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. Example 2.3 This example illustrates that a semistrictly semilocally prequasi- invex function is not necessarily a strictly semilocally prequasi-invex function. Let K = [−2, 2] ⊂ R, and η(x, y)=0, if x = −y; η(x, y)= x − y, otherwise, f(x)= |x|. Then, it is easy to see that f is semistrictly semilocally prequasi-invex on K. But since η(x, −x) = 0, and f(−x + λη(x, −x)) = f(−x)= |x| = max{f(x),f(−x)}, so f is not strictly semilocally prequasi-invex on K with respect to map η.
3. Basic characterizations. First, we discuss the conditions for differentiable and semilocally prequasi-invex functions. Theorem 3.1 (Conditions for a differentiable and semilocally prequasi-invex func- tion). Suppose that f : K → R is differentiable on an open locally invex set K with respect to map η. Then: (i) If f is a semilocally prequasi-invex function, then f(x) ≤ f(y), x, y ∈ K =⇒▽f(y)T η(x, y) ≤ 0. (1) (ii) If “f(x) ≤ f(y),x,y ∈ K =⇒▽f(y)T η(x, y) < 0”, then f is strictly semilocally prequasi-invex. Proof of Theorem 3.1. (i) Let x, y ∈ K and f(x) ≤ f(y). Then from Taylor ex- pansion and the semilocal prequasi-invexity of f, one knows that f(y) = max{f(x),f(y)}≥ f(y + λη(x, y)) = f(y)+ λ ▽ f(y)T η(x, y)+ o(λ) holds for positive and sufficiently small λ, so λ ▽ f(y)T η(x, y)+ o(λ) ≤ 0, this further shows that ▽f(y)T η(x, y) ≤ 0. (ii) Let x, y ∈ K and x 6= y. If f(x) > f(y), then, by the continuity of f, it follows that f(y + λη(x, y))
Definition 3.2. Suppose that K ⊆ Rn is an open locally invex set with respect to map η, and f : K → R is differentiable. We call f a semilocally prepseudo-invex function with respect to map η, if ▽f(y)T η(x, y) ≥ 0, x,y ∈ K =⇒ f(x) ≥ f(y). In order to further discuss more hidden characters of the proposed generalized convexity, two conditions as follows are needed, which are generalizations of the ones in [9]. Condition C Let set K be locally invex with respect to maps η : K ×K −→ Rn and a(x, y). We say that map η satisfies Condition C, if η(y,y + λη(x, y)) = −λη(x, y), ∀x y ∈ K, ∀λ ∈ [0,a(x, y)], (2) η(x, y + λη(x, y)) = (1 − λ)η(x, y), ∀x y ∈ K, ∀λ ∈ [0,a(x, y)]. (3) Condition D Let set K be locally invex with respect to maps η and a(x, y), f : K → R, and a0 = inf{a(x, y) : x, y ∈ K}. Function f is said to satisfy Condition D if f(y + a0η(x, y)) ≤ f(x), ∀x, y ∈ K. (4) It is obvious that Condition C is satisfied if η(x, y) = x − y, additionally, if a0 = 1, then Condition D also holds. On the other hand, Condition C implies that η(x, x) ≡ 0, ∀x ∈ K. In the rest of this paper we always assume that: (i) K ⊆ Rn is a locally invex set with respect to maps η(x, y) and a(x, y), f : K → R; (ii) If f is (semistrictly/strictly) semilocally prequasi-invex, then d(x, y) is the relative positive real number. Basing on the assumptions above, we present a lemma as follows, which is useful in proving our main result–Theorem 3.2. Lemma 3.3. Suppose that Conditions C and D hold, and assume there exists an α ∈ (0,a0) (this implies a0 > 0 and K is uniformly locally invex), such that f(y + αη(x, y)) ≤ max{f(x),f(y)}, ∀x, y ∈ K. (5)
Then, the set A = {λ ∈ [0,a0] : f(y + λη(x, y)) ≤ max{f(x),f(y)}, ∀x, y ∈ K} is dense in (0,a0), that is for any neighborhood N(λ) of any point λ ∈ (0,a0), A ∩ N(λ) 6= ∅. Proof of Lemma 3.3. Suppose that the conclusion does not hold. Then, there exists an λ0 ∈ (0,a0) and a neighborhood N(λ0) of λ0 such that
N(λ0) ∩ A = ∅. (6) From Condition D, we have
a0 ∈{λ ∈ A : λ ≥ λ0} 6= ∅, 0 ∈{λ ∈ A : λ ≤ λ0} 6= ∅. Define λ1 = inf{λ ∈ A : λ ≥ λ0}, λ2 = sup{λ ∈ A : λ ≤ λ0}. (7)
Then, by (6) and (7), we have 0 ≤ λ2 < λ1 ≤ a0 ≤ 1. It is easy to see that λ1−λ2 1 max{α, (1 − α)} ∈ (0, 1). Let ε = 2 ( max{α,(1−α)} − 1) > 0. Then from the definitions of λ1 and λ2, there are two numbers u1,u2 ∈ A such that
λ0 ≤ u1, u2 ≤ λ0, 0 ≤ u1 − λ1 < ε, 0 ≤ u2 − λ2 < ε. These relationships imply that
max{α, (1 − α)}(u1 − u2) < λ1 − λ2. (8) 508 JIN-BAO JIAN, HUA-QIN PAN AND HAN-JUN ZENG
Again, it is easy to see that
u1 − u2 −(a0 − u1) − u2(1 − a0) u1 − u2 − a0 = ≤ 0, i.e., ≤ a0. 1 − u2 1 − u2 1 − u2
Next, let us consider λ3 = (1−α)u1 +αu2 (3) u1−u2 (u2 − u1)η(x, y) = (− 1−u2 )η(x, y + u2η(x, y)) (2) u1−u2 = η(y + u2η(x, y),y + u2η(x, y) + ( 1−u2 )η(x, y + u2η(x, y))) (3) = η(y + u2η(x, y),y + u2η(x, y) + (u1 − u2)η(x, y)) = η(y + u2η(x, y),y + u1η(x, y)). So y + λ3η(x, y) = y + u1η(x, y)+ α(u2 − u1)η(x, y) = y + u1η(x, y)+ αη(y + u2η(x, y),y + u1η(x, y)). Hence, from (5) and u1,u2 ∈ A, we obtain f(y + λ3η(x, y)) = f(y + u1η(x, y)+ αη(y + u2η(x, y),y + u1η(x, y))) ≤ max{f(y + u2η(x, y)),f(y + u1η(x, y))} ≤ max{max{f(x),f(y)}, max{f(x),f(y)}} = max{f(x),f(y)}. This implies that λ3 ∈ A. If λ3 ≥ λ0, then it follows from (8) that λ3 − u2 = (1 − α)(u1 − u2) < λ1 − λ2, therefore λ3 < λ1 + (u2 − λ2) ≤ λ1. On the other hand, λ3 ≥ λ0 and λ3 ∈ A show that λ3 ≥ λ1, this is a contradiction. Similarly, if λ3 < λ0, then λ3 − u1 = α(u2−u1) > λ2−λ1, thus λ3 > λ2+(u1−λ1) ≥ λ2, this also provides a contradiction. Consequently, the proof is completed. Theorem 3.4. Let K be a locally invex set and let f : K → R be upper semicon- tinuous. Suppose that Conditions C and D hold. If there is an α ∈ (0,a0) such that f satisfies the inequality 5, then f is a semilocally prequasi-invex function on K. Proof of Theorem 3.4. Suppose, by contradiction, the conclusion does not hold. 1 Then, there exists a pair of points x0,y0 ∈ K, such that for any k ∈ (0,a(x0,y0)] 1 (k large enough), there is a number δk ∈ (0, k ) such that f(y0 + δkη(x0,y0)) > max{f(x0),f(y0)}. (9) k−1 k−1 1 Since k δk ∈ (0,a0) (k large enough), Lemma 3.3 ensures that N( k δk; 2k )∩A 6= ∅, where N(x; ε) = {y : ky − xk < ε}. Therefore, we can choose an λk from k−1 1 N( k δk; 2k ) ∩ A. Therefore, 0 < λk <δk, λk ∈ A, λk → 0, δk → 0, k → ∞. Define λk zk = y0 + δkη(x0,y0), yk = zk − η(x0,zk). 1 − λk Therefore, from Condition C, we have δk − λk η(x0,zk)= η(x0,y0 + δkη(x0,y0)) = (1 − δk)η(x0,y0), yk = y0 + η(x0,y0). 1 − λk SEMILOCALLY PREQUASI-INVEX FUNCTIONS AND CHARACTERIZATIONS 509 δk−λk So, taking into account the local invexity of K and 1−λk ∈ (0,a0) (k large enough), δk −λk one can conclude that yk = y0 + 1−λk η(x0,y0) ∈ K. Further, from Condition C, one has δk −λk δk−λk yk + λkη(x ,yk) = y + η(x ,y )+ λkη(x ,y + η(x ,y )) 0 0 1−λk 0 0 0 0 1−λk 0 0 (10) = y0 + δkη(x0,y0)= zk. On the other hand, by yk → y0 and the upper semicontinuity of f, for any ε > 0, there exists a positive integer kε > 0 such that f(yk) ≤ f(y0)+ ε, ∀k > kε. Therefore, from (10), yk,y0 ∈ K and λk ∈ A, we have f(zk)= f(yk + λkη(x0,yk)) ≤ max{f(x0),f(yk)}≤ max{f(x0),f(y0)+ ε}. Since ε> 0 is arbitrarily small, the inequalities above imply that f(zk) ≤ max{f(x0),f(y0)}, which contradicts the inequality (9). Thus f is a semilocally prequasi-invex function on K, and the proof is completed. Remark 4. It is known that Theorem 3.4 above is a generalization of Theorem 2.1 in [9]. However, it is interesting that, unlike the latter, the former no longer requests the open property of the set K. To given an application of Theorem 3.4, we present another generalized convex function as follows. Definition 3.5. (i) (see [11]) A subset K ⊆ Rn is said to be a locally starshaped set if for each pair of points x, y ∈ K, there exists a maximal positive number a(x, y) ≤ 1 such that y + λ(x − y) ∈ K for λ ∈ (0,a(x, y)). (ii) A function f : K → R is called semilocally quasi-convex on a locally starshaped set K if corresponding to each pair of points x, y ∈ K, there exists a maximal positive number d(x, y) ≤ a(x, y) such that f(y + λ(x − y)) ≤ max{f(x),f(y)}, ∀ λ ∈ (0, d(x, y)). Corollary 1. Let K ⊆ Rn be a locally starshaped set, and f : K → R be upper semicontinuous. Suppose that Condition D holds with η(x, y) = x − y. Then, f is semilocally quasiconvex on K if there exists an α ∈ (0,a0) such that f(αx + (1 − α)y) ≤ max{f(x),f(y)}, ∀x, y ∈ K. 4. Relationships among the generalized convexities. In the section 3, we discussed some basic properties of semilocal prequasi-invexity. Now we present some relationships among the proposed generalized convex functions. For this goal, we give a supplement to Condition C: η(y + λη(x, y),y)= λη(x, y), ∀x, y ∈ K, ∀λ ∈ [0,a(x, y)]. (11) And we say Condition C+ is satisfied if both (1)–(2) and (11) are satisfied. First, similar to the semistrictly quasiconvex functions, we get an interesting relationship between semistrict semilocal prequas-invexity and semilocal prequasi- invexity as follows. Theorem 4.1. Suppose that f is semistrictly semilocally prequasi-invex on K ⊆ Rn and Condition C is satisfied. Then, f is semilocally prequasi-invex if f is lower semicontinuous. 510 JIN-BAO JIAN, HUA-QIN PAN AND HAN-JUN ZENG Proof of Theorem 4.1. By Contradiction, suppose that there exist two points x, y ∈ K and a sequence {λk} such that f(y + λkη(x, y)) > max{f(x),f(y)}, λk > 0, λk → 0. (12) So, by the semistrict semilocal prequasi-invexity of f, we know that f(x) = f(y). Let zk = y + λkη(x, y), yλ = zk + λη(y,zk), where λ > 0 is sufficiently small. Then zk, yλ ∈ K. Furthermore, from Condition C (2), one gets y = z + λη(y,z )= y + λ η(x, y)+ λη(y,y + λ η(x, y)) λ k k k k (13) = y + (1 − λ)λkη(x, y). On the other hand, for a given sufficiently large k. Let εk = f(zk) − f(y) > 0. Then, from the lower semicontinuity of f, it follows that f(yλ) − f(zk) > −εk = f(y) − f(zk), if λ > 0 is small enough. Thus, f(yλ) > f(y) = f(x), if λ is small enough. This together with the semistrict semilocal prequasi-invexity of f implies that f(x)= f(y) λλk Again, since 0 < 1−(1−λ)λk → 0, then from (13) and Condition C (3), one has λλk (13) λλk yλ + 1−(1−λ)λk η(x, yλ) = yλ + 1−(1−λ)λk η(x, y + (1 − λ)λkη(x, y)) (3) λλk = yλ + 1−(1−λ)λk (1 − (1 − λ)λk)η(x, y) = y + λkη(x, y)= zk. So, from f(yλ) >f(x) and the semistrict semilocal prequasi-invexity of f, one has λλk f(zk)= f(yλ + η(x, yλ)) < max{f(x),f(yλ)} = f(yλ). 1 − (1 − λ)λk This inequality contradicts (14), and the proof is completed. Remark 5. In a degree, Theorem 4.1 above is an extension of Theorem 3.3 in [9], however, the proof of the former is much simpler than the one of the latter, and the request conditions are not completely similar. Theorem 4.2. Let f be a semilocally prequasi-invex function on K ⊆ Rn, and + let d0 = inf{d(x, y) : x, y ∈ K}. Assume that Condition C is satisfied. If there exists an σ ∈ (0, d0) such that f(y + ση(x, y)) < max{f(x),f(y)}, ∀x, y ∈ K, f(x) 6= f(y). (15) Then f(y + λη(x, y)) < max{f(x), f(y)}, ∀x y ∈ K, f(x) 6= f(y) ∀λ ∈ (0, d0), (16) and f is an uniformly semistrictly semilocally prequasi-invex function on K. Remark 6. It is found that Theorem 4.2 above is really an extension of Theorem 3.1 in [9], moreover, the request conditions in the former are weaker than the one in the latter. SEMILOCALLY PREQUASI-INVEX FUNCTIONS AND CHARACTERIZATIONS 511 Proof of Theorem 4.2: By contradiction, assume that relationship (16) does not hold. Then there exists a pair of points x0,y0 ∈ K, f(x0) 6= f(y0), and a constant λ0 ∈ (0, d0) satisfying: △ z = y0 + λ0η(x0,y0), f(z) ≥ max{f(x0),f(y0)}. (17) There are two cases: f(y0) f(z + ση(y0,z)) < max{f(y0),f(z)} = f(z). (19) If one denotes w1 = z +ση(y0,z), then w1 ∈ K and f(w1) w1 = z + ση(y0,z), w2 = w1 + ση(z, w1), ..., wk+1 = wk + ση(z, wk), ∀k ∈ N. (20) Next, we analyze the properties of {wk}. (2) w1 = y0 + λ0η(x0,y0)+ ση(y0,y0 + λ0η(x0,y0)) = y0 + (1 − σ)λ0η(x0,y0), w2 = y0 + (1 − σ)λ0η(x0,y0) +ση(y0 + λ0η(x0,y0),y0 + λ0η(x0,y0) − σλ0η(x0,y0)) (2) = y0 + (1 − σ)λ0η(x0,y0) +ση(y0 + λ0η(x0,y0),y0 + λ0η(x0,y0)+ ση(y0,y0 + λ0η(x0,y0))) (2) 2 = y0 + (1 − σ)λ0η(x0,y0) − σ η(y0,y0 + λ0η(x0,y0)) (2) 2 = y0 + (1 − σ + σ )λ0η(x0,y0). Generally, suppose that wk = y0 + (1 − ξk)λ0η(x0,y0), 0 < ξk < σ. Then ξ1 = σ, ξ2 = σ(1 − σ), and wk+1 = wk + ση(z, wk)= wk + ση(z,z − ξkλ0η(x0,y0)) (2) (2) = wk + ση(z,z + ξkη(y0,z)) = wk − σξkη(y0,y0 + λ0η(x0,y0)) (2) = y0 + (1 − ξk + σξk)λ0η(x0,y0)= y0 + (1 − (1 − σ)ξk)λ0η(x0,y0). So k k ξk+1 = (1 − σ)ξk = · · · = (1 − σ) ξ1 = σ(1 − σ) < σ, k−1 wk = y0 + (1 − σ(1 − σ) )λ0η(x0,y0), ∀k =1, 2, ... . (21) Furthermore, by f(w1) In view of σ ∈ (0, 1), λ0 ∈ (0, d0), we can conclude, if k is sufficiently large enough, that 0 <γ1k < λ0 <γ2k < d0 ≤ 1, λ0 = σγ1k +(1−σ)γ2k, wk = y0 +γ1kη(x0,y0). (24) So, from (21)–(24), one has f(wk)= f(y0 + γ1kη(x0,y0)) ≤ max{f(x0),f(y0)} = f(x0)= f(z). (25) γ2k −γ1k On the other hand, one has γ2k −d0 < 0 ≤ (1−d0)γ1k from (24), so 0 < 1−γ1k < d0. Therefore, from Condition C and (24), it follows that sk + ση(wk,sk) = sk + ση(wk, wk + (γ2k − γ1k)η(x0,y0)) (3) γ2k−γ1k = sk + ση(wk, wk + 1−γ1k η(x0, wk)) (2) γ2k −γ1k (3) = sk − σ 1−γ1k η(x0, wk) = sk − σ(γ2k − γ1k)η(x0,y0) = y0 + γ2kη(x0,y0) − σ(γ2k − γ1k)η(x0,y0) (24) = y0 + λ0η(x0,y0)= z. Now we bring out a final contradiction as follows. If f(wk) ≥ f(sk), then, from the relationship above and the semilocal prequasi- invexity of f, one knows f(z)= f(sk + ση(wk,sk)) ≤ max{f(sk),f(wk)} = f(wk), which contradicts (22). If f(wk) < f(sk), in view of z = sk + ση(wk,sk), sk, wk ∈ K, we have f(z) < max{f(sk),f(wk)} = f(sk) = f(y0 + γ2kη(x0,y0)) ≤ max{f(x0),f(y0)} = f(x0). This contradicts (18). Case B: Suppose that f(y0) >f(x0). Since f is semilocally prequasi-invex, we have f(z) ≤ max{f(x0),f(y0)} = f(y0), which together with the inequality (17) leads to f(z) = max{f(x0,f(y0))} = f(y0) >f(x0). (26) Similar to the analysis in Case A, we yield a sequence {zk} as follows. z0 = x0 ∈ K, z1 = z +ση(x0,z) ∈ K, ..., zk = z +ση(zk−1,z) ∈ K, ∀k ∈ N. (27) According to (17), f(z) >f(x0) implies f(z1)= f(z + ση(x0,z)) Let k1 be a positive integer such that k1−1 k1 [σ /(1 − σ)] < [λ0/(1 − λ0)], σ (1 − λ0) < d0 − λ0. (30) And let k1 (k1−1) β1 = λ0 + σ (1 − λ0), β2 = λ0 − σ (1 − σ)(1 − λ0), y1 = y0 + β1η(x0,y0), x1 = y0 + β2η(x0,y0). Then 0 <β2 < λ0 <β1 < d0, and y1, x1 ∈ K. Furthermore, one gets from (3) k1 k1 z + σ η(x0,z) = y0 + λ0η(x0,y0)+ σ η(x0,y0 + λ0η(x0,y0)) (3) k1 = y0 + [λ0 + σ (1 − λ0)]η(x0,y0)= y0 + β1η(x0,y0)= y1. SEMILOCALLY PREQUASI-INVEX FUNCTIONS AND CHARACTERIZATIONS 513 So, according to (28) and the equality above, we obtain k1 f(y1)= f(z + σ η(x0,z)) = f(zk1 ) Again, since β1 − d0 < 0 ≤ (1 − d0)β2, it follows that (β1 − β2)/(1 − β2) < d0. So, from Condition C, one knows that y1 + ση(x1,y1) = y1 + ση(x1, x1 + (β1 − β2)η(x0,y0)) (3) β1−β2 (2) β1−β2 = y1 + αη(x1, x1 + 1−β2 η(x0, x1) = y1 − σ 1−β2 η(x0, x1) (3) = y0 + [β1 − σ(β1 − β2)]η(x0,y0)= y0 + λ0η(x0,y0)= z. (32) Now we shall finish the rest proof in two cases as follows: f(y1) ≥ f(x1) and f(y1) f(z)= f(y1 + ση(x1, y1)) ≤ max{f(y1), f(x1)} = f(y1), which contradicts (31). (ii) Assume that f(y1) f(z) < max{f(x1),f(y1)} = f(x1). (33) Also taking into account x1 = y0 + β2η(x0,y0), β2 < d0 and f being semilocally prequasi-invex, we have f(x1) ≤ max{f(x0),f(y0)} = f(y0). (34) So, (33) and (34) show that f(z) Theorem 4.3. Suppose that f is a semilocally prequasi-invex function on K, and d0 = inf{d(x, y) : x, y ∈ K}. Assume that Condition C is satisfied. Assume that there exist an σ ∈ (0, d0) such that f(y + ση(x, y)) < max{f(x),f(y)}, ∀x, y ∈ K, x 6= y. (35) Then f(y + λη(x, y)) < max{f(x),f(y)}, ∀x, y ∈ K, x 6= y, ∀λ ∈ (0, d0). Therefore, f is an uniformly strictly semilocally prequasi-invex function on K. Proof of Theorem 4.3. By contradiction, we assume that there exists x0,y0 ∈ K, x0 6= y0, and a real number λ0 ∈ (0, d0) such that f(y0 + λ0η(x0,y0)) ≥ max{f(x0),f(y0)}. Since f is semilocally prequasi-invex, we have f(y0 + λ0η(x0,y0)) ≤ max{f(x0),f(y0)}, hence f(y0 + λ0η(x0,y0)) = max{f(x0),f(y0)}. (36) Taking into account λ0 > (λ0 − (1 − σ)d0)/σ, one can choose a number β2 such that λ0 >β2 > 0, β2 > (λ0 − (1 − σ)d0)/σ. Let β1 = (λ0 − σβ2)/(1 − σ). Then, λ0 = (1 − σ)β1 + σβ2, 0 <β2 < λ0 <β1 < d0. (37) 514 JIN-BAO JIAN, HUA-QIN PAN AND HAN-JUN ZENG Define x1 = y0 + β1η(x0,y0), y1 = y0 + β2η(x0,y0). Further, by β1 − d0 < 0 ≤ (1 − d0)β2 and β2 <β1 < 1, one has (β1 − β2)/(1 − β2) ∈ (0, d0). So, using Condition C, similar to the analysis of (32), we can obtain y1 + ση(x1,y1)= y0 + λ0η(x0,y0). (38) Again, since f is semilocally prequasi-invex and β2 <β1 < d0, one gets f(x1) ≤ max{f(x0),f(y0)}, f(y1) ≤ max{f(x0),f(y0)}. (39) On the other hand, (35) and (36) imply that η(x0, y0) 6=0, so x1 6= y1. Therefore, by (38), (35) and (39), we have f(y0 + λ0η(x0,y0)) = f(y1 + ση(x1,y1)) < max{f(x1),f(y1)}≤ max{f(x0),f(y0)}, which contradicts the inequality (36). The proof is completed. Theorem 4.4. Let f be a semistrictly semilocally prequasi-invex function on K + + and let d0 = inf{d(x, y) : x, y ∈ K,f(x) 6= f(y)}. Suppose Condition C is satisfied. Then f is a semilocally prequasi-invex function on K if there is an α ∈ + (0, min{d0 ,a0}) such that f satisfies the inequality (5). Proof of Theorem 4.4. Suppose, by contradiction, f is not semilocally prequasi- 1 invex on K. Then there exists x0,y0 ∈ K corresponding to k ∈ (0,a(x0,y0)], 1 there is an λk ∈ (0, k ), such that f(y0 + λkη(x0,y0)) > max{f(x0),f(y0)} (k large enough). Therefore, by the semistrict semilocal prequasi-invexity of f and λk → 0, we know that f(x0)= f(y0). Let λ z = y + λ η(x ,y ), y = y + k η(x ,y ). k 0 k 0 0 k 0 α 0 0 Then f(zk) > f(x0) = f(y0). Without loss of generality, assume that λk ≤ αa0. + From Condition C , we have yk ∈ K and λ (11) λ y + αη(y ,y )= y + αη(y + k η(x ,y ),y ) = y + α k η(x ,y )= z . 0 k 0 0 0 α 0 0 0 0 α 0 0 k So, according to (5), we have f(zk) ≤ max{f(yk),f(y0)}. Therefore, f(x0)= f(y0) max{f(yk),f(y0)} = f(yk), f(zk) ≤ f(yk). (40) λk (1−α) + Let bk = α(1−λk ) . Then bk → 0 and 0 zk + bkη(x0,zk) = y0 + λkη(x0,y0)+ bkη(x0,y0 + λkη(x0,y0)) (3) = y0 + [λk + bk(1 − λk)]η(x0,y0) λk (1−α) λk = y0 + [λk + α ]η(x0,y0)= y0 + α η(x0,y0)= yk. Therefore, this relationship together with the semistrict semilocal prequasi-invexity of f and f(zk) >f(x0) shows that f(yk)= f(zk + bkη(x0,zk)) < max{f(x0),f(zk)} = f(zk), which contradicts (40). The proof is completed. SEMILOCALLY PREQUASI-INVEX FUNCTIONS AND CHARACTERIZATIONS 515 Remark 7. In a degree, Theorem 4.4 above is an extension of Theorem 2.4 in [9]. Furthermore, we guess that f may be uniformly semilocally prequasi-invex under the conditions requested in Theorem 4.4, but it is a regret that we can not finish the whole proof up to now (part proof has been finished). 5. Semilocally prequasi-invex programs. In this section, some important properties of optimal solutions to a semilocally prequasi-invex program problem are discussed. First, we discuss a property of the level set of a semilocally prequasi-invex function. Theorem 5.1. Let K ⊆ Rn be locally invex with respect to map η and let f : K → R. Then f is a semilocally prequasi-invex function on K if and only if the set Kα = {x ∈ K : f(x) ≤ α} is locally invex with respect to the same map η for each α ∈ R. Proof of Theorem 5.1. We first prove the necessity. Let f be a semilocally prequai- invex function on K. For any x, y ∈ Kα, one knows f(x) ≤ α and f(y) ≤ α. Then for each λ ∈ [0, d(x, y)], we have f(y + λη(x, y)) ≤ max{f(x),f(y)} ≤ α, so y + λη(x, y) ∈ Kα, that is to say, Kα is a locally invex set with respect to the same map η. Second, we prove the sufficiency. For any x, y ∈ K, let α = max{f(x),f(y)}. It is obvious that x, y ∈ Kα. Note that Kα is a locally invex set, corresponding to map η, there is a maximal positive number a(x, y) ≤ 1 such that y + λη(x, y) ∈ Kα, ∀λ ∈ [0,a(x, y)]. That is f(y + λη(x, y)) ≤ α = max{f(x),f(y)}, ∀λ ∈ [0,a(x, y)]. So the function f is semilocally prequasi-invex on K. The proof is completed. Theorem 5.2. Let f be a semistrictly semilocally prequasi-invex function on a locally invex set K with respect to map η. Consider the optimization problem (P): min{f(x) : x ∈ K}. Then the following three statements hold true. (i) Each local optimal solution x¯ of the problem (P) is also a global one. (ii) The optimal solution set Ω = {x ∈ K : x is an optimal solution of (P)} is locally invex with respect to the map η. (iii) If f is strictly semilocally prequasi-invex on K, then the optimal solution of (P) is unique. Proof of Theorem 5.2. (i) By contradiction, suppose thatx ¯ is not a global solution of (P). Then there exists x′ ∈ K such that f(x′) x2 + λη(x1, x2) ∈ K, f(x2 + λη(x1, x2)) < max{ f(x1),f(x2)} = f(x2)= f(x1), ∀λ ∈ (0, d(x1, x2)), which contradicts the optimal property of x1 and x2. 516 JIN-BAO JIAN, HUA-QIN PAN AND HAN-JUN ZENG Remark 8. The following example shows that the semistrict semilocal prequasi- invexity of function f in Theorem 5.2 can’t be weaken as the semilocal prequasi- invexity. Example 5.1 In program (P), let K = R, η(x, y) ≡ 0, f(x)= −1, if x = 0; f(x)= x, otherwise. Then f is semilocally prequasi-invex, but not semistrictly semilocally prequasi- invex. Moreover, Theorem 5.2 doesn’t hold true, since x = 0 is a local optimal solution, but not a global one. Now, let’s consider the inequality constrained program : min f(x) (ICP) n s.t.Kg = {x ∈ R : gi(x) ≤ 0,i =1, .., m}, and apply the results of Theorem 5.2 to it. Theorem 5.3. For the problem (ICP), suppose that each constraint function gi (i = 1, ..., m) is semilocally prequasi-invex function with respect to a same map η : Rn × Rn → Rn. Then the following four statements hold true. (i) The feasible set Kg of (ICP) is locally invex with the map η. (ii) If the objective function f is semistrictly semilocally prequasi-invex on Kg, then each local optimal solution of the problem (ICP) is also a global one. (iii) If the objective function f is semistrictly semilocally prequasi-invex on Kg, then the optimal solution set Ωg = {x ∈ Kg : x is an optimal solution of (ICP)} is locally invex with respect to the map η. (iv) Additionally, if f is strictly semilocally prequasi-invex on Kg, then the opti- mal solution of (ICP) is unique. m n Proof of Theorem 5.3. Let Ki = {x ∈ R : gi(x) ≤ 0}. Then Kg = Ki. In iT=1 view of Theorem 5.1, we know that the sets Ki are all locally invex corresponding to the same map η, furthermore, Kg is locally invex by Proposition 2.1. Conclusions (ii)–(iv) are direct applications of Theorem 5.2. Theorem 5.4. Suppose that the functions f and gi in (ICP) are all differentiable, and assume that x¯ is a KKT point of the problem (ICP), that is, there exist multi- pliers u¯i such that ∇f(¯x)+ u¯i∇gi(¯x)=0, u¯i ≥ 0, i ∈ I = {i : gi(¯x)=0}. (41) Xi∈I n n n If there exists a map η : R × R → R such that functions gi (i ∈ I) are all semilocally prequasi-invex and f is semilocally prepseudo-invex (see Definition 3.1) with respect to the same map η, then x¯ is a global optimal solution of (ICP). Proof of Theorem 5.4. For each feasible point x ∈ Kg, it follows that gi(x) ≤ 0 = gi(¯x), i ∈ I. So, from Theorem 3.1(i), one gets T ∇gi(¯x) η(x, x¯) ≤ 0, ∀i ∈ I. Which together with (41) implies that T T ∇f(¯x) η(x, x¯)= − u¯i∇gi(¯x) η(x, x¯) ≥ 0. Xi∈I SEMILOCALLY PREQUASI-INVEX FUNCTIONS AND CHARACTERIZATIONS 517 Therefore, by Definition 3.1, one knows that f(x) ≥ f(¯x), sox ¯ is a global optimal solution of (ICP), and the proof is completed. Acknowledgements. 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