)}, ∀x, y ∈ K, λ ∈ [0, 1]. Definition 2.4 (see [9]). Let K ⊆ Rn be an invex set with respect to η : Rn ×Rn → Rn, and let f : K → R. We say that f is strictly prequasi-invex if f(y + λη(x, y)) < max{f(x),f(y)}, ∀x, y ∈ K, x 6= y, λ ∈ (0, 1). Definition 2.5 (see [9]). Let K ⊆ Rn be an invex set with respect to η : Rn ×Rn → Rn, and let f : K → R. We say that f is semistrictly prequasi-invex if f(y + λη(x, y)) < max{f(x),f(y)}, ∀x, y ∈ K,f(x) 6= f(y), λ ∈ (0, 1). Now, we give our main definitions as follows. Definition 2.6. A set K ⊆ Rn is said to be a locally invex set if there exists a vector function η : K × K → Rn such that for each pair of points x, y ∈ K, there is a maximal positive number a(x, y) ≤ 1, such that y +λη(x, y) ∈ K, ∀λ ∈ [0,a(x, y)]. △ In addition, if a0 = inf{a(x, y) : x, y ∈ K} > 0,K is said to be an uniformly locally invex set. Remark 1. A similar definition of a locally invex set was also given in [10], where it is called a η-locally starshaped set. It is obvious to show that an invex set with respect to map η is locally invex with respect to the same map η by taking a(x, y) ≡ 1. But the converse is not necessarily true, see the following example. Example 2.1 Let K = [1, 2] ∪{3}⊂ R, and η(x, y)=0, if y =2 or y = 3; η(x, y)= x + y, otherwise. Then K is locally invex with respect to map a(x, y)=1, if y =2 or y = 3; a(x, y)=(2 − y)/(x + y), otherwise. But it is not difficult to see that K is not invex, in fact, for x =2, y =1, 1+ λ(2 + 1 1)=1+3λ∈ / K for λ = 2 . SEMILOCALLY PREQUASI-INVEX FUNCTIONS AND CHARACTERIZATIONS 505 Proposition 1. If sets K1, K2,..., Km are all locally invex with respect to a same m n n n map η : R × R → R , then the intersection K = Ki is also a locally invex i=1 set with respect to map η. T The proof is elementary and is omitted here. Definition 2.7. Let K ⊆ Rn be a locally invex set with respect to maps η(x, y) and a(x, y), and let f : K → R. Function f is said to be semilocally prequasi-invex, if corresponding to each pair of points x, y ∈ K, there is a maximal positive number d(x, y) ≤ a(x, y), such that f(y + λη(x, y)) ≤ max{f(x),f(y)}, ∀λ ∈ [0, d(x, y)]. △ Additionally, if d0 = inf{d(x, y) : x, y ∈ K} > 0, we say that f is uniformly semilocally prequasi-invex. Definition 2.8. Let K ⊆ Rn be a locally invex set with respect to maps η(x, y) and a(x, y), and let f : K → R. Function f is said to be semistrictly semilocally prequasi-invex, if corresponding to each pair of points x, y ∈ K and f(x) 6= f(y), there is a maximal positive number d(x, y) ≤ a(x, y), such that f(y + λη(x, y)) < max{f(x),f(y)}, ∀λ ∈ (0, d(x, y)). + △ Additionally, if d0 = inf{d(x, y) : x, y ∈ K,f(x) 6= f(y)} > 0, we say that f is uniformly semistrictly semilocally prequasi-invex. Definition 2.9. Let K ⊆ Rn be a locally invex set with respect to maps η(x, y) and a(x, y), and let f : K → R. Function f is said to be strictly semilocally prequasi-invex, if corresponding to each pair of points x, y ∈ K and x 6= y, there is a maximal positive number d(x, y) ≤ a(x, y), such that f(y + λη(x, y)) < max{f(x),f(y)}, ∀λ ∈ (0, d(x, y)). ++ △ Additionally, if d0 = inf{d(x, y) : x, y ∈ K, x 6= y} > 0, we say that f is uniformly strictly semilocally prequasi-invex. Remark 2. (1) It is not difficult to show that the semilocal prequasi-invexity implies neither the semistrict semilocal prequasi-invexity nor the strict semilocal prequasi-invexity; (2) It is obvious that the strict semilocal prequasi-invexity implies the semistrict semilocal prequasi-invexity, but it does not necessarily imply the semilocal prequasi- invexity (see Example 2.2 below), so the semistrict semilocal prequasi-incexity does not necessarily imply the semiloal prequasi-invexity. Of course, if η(x, x) = 0, then the strict semilocal prequasi-invexity implies the semilocal prequasi-invexity. And the following Example 2.3 show that semistrict semilocal prequasi-invexity does not imply the strict semilocal prequasi-invexity. Example 2.2 This example illustrates that a strictly semilocally prequasi-invex function is not necessarily a semilocally prequasi-invex function. Let K = [−2, 2] ⊂ R. Then K is locally invex with respect to maps x, if x = y and |x| < 2; η(x, y)= x − y, otherwise, min { 2 − 1, 1}, if x = y and 0 < |x| < 2; a(x, y)= |x| f(x)= x2. 1, otherwise, 506 JIN-BAO JIAN, HUA-QIN PAN AND HAN-JUN ZENG It is not difficult to see that f is strictly semilocally prequasi-invex on K. However, for each x ∈ K and |x| < 2, one knows that f(x + λη(x, x)) = f((1 + λ)x) = (1 + λ)2x2 > x2 = max{f(x),f(x)} holds for all λ ∈ (0, 1). So f is not semilocally prequasi-invex on K with respect to maps η and a.
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