JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 168172 10.1155/2012/168172 168172 Research Article A Class of G-Semipreinvex Functions and Optimality Liu Xue Wen 1 Zhao Ke Quan 1 Chen Zhe 1, 2 Huang Xue-Xiang 1 Department of Mathematics Chongqing Normal University, Chongqing 400047 China cqnu.edu.cn 2 School of Economics and Management Tsinghua University, Beijing 100084 China tsinghua.edu.cn 2012 9 12 2012 2012 31 05 2012 27 10 2012 2012 Copyright © 2012 Xue Wen Liu 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 G-semipreinvex functions, which are some generalizations of the semipreinvex functions, and the G-convex functions, is introduced. Examples are given to show their relations among G-semipreinvex functions, semipreinvex functions and G-convex functions. Some characterizations of G-semipreinvex functions are also obtained, and some optimality results are given for a class of G-semipreinvex functions. Ours results improve and generalize some known results.

1. Introduction

Generalized convexity has been playing a central role in mathematical programming and optimization theory. The research on characterizations of generalized convexity is one of most important parts in mathematical programming and optimization theory. Many papers have been published to study the problems of how to weaken the convex condition to guarantee the optimality results. Schaible and Ziemba  introduced G-convex function which is a generalization of convex function and studied some characterizations of G-convex functions. Hanson  introduced invexity which is an extension of differentiable convex function. Ben-Israel and Mond  considered the functions for which there exists η:Rn×RnRn such that, for any x,yRn,  λ[0,1], (1.1)f(y+λη(x,y))λf(x)+(1-λ)f(y). Weir et al. [4, 5] named such kinds of functions which satisfied the condition (1.1) as preinvex functions with respect to η. Further study on characterizations and generalizations of convexity and preinvexity, including their applications in mathematical programming, has been done by many authors (see ). As a generalization of preinvexity, Yang and Chen  introduced semipreinvex functions and discussed the applications in prevariational inequality. Yang et al.  investigated some properties of semipreinvex functions. As a generalization of G-convex functions and preinvex functions, Antczak  introduced G-preinvex functions and obtained some optimality results for a class of constrained optimization problems. As a generalization of B-vexity and semipreinvexity, Long and Peng  introduced the concept of semi-B-preinvex functions. Zhao et al.  introduced r-semipreinvex functions and established some optimality results for a class of nonlinear programming problems.

Motivated by the results in , in this paper, we propose the concept of G-semipreinvex functions and obtain some important characterizations of G-semipreinvexity. At the same time, we study some optimality results under G-semipreinvexity. Our results unify the concepts of G-convexity, preinvexity, G-preinvexity, semipreinvexity, and r-semipreinvexity.

2. Preliminaries and Definitions Definition 2.1 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

Let G be a continuous real-valued strictly monotonic function defined on DR. A real-valued function f defined on a convex set XRn is said to be G-convex if for any x,yX,  λ[0,1], (2.1)f(y+λ(x-y))G-1(λG(f(x))+(1-λ)G(f(y))), where G-1 is the inverse of G, f(X)D.

Remark 2.2.

Every convex functions is G-convex, but the converse is not necessarily true.

Example 2.3.

Let X=[-1,1],  f:XR, If(X) be the range of real-valued function f, and let G:If(X)R be defined by (2.2)f(x)=arctan(|x|+1),G(t)=tan(t). Then, we can verify that f is a G-convex function. But f is not a convex function because the following inequality (2.3)f(y+λ(x-y))>λf(x)+(1-λ)f(y) holds for x=1/4,  y=3/4,  and  λ=1/2.

Weir et al. [4, 5] presented the concepts of invex sets and preinvex functions as follows.

Definition 2.4 (see [<xref ref-type="bibr" rid="B4">4</xref>, <xref ref-type="bibr" rid="B5">5</xref>]).

A set XRn is said to be invex if there exists a vector-valued function η:X×XRn such that for any x,yX,  λ[0,1], (2.4)y+λη(x,y)X.

Definition 2.5 (see [<xref ref-type="bibr" rid="B4">4</xref>, <xref ref-type="bibr" rid="B5">5</xref>]).

Let XRn be invex with respect to vector-valued function η:X×XRn. Function f(x) is said to be preinvex with respect to η if for any x,yX,  λ[0,1], (2.5)f(y+λη(x,y))λf(x)+(1-λ)f(y).

Remark 2.6.

Every convex function is a preinvex function with respect to η=x-y, but the converse is not necessarily true.

Example 2.7.

Let X=[-1,1]. f:XR be defined by (2.6)f(x)=arctan(|x|+1). Then, we can verify that f is a preinvex function with respect to η, where (2.7)η(x,y)={-y-x2+2x,0x1,0y1,-y-x,-1x<0,0y1,-y-x,0x1,-1y<0,-y+x,-1x<0,-1y<0. But f is not convex a function in Example 2.3.

Antczak  introduced the concept of G-preinvex functions as follows.

Definition 2.8 (see [<xref ref-type="bibr" rid="B17">17</xref>]).

Let X be a nonempty invex (with respect to η) subset of Rn. A function f:XR is said to be (strictly) G-preinvex at y with respect to η if there exists a continuous real-valued increasing function G:If(X)R such that for all xX(xy), λ[0,1], (2.8)f(y+λη(x,y))G-1(λ(G(f(x)))+(1-λ)G(f(y))),(f(y+λη(x,y))<G-1(λ(G(f(x)))+(1-λ)G(f(y)))). If (2.8) is satisfied for any yX, then f is said to be (strictly) a G-preinvex function on X with respect to η.

Remark 2.9.

Every preinvex function with respect to η is G-preinvex function with respect to the same η, where G(x)=x. Every G-convex function is G-preinvex function with respect to η(x,y,λ)=x-y. However, the converse is not necessarily true.

Example 2.10.

Let X=[-1,1]. f:XR, G:If(X)R be defined by (2.9)f(x)=arctan(2-|x|),G(t)=tant. Then, we can verify that f is a G-preinvex function with respect to η, where (2.10)η(x,y)={-y-x2+2x,0x1,0y1,-y-x2-2x,-1x<0,0y1,-y-x,0x1,-1y<0,-y+x,-1x<0,-1y<0. But f is not a preinvex function because the following inequality (2.11)f(y+λη(x,y))>λf(x)+(1-λ)f(y) holds for x=0,  y=1, and  λ=1/2.

And f(x) is not a G-convex function because the following inequality (2.12)f(y+λ(x-y))>G-1(λG(f(x))+(1-λ)G(f(y))) holds for x=1,  y=-1, and  λ=1/2.

Definition 2.11 (see [<xref ref-type="bibr" rid="B15">15</xref>]).

A set XRn is said to be a semiconnected set if there exists a vector-valued function η:X×X×[0,1]Rn such that for any x,yX,  λ[0,1], (2.13)y+λη(x,y,λ)X.

Definition 2.12 (see [<xref ref-type="bibr" rid="B15">15</xref>]).

Let XRn be a semiconnected set with respect to a vector-valued function η:X×X×[0,1]Rn. Function f is said to be semipreinvex with respect to η if for any x,yX,  λ[0,1], limλ0λη(x,y,λ)=0, (2.14)f(y+λη(x,y,λ))λf(x)+(1-λ)f(y). Next we present the definition of G-semipreinvex functions as follows.

Definition 2.13.

Let XRn be semiconnected set with respect to vector-valued function η:X×X×[0,1]Rn. A function f:XR is said to be (strictly) G-semipreinvex at y with respect to η if there exists a continuous real-valued strictly increasing function G:If(X)R such that for all xX(xy), λ[0,1], limλ0λη(x,y,λ)=0,(2.15)f(y+λη(x,y,λ))G-1(λG(f(x))+(1-λ)G(f(y))),(f(y+λη(x,y,λ))<G-1(λG(f(x))+(1-λ)G(f(y)))). If (2.15) is satisfied for any yX, then f is said to be (strictly) G-semipreinvex on X with respect to η.

Remark 2.14.

Every semipreinvex function with respect to η is a G-semipreinvex function with respect to the same η, where G(x)=x. However, the converse is not true.

Example 2.15.

Let X=[-6,6]. Then X is a semiconnected set with respect to η(x,y,λ) and limλ0λη(x,y,λ)=0, where (2.16)η(x,y,λ)={x-yλ3,-6x<0,-6y<0,x>y,0<λ1,λ2(x-y),0x6,0y6,xy,λ2(x-y),-6x<0,-6y<0,xy,x-y,0x6,0y6,x<y,x-y,0x6,-6y<0,x<-y,x-y,-6x<0,0y6,x>-y,0,0x6,-6y<0,x-y,0,-6x<0,0y6,x-y. Let f:XR, G:If(X)R be defined by (2.17)f(x)=arctan(x2+2),G(t)=tant. Then, we can verify that f is a G-semipreinvex function with respect to η. But f is not a semipreinvex function with respect to η because the following inequality (2.18)f(y+λη(x,y,λ))>λf(x)+(1-λ)f(y) holds for x=2, and  y=4,  λ=1/2.

Example 2.16.

Let X=[-6,6]. Then X is a semiconnected set with respect to η(x,y,λ) and limλ0λη(x,y,λ)=0, where (2.19)η(x,y,λ)={x-yφ(λ),-6x<0,-6y<0,x>y,0<λ1,φ(λ)(x-y),0x6,0y6,xy,φ(λ)(x-y),-6x<0,-6y<0,xy,x-y,0x6,0y6,x<y,x-y,0x6,-6y<0,x<-y,λ<φ(λ)<1x-y,-6x<0,0y6,x>-y,0,0x6,-6y<0,x-y,0,-6x<0,0y6,x-y. Let f:XR,  G:If(X)R be defined by (2.20)f(x)=arctan(x2+k),G(t)=tant,kR. Then, we can verify that f(x) is a G-semipreinvex function with respect to classes of functions η. But f(x) is not semipreinvex function with respect to η because the following inequality (2.21)f(y+λη(x,y,λ))>λf(x)+(1-λ)f(y) holds for x=2,  y=4,  and  λ=1/2.

Remark 2.17.

Every a G-convex function is G-semipreinvex function with respect to η(x,y,λ)=x-y. But the converse is not true.

Example 2.18.

Let X=(-6,6), it is easy to check that X is a semiconnected set with respect to η(x,y,λ) and limλ0λη(x,y,λ)=0, where (2.22)η(x,y,λ)={λ(x-y),0x<6,0y<6,x<y,λ(x-y),-6<x<0,-6<y<0,x>y,x-yλ,0x<6,0y<6,xy,0<λ1,x-yλ,-6<x<0,-6<y<0,xy,0<λ1,-x-y,0x<6,-6<y<0,x-y,-x-y,-6<x<0,0y<6,x-y,0,  0x<6,-6<y<0,x<-y,0,-6<x<0,0y<6,x>-y. Let f:XR,  G:If(X)R be defined by (2.23)f(x)=arctan(6-|x|),G(t)=tant. Then, we can verify that f is a G-semipreinvex function with respect to η. But f is not a G-convex function, because the following inequality (2.24)f(y+λ(x-y))>G-1(λG(f(x))+(1-λ)G(f(y))) holds for  x=1,  y=-1,  and  λ=1/2.

3. Some Properties of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M202"><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:math></inline-formula>-Semipreinvex Functions

In this section, we give some basic characterizations of G-semipreinvex functions.

Theorem 3.1.

Let f be a G1-semipreinvex function with respect to η on a nonempty semiconnected set XRn with respect to η, and let G2 be a continuous strictly increasing function on If(X). If the function g(t)=G2G1-1(t) is convex on the image under G1 of the range of f, then f is also G2-semipreinvex function on X with respect to the same function η.

Proof.

Let X be a nonempty semiconnected subset of Rn with respect to η, and we assume that f is G1-semipreinvex with respect to η. Then, for any x,yX,  λ[0,1], (3.1)f(y+λη(x,y,λ))G1-1(λG1(f(x))+(1-λ)G1(f(y))). Let G2 be a continuous strictly increasing function on If(X). Then, (3.2)G2(f(y+λη(x,y,λ)))G2G-1(λG1(f(x))+(1-λ)G1(f(y))). By the convexity of g(t)=G2G1-1, it follows the following inequality (3.3)G2G-1(λG1(f(x))+(1-λ)G1(f(y)))λG2G1-1(G1(f(x))+(1-λ)G2G1-1(G1f(y)))=λG2(f(x))+(1-λ)G2(f(y)) for all x,yX,  λ[0,1]. Therefore, (3.4)G1-1[λ(G1(f(x)))+(1-λ)G1(f(y))]G2-1[λ(G2(f(x)))+(1-λ)G2(f(y))]. Thus, we have (3.5)f(y+λη(x,y,λ))G2-1(λG2(f(x))+(1-λ)G2(f(y))).

Theorem 3.2.

Let f be a G-semipreinvex function with respect to η on a nonempty semiconnected set XRn with respect to η. If the function G is concave on If(X), then f is semipreinvex function with respect to the same function η.

Proof.

Let y,zIf(X), from the assumption G is concave on If(X), we have (3.6)G(z+λ(y-z))λG(y)+(1-λ)G(z),λ[0,1]. Let (3.7)G(y)=x,G(z)=u,y=G-1(x),z=G-1(u), then (3.8)G(G-1(u)+λ(G-1(x)-G-1(u)))λG(G-1(x))+(1-λ)G(G-1(u))=λx+(1-λ)u. It follows that (3.9)G-1G(λG-1(x)+(1-λ)G-1(u))G-1(λx+(1-λ)u). Then, (3.10)λG-1(x)+(1-λ)G-1(u)G-1(λx+(1-λ)u). This means that G-1 is convex. Let G1=G,  G2=t, then g(t)=G2G1-1(t) is convex. Hence by Theorem 3.1, f is G2-semipreinvex with respect to η. But G2 is the identity function; hence, f is a semipreinvex function with respect to the same function η.

Theorem 3.3.

Let X be a nonempty semiconnected set with respect to η subset of Rn and let fi:XR,  iI, be finite collection of G-semipreinvex function with respect to the same η and G on X. Define f(x)=sup(fi(x):iI), for every xX. Further, assume that for every xX, there exists i*=i(x)I, such that f(x)=fi*(x). Then f is G-semipreinvex function with respect to the same function η.

Proof.

Suppose that the result is not true, that is, f is not G-semipreinvex function with respect to η on X. Then, there exists x,yX,  λ[0,1] such that (3.11)f(y+λη(x,y,λ))>G-1(λG(f(x))+(1-λ)G(f(y))). We denote z=y+λη(x,y,λ) there exist i(z):=izI,  i(x):=ixI,  and  i(y):=iyI, satisfying (3.12)f(z)=fiz(z),f(x)=fix(x),f(y)=fiy(y). Therefore, by (3.11), (3.13)fiz(z)>G-1(λG(fix(x))+(1-λ)G(fiy(y))). By the condition, we obtain (3.14)fiz(z)G-1(λG(fiz(x))+(1-λ)G(fiz(y))). From the definition of G-semipreinvexity, G is an increasing function on its domain. Then, G-1 is increasing. Since fiz(x)fix(x),  fiz(y)fiy(y), then (3.14) gives (3.15)fiz(z)G-1(λG(fix(x))+(1-λ)G(fiy(y))). The inequality (3.15) above contradicts (3.13).

Theorem 3.4.

Let f be a G-semipreinvex function with respect to η on a nonempty semiconnected set XRn with respect to η. Then, the level set Sα={xX:f(x)α} is a semiconnected set with respect to η, for each αR.

Proof.

Let x,ySα, for any arbitrary real number α. Then, f(x)α,  f(y)α. Hence, it follows that (3.16)f(y+λη(x,y,λ))G-1(λG(f(x))+(1-λ)G(f(y)))G-1(G(α))=α. Then, by the definition of level set we conclude that y+λη(x,y,λ)Sα, for any λ[0,1], we conclude that Sα is a semiconnected set with respect to η.

Let f is a G-semipreinvex function with respect to η, its epigraph Ef={(x,α):xX,αR,f(x)α} is said to be G-semiconnected set with respect to η if for any (x,α)Ef,  (y,β)Ef,  λ[0,1], (3.17)(y+λη(x,y,λ),G-1(λG(α)+(1-λ)G(β)))Ef.

Theorem 3.5.

Let XRn with respect to η be a nonempty semiconnected set, and let f be a real-valued function defined on X. Then, f is a G-semipreinvex function with respect to η if and only if its epigraph Ef={(x,α):xX,αR,f(x)α} is a G-semiconnected set with respect to η.

Proof.

Let (x,α)Ef,(y,β)Ef, then f(x)α,f(y)β. Thus, for any λ[0,1], (3.18)f(y+λη(x,y,λ))G-1(λG(f(x))+(1-λ)G(f(y)))G-1(λG(α)+(1-λ)G(β)). By the definition of an epigraph of f, this means that (3.19)(y+λη(x,y,λ),G-1(λG(α)+(1-λ)G(β)))Ef. Thus, we conclude that Ef is a G semiconnected set with respect to η.

Conversely, let Ef be a G semiconnected set. Then, for any x,yX, we have (x,f(x))Ef,  (y,f(y))Ef. By the definition of an epigraph of f, the following inequality (3.20)f(y+λη(x,y,λ))G-1(λG(f(x))+(1-λ)G(f(y))) holds for any λ[0,1]. This implies that f is a G-semipreinvex function on X with respect to η.

The following results characterize the class of G-semipreinvex functions.

Theorem 3.6.

Let XRn be a semiconnected set with respect to η:X×X×[0,1]Rn; f:XR is a G-semipreinvex function with respect to the same η if and only if for all x,yX,  λ[0,1], and u,vR, (3.21)f(x)u,f(y)vf(y+λη(x,y,λ))G-1(λG(u)+(1-λ)G(v)).

Proof.

Let f be G-semipreinvex functions with respect to η, and let f(x)u,  f(y)v,  0<λ<1. From the definition of G-semipreinvexity, we have (3.22)f(y+λη(x,y,λ))G-1(λG(f(x))+(1-λ)G(f(y)))G-1(λG(u)+(1-λ)G(v)). Conversely, let x,yX,  λ[0,1]. For any δ>0, (3.23)f(x)<f(x)+δ,f(y)<f(y)+δ. By the assumption of theorem, we can get that for 0<λ<1, (3.24)f(y+λη(x,y,λ))G-1(λG(f(x))+(1-λ)G(f(y)))G-1(λG(f(x)+δ)+(1-λ)G(f(y)+δ)). Since G is a continuous real-valued increasing function, and δ>0 can be arbitrarily small, let δ0, it follows that (3.25)f(y+λη(x,y,λ))G-1(λG(u)+(1-λ)G(v)).

4. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M377"><mml:mrow><mml:mi>G</mml:mi></mml:mrow></mml:math></inline-formula>-Semipreinvexity and Optimality

In this section, we will give some optimality results for a class of G-semipreinvex functions.

Theorem 4.1.

Let f:XR be a G-semipreinvex function with respect to η, and we assume that η satisfies the following condition: η(x,y,λ)0, when xy. Then, each local minimum point of the function f is its point of global minimum.

Proof.

Assume that y-X is a local minimum point of f which is not a global minimum point. Hence, there exists a point x-X such that f(x-)<f(y-). By the G-semipreinvexity of f with respect to η, we have (4.1)f(y-+λη(x-,y-,λ))G-1(λG(f(x-))+(1-λ)G(f(y-))),λ[0,1]. Then, for λ[0,1], (4.2)f(y-+λη(x-,y-,λ))<G-1(λG(f(y-))+(1-λ)G(f(y-)))=G-1(G(f(y¯)))=f(y¯). Thus, we have (4.3)f(y-+λη(x-,y-,λ))<f(y-). This is a contradiction with the assumption.

Theorem 4.2.

The set of points which are global minimum of f is a semiconnected set with respect to η.

Proof.

Denote by A the set of points of global minimum of f, and let x,yA. Since f is G-semipreinvex with respect to η, then (4.4)f(y+λη(x,y,λ))G-1(λG(f(x))+(1-λ)G(f(y))),λ[0,1] is satisfied. Since f(x)=f(y), we have (4.5)f(y+λη(x,y,λ))G-1(λG(f(y))+(1-λ)G(f(y))). So, for any λ[0,1], (4.6)f(y+λη(x,y,λ))G-1(G(f(y)))=f(y)=f(x). Since x,yA are points of a global minimum of f, it follows that, for any λ[0,1], the following relation: (4.7)y+λη(x,y,λ)A is satisfied. Then, A is a semiconnected set with respect to η.

Acknowledgments

This work is supported by the National Science Foundation of China (Grants nos. 10831009, 11271391, and 11001289) and Research Grant of Chongqing Key Laboratory of Operations Research and System Engineering. The authors are thankful to Professor Xinmin Yang, Chongqing Normal University, for his valuable comments on the original version of this paper.

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