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Semistrict Gpreinvexity and its application
Journal of Inequalities and Applications volume 2012, Article number: 198 (2012)
Abstract
In this paper, a class of semistrictly Gpreinvex functions introduced by Luo and Wu (J. Comput. Appl. Math. 222:372380, 2008) is further considered. Some properties of semistrictly Gpreinvex functions are obtained, especially those containing an interesting gradient property. Then, some optimality results, which extend the corresponding results in the literature (Yang and Li in J. Math. Anal. Appl. 256:229241, 2001; Yang and Li in J. Math. Anal. Appl. 258:287308, 2001; Antczak in J. Glob. Optim. 43:97109, 2009; Luo and Wu in J. Comput. Appl. Math. 222:372380, 2008), are derived in multiobjective optimization problems.
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 on characterizations and generalizations of convexity and generalized convexity is 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. In 1981, Hanson [5] introduced the concept of invexity which is an extension of differentiable convex functions and proved the sufficiency of KuhnTucker condition. Later, Weir and Mond [6] and Weir and Jeyakumar [7] introduced preinvex functions, and they also studied how and where preinvex functions can replace convex functions in an optimization problem. Then, Yang and Li [8] obtained some properties of a preinvex function in 2001. Yang and Li [9] also 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 of invex functions and (semistrictly) preinvex functions and their applications in mathematical programming are discussed in some existing literature (see [6–11]).
On the other hand, Avriel et al. [12] introduced the definition of Gconvex functions, which is another generalization of convex functions, where G is a continuous realvalued increasing function. As a generalization of Gconvex functions and invex functions, Antczak [13] introduced the concept of Ginvex functions and derived some optimality conditions for constrained optimization problems under the assumption of Ginvexity. Antczak [14] introduced a class of Gpreinvex functions, which is a generalization of Ginvex [13], preinvex functions [6, 8] and rpreinvex functions [15]. Then, Luo and Wu [16] introduced the concept of semistrictly Gpreinvex functions, which includes semistrictly preinvex functions [9] as a special case, and investigated the relations between semistrictly Gpreinvex functions and Gpreinvex functions.
However, to the best of our knowledge, it appears that there are no results on the properties and applications of semistrictly Gpreinvex functions in literature. So, in this paper we study some properties of semistrictly Gpreinvex functions and applications in a multiobjective optimization problem. The rest of the paper is organized as follows. In Section 2, we recall some definitions and give some examples to show that semistrictly Gpreinvex functions are different from preinvex functions, Ginvex functions, Gpreinvex functions and strictly Gpreinvex functions. In Section 3, we obtain some properties of semistrictly Gpreinvex functions, especially those containing an interesting gradient property. Finally, optimality results for multiobjective optimization problems are obtained in Section 4. Our results extend and generalize the corresponding ones in [8, 9, 13, 14, 16].
2 Preliminaries
Throughout this paper, let K be a nonempty subset of {R}^{n}. Let f:K\to R be a realvalued function and \eta :K\times K\to {R}^{n} be a vectorvalued function. And let {I}_{f}(K) be the range of f, i.e., the image of K under f, and {f}^{1} be the inverse of f.
Now we recall some definitions.
Definition 2.1 ([5])
A set K is said to be invex at y with respect to η if for all x\in K, \lambda \in [0,1] such that
The set K is said to be invex with respect to η if K is invex at each y\in K.
Definition 2.2 ([6])
Let K\subseteq {R}^{n} be an invex set with respect to η. The function f is said to be preinvex on K with respect to η iff
Remark 2.1 Any convex function is a preinvex function with \eta (x,y)=xy. But the converse is not true.
Definition 2.3 ([9])
Let K\subseteq {R}^{n} be an invex set with respect to η. The function f is said to be semistrictly preinvex on K with respect to η if, for all x,y\in K, f(x)\ne f(y), we have
Remark 2.2 Any semistrictly (or strong) convex function is a semistrictly preinvex function with \eta (x,y)=xy. But the converse is not true.
Definition 2.4 ([14])
Let K\subseteq {R}^{n} be an invex set with respect to η. The function f is said to be Gpreinvex on K with respect to η if there exists a continuous realvalued increasing function G:{I}_{f}(K)\to R such that for all x,y\in K and \lambda \in [0,1], we have
f:K\to R is said to be strictly Gpreinvex on K, if the inequality (2.1) is strict for all x,y\in K, x\ne y and \lambda \in (0,1).
Definition 2.5 ([16])
Let K\subseteq {R}^{n} be an invex set with respect to η. The function f is said to be semistrictly Gpreinvex on K with respect to η if there exists a continuous realvalued increasing function G:{I}_{f}(K)\to R such that for all x,y\in K, f(x)\ne f(y) and \lambda \in (0,1),
Remark 2.3 It is clear that the semistrictly Gpreinvex function is a generalization of semistrictly preinvex function.
Example 2.1 This example illustrates that a semistrictly Gpreinvex function is not necessarily a (strictly) Gpreinvex function with respect to the same η. Let
Then, we can verify that f is a semistrictly Gpreinvex function with respect to η, where G(t)={e}^{t}. However, by letting x=2, y=\frac{3}{2} (x\ne y), \lambda =\frac{1}{2}, we have
Thus, f is not a Gpreinvex function with respect to the same η, and it is also not a strictly Gpreinvex function with respect to the same η.
Example 2.2 This example illustrates that a semistrictly Gpreinvex function is not necessarily a Ginvex function. Let
Then, by [16], Example 2], f is a semistrictly Gpreinvex function with respect to η, where G(t)={e}^{t}. It can be easily noticed that f is not differentiable at x=0. Thus, f is not a Ginvex function (see [13]) with respect to η.
Example 2.3 Seeing the function f and η in Example 2.1, it is obvious that f is a semistrictly Gpreinvex function with respect to η, where G(t)={e}^{t}. However, by letting x=4, y=3, \lambda =\frac{1}{2}, we have
Thus, f is not a preinvex function with respect to the same η.
Remark 2.4 From Examples 2.12.3, we know that semistrictly Gpreinvex functions are different from Gpreinvex functions, strictly Gpreinvex functions, Ginvex functions and preinvex functions with respect to the same η.
In order to discuss the properties of semistrictly Gpreinvex functions, we recall the definition of Condition C as follows.
The vectorvalued function \eta :K\times K\to {R}^{n} is said to satisfy Condition C if for any x,y\in K and \lambda \in [0,1],
In the sequel, we will use the following lemma.
Lemma 2.1 ([14])
{G}^{1} is (strictly) increasing if and only if G is (strictly) increasing.
The next lemma can be easily proved by Lemma 2.1 and the definitions of a concave function and a convex function, so we omit it.
Lemma 2.2 If G is increasing and concave, then {G}^{1} is convex.
3 Some properties of semistrictly Gpreinvex functions
In this section, we derive some interesting properties of semistrictly Gpreinvex functions.
Theorem 3.1 Let K be a nonempty invex set with respect to η, where η satisfies Condition C, and let {f}_{i}:K\to R (i\in I) be a finite or infinite collection of both semistrictly Gpreinvex and Gpreinvex functions for the same η on K. Define f(x)=sup\{{f}_{i}(x),i\in I\}, for every x\in K. Assume that for every x\in K, there exists an {i}_{0}:=i(x)\in I such that f(x)={f}_{{i}_{0}}(x). Then f is both a semistrictly Gpreinvex and Gpreinvex function with respect to the same η on K.
Proof By Proposition 12 in [14], we know that f is Gpreinvex on K. We need to show that f is a semistrictly Gpreinvex function on K. Assume that f is not a semistrictly Gpreinvex function. Then, there exist x,y\in K with f(x)\ne f(y) and \alpha \in (0,1) such that
By the Gpreinvexity of f, we have
It follows that
Let z=y+\alpha \eta (x,y). By the assumptions, there exist i(z)={i}_{0}, i(x)={i}_{1}, i(y)={i}_{2}, satisfying
This fact together with (3.1) yields

(i)
If {f}_{{i}_{0}}(x)\ne {f}_{{i}_{0}}(y), then by the semistrict Gpreinvexity of {f}_{{i}_{0}},
{f}_{{i}_{0}}(z)<{G}^{1}(\alpha G({f}_{{i}_{0}}(x))+(1\alpha )G({f}_{{i}_{0}}(y))).(3.3)
From {f}_{{i}_{0}}(x)\le {f}_{{i}_{1}}(x), {f}_{{i}_{0}}(y)\le {f}_{{i}_{2}}(y) and (3.3), we obtain
which contradicts (3.2).

(ii)
If {f}_{{i}_{0}}(x)={f}_{{i}_{0}}(y), then by the Gpreinvexity of {f}_{{i}_{0}},
{f}_{{i}_{0}}(z)\le {G}^{1}(\alpha G({f}_{{i}_{0}}(x))+(1\alpha )G({f}_{{i}_{0}}(y))).(3.4)
Since f(x)\ne f(y), at least one of the inequalities {f}_{{i}_{0}}(x)\le {f}_{{i}_{1}}(x)=f(x) and {f}_{{i}_{0}}(y)\le {f}_{{i}_{2}}(y)=f(y) has to be a strict inequality. From (3.4) and the continuity and increasing property of G, we obtain
which contradicts (3.2). This completes the proof. □
Remark 3.1 Theorem 3.1 generalizes Theorem 3.8 [9] from a semistrictly preinvex case to a semistrictly Gpreinvex case.
Next, we will establish an important gradient property of semistrictly Gpreinvex functions. Before showing the property in Theorem 3.3, we first derive a result of Gpreinvex functions.
Theorem 3.2 Let K be a nonempty invex set in {R}^{n} with respect to \eta :K\times K\to {R}^{n}, and f be a Gpreinvex function with respect to the same η on K. Assume that η satisfies Condition C. For any x,y\in K and \lambda \in [0,1], let g(\lambda )=G(f(x+\lambda \eta (y,x))). Then
or equivalently,
Proof For 0<\alpha <\beta \le 1, let {z}_{\alpha}=x+\alpha \eta (y,x), {z}_{\beta}=x+\beta \eta (y,x), u=1\frac{\alpha}{\beta}. By Condition C,
We have
Therefore, we obtain
This completes the proof. □
From Definitions 2.4 and 2.5, we can obtain the following lemma.
Lemma 3.1 Let K be a nonempty invex set with respect to η, where η satisfies Condition C. Let f:K\to R be continuous and semistrictly Gpreinvex with respect to η on K, and satisfy f(y+\eta (x,y))\le f(x) (\mathrm{\forall}x\in K). Then f is a Gpreinvex function on K.
Theorem 3.3 Let K be a nonempty invex set with respect to η, where η satisfies Condition C. Assume that f:K\to R is differentiable and semistrictly Gpreinvex with respect to η on K, and satisfies f(y+\eta (x,y))\le f(x) (\mathrm{\forall}x\in K), where G is a differentiable function. Then for any x,y\in K with f(x)\ne f(y), we have

(i)
G(f(y))>G(f(x))+{G}^{\mathrm{\prime}}(f(x))\eta {(y,x)}^{T}\mathrm{\nabla}f(x),

(ii)
{G}^{\mathrm{\prime}}(f(x))\eta {(y,x)}^{T}\mathrm{\nabla}f(x)+{G}^{\mathrm{\prime}}(f(y))\eta {(x,y)}^{T}\mathrm{\nabla}f(y)<0.
Proof (i) Suppose that f is a semistrictly Gpreinvex function on K. By Definition 2.4, for any x,y\in K with f(x)\ne f(y), we have
which implies
It follows that
From Lemma 3.1 and Theorem 3.2, we get
that is

(ii)
Since f is a semistrictly Gpreinvex function on K, for x,y\in K with f(x)\ne f(y), it follows form the above results that
G(f(y))>G(f(x))+{G}^{\mathrm{\prime}}(f(x))\eta {(y,x)}^{T}\mathrm{\nabla}f(x).(3.5)
From (3.5) and (3.6), we can obtain
This completes the proof. □
Remark 3.2 As a matter of fact, the assumption of continuity for f can be extended to lower semicontinuity in Lemma 3.1.
4 Semistrict Gpreinvexity and optimality
In the section, we consider a class of multiobjective optimization problems and obtain an important optimality result under semistrict Gpreinvexity.
From now on, we suppose that f=({f}_{1},\dots ,{f}_{n}):X\to {R}^{n} is a vectorvalued mapping, where X is an invex subset of {R}^{n} endowed with the Euclidean norm \parallel \cdot \parallel.
We consider the following multiobjective optimization problem:
In the sequel, we use the following notations. For x,y\in X

(i)
f(x)<f(y)\u27fa{f}_{i}(x)<{f}_{i}(y) for every i=1,2,\dots ,n;

(ii)
f(x)\nless f(y) is the negation of f(x)<f(y).
Definition 4.1 Let m\ge 1 be an integer. A point \overline{x}\in X is said to be a strictly local minimizer of order m for (MOP) if there exist an \epsilon >0 and a vector c\in int{R}_{+}^{n} such that
where N(\overline{x},\epsilon ) is an ε neighbor of \overline{x}.
Now, we give the notion of a strict minimizer in the global sense if the neighbor N(\overline{x},\epsilon ) is replaced by the whole space {R}^{n}.
Definition 4.2 Let m\ge 1 be an integer. A point \overline{x}\in X is said to be a strictly global minimizer of order m for (MOP) if there exists a vector c\in int{R}_{+}^{n} such that
Remark 4.1 If \eta (x,y)=xy, then Definitions 4.14.2 reduce to Definitions 4.24.3 introduced by Bhatia [17], respectively.
Remark 4.2 From Definitions 4.1 and 4.2, we know that the concepts of a strictly local minimizer of order m and a strictly global minimizer of order m for (MOP) are stronger than the concepts of a strictly local minimizer and a strictly global minimizer for (MOP), respectively.
It is clear that any strictly global minimizer of order m is a strictly global minimizer. But the converse may not be true. We can see the case in the following example.
Example 4.1 Let f:R\to {R}^{3} be defined as
Consider the multiobjective optimization problem,
\overline{x}=0 is a strictly global minimizer but is not a strictly global minimizer of order m, because for any m>0, c=({c}_{1},{c}_{2},{c}_{3})\in int{R}_{+}^{3} and x>0 sufficiently small, we have f(x)<c{x}^{m}, where \eta (x,y)=xy or \eta (x,y)=x+y.
Theorem 4.1 Let X be an invex set with respect to η. Suppose the following conditions are satisfied:

(i)
Let G be increasing and concave on {I}_{f}(X);

(ii)
\eta :X\times X\to {R}^{n} satisfies Condition C;

(iii)
\overline{x}\in X is a strictly local minimizer of order m for (MOP).
If {f}_{i}:X\to R, i=1,2,\dots ,n, are semistrictly Gpreinvex on X with respect to η, then \overline{x} is a strictly global minimizer of order m for (MOP).
Proof Let \overline{x}\in X be a strictly local minimizer of order m for (MOP). Then there exist an ε neighborhood N(\overline{x},\epsilon ) of \overline{x} and a vector c\in int{R}_{+}^{n} such that
Hence, there exists no x\in N(\overline{x},\epsilon )\cap X such that
where c=({c}_{1},{c}_{2},\dots ,{c}_{n}).
Suppose by contradiction that \overline{x} is not a strictly global minimizer of order m for (VP), then there exists {x}^{\ast}\in X with f({x}^{\ast})\ne f(\overline{x}) such that
for any {c}^{\ast}=({c}_{1}^{\ast},{c}_{2}^{\ast},\dots ,{c}_{n}^{\ast})\in int{R}_{+}^{n}. Since X is an invex set with respect to η,
Because {f}_{i}:X\to R, i=1,2,\dots ,n, are semistrictly Gpreinvex on X with respect to η, G is increasing on {I}_{f} and {G}^{1} is convex, it follows that for any \lambda \in (0,1)
where Lemma 2.2 is used in the second inequality.
According to (4.2), (4.3) and Condition C, we have
where {d}_{i}={\lambda}^{1m}{c}_{i}^{\ast}. For a sufficiently small \lambda >0, we obtain
Let d=({d}_{1},\dots ,{d}_{n}). Since {c}^{\ast}=({c}_{1}^{\ast},{c}_{2}^{\ast},\dots ,{c}_{n}^{\ast})\in int{R}_{+}^{n} is arbitrary, d=({d}_{1},\dots ,{d}_{n})\in int{R}_{+}^{n} is also arbitrary. Therefore,
or
which implies that \overline{x} is not a strictly local minimizer of order m. It is a contradiction. Hence, \overline{x} is a strict minimizer of order m for (MOP). □
Now, we give an example of an optimization problem to illustrate Theorem 4.1.
Example 4.2 Let f:R\to {R}^{3} be defined as
where
From Definition 2.5, we can verify that {f}_{i} (i=1,2,3) are semistrictly Gpreinvex functions with respect to η, where G(t)={e}^{t}. \overline{x}=1 is a strictly local minimizer of order m for (MOP). From Theorem 4.1, we can get \overline{x}=1 is also a strictly global minimizer of order m for (MOP), and f(\overline{x})=(0,0,0) is a global minimal value of (MOP).
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Acknowledgements
The author is very grateful to the three anonymous referees for valuable comments and suggestions which helped to improve the paper. This work was supported by the Natural Science Foundation of China (No. 11271389, 11201509, 71271226), the Natural Science Foundation Project of Chongqing (No. CSTC, 2011AC6104.2012jjA00016) and the Education Committee Project Research Foundation of Chongqing (No. KJ100711).
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Peng, Z.Y. Semistrict Gpreinvexity and its application. J Inequal Appl 2012, 198 (2012). https://doi.org/10.1186/1029242X2012198
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DOI: https://doi.org/10.1186/1029242X2012198
Keywords
 Gpreinvex function
 semistrictly Gpreinvex function
 optimality
 multiobjective optimization