A class of generalized invex functions and vector variational-like inequalities
- Ru Li^{1} and
- Guolin Yu^{1}Email author
https://doi.org/10.1186/s13660-017-1345-8
© The Author(s) 2017
Received: 21 January 2017
Accepted: 22 March 2017
Published: 7 April 2017
Abstract
In this paper, a class of generalized invex functions, called \((\alpha,\rho,\eta)\)-invex functions, is introduced, and some examples are presented to illustrate their existence. Then we consider the relationships of solutions between two types of vector variational-like inequalities and multi-objective programming problem. Finally, the existence results for the discussed variational-like inequalities are proposed by using the KKM-Fan theorem.
Keywords
vector variational-like inequality multi-objective programming invex function KKM-Fan theoremMSC
90C29 90C46 26B251 Introduction
As we all know, convexity and generalized convexity of functions play an important role in the field of optimization theory and its application. Among many generalized convexities given in the literature, a meaningful notion is called invex function, which was firstly introduced by Hanson [1]. Then Jeyakumar and Mond [2] popularized it and put forward the concept of V-invex function, and they studied the optimality conditions for nonlinear programming problem. Ivanov [3] provided the concept of second-order invex functions and dealt with the optimal solution of nonlinear programming problem. Yu and Liu [4] investigated a class of generalized invex type I functions and presented the optimality conditions of multi-objective programming. One aim of this paper is to present a new kind of generalized invex functions, termed \((\alpha,\rho,\eta)\)-invex functions, which is weaker than the above mentioned generalized invexities.
It is well known that vector variational-like inequalities are of importance in the field of applied mathematics. There are several interesting and important topics, for instance, existence results of vector variational inequalities, which ensure the existence of efficient solutions of vector optimization problems and establish the relations between both problems. In order to do so, a great quantity of researchers have been attracted towards this direction, see [5–19]. In this note, we shall investigate the relations between vector variational-like inequalities and multi-objective programming problems under the hypothesis of \((\alpha,\rho,\eta)\)-invexity. We also define a kind of generalized monotonicity functions, called \((\alpha,\rho,\eta)\)-monotone functions. Then, by using the KKM-Fan theorem, we present the existence theorems for vector variational-like inequalities under the assumption of \((\alpha,\rho,\eta)\)-monotonicity.
The content of the present work can be organized into four sections of which this introduction is the first. In Section 2, some notations and the concepts are recalled. Besides, a new class of generalized invex functions, called \((\alpha,\rho,\eta)\)-invex functions, is introduced, and examples are provided in the support of this generalization. In Section 3, the relationships between Stampacchia and Minty invex vector variational-like inequalities and \((\alpha,\rho,\eta)\)-invex multi-objective programming problem are discussed. Section 4 gives the existence theorems of solution of Stampacchia and Minty vector variational-like inequality under the hypothesis of \((\alpha,\rho,\eta)\)-monotonicity.
2 Notations and preliminaries
We begin with recalling some known definitions, which will be applicable in the sequel of the paper. Suppose that \(X\subseteq\mathbb{R}^{n}\) is a nonempty open subset and \(\varphi:X\rightarrow\mathbb{R}\) is a real-valued function. In the rest of paper, we always assume that \(\alpha,\rho: X\times X\rightarrow \mathbb{R}\) are real-valued functions and \(\eta: X\times X\rightarrow \mathbb{R}^{n}\) is a vector-valued function.
Definition 2.1
see [9]
Definition 2.2
see [9]
Definition 2.3
see [9]
Motivated by the above definition of η-invex function, we make two extensions of invexity to \((\alpha,\rho,\eta)\)-invexity and pseudo-\((\alpha, \rho, \eta)\)-invexity as follows.
Definition 2.4
Remark 2.1
Setting \(\alpha=1\) and \(\rho=0\) in Definition 2.4, we arrive at the notion of η-invexity, defined by Farajzadeh and Lee (see [9]).
Definition 2.5
The following two examples illustrate the existences of \((\alpha,\rho,\eta)\)-invex functions and \((\alpha, \rho, \eta )\)-pseudoinvex functions, respectively.
Example 2.1
Example 2.2
In view of the concepts of monotonicities given by Al-Homidan et al. [20], we introduce \((\alpha, \rho, \eta )\)-monotonicity for a real-valued function, which will be helpful in proving our results.
Definition 2.6
From now on, unless otherwise specified, we always assume that \(X\subseteq\mathbb{R}^{n}\) is an η-invex set, \(f(x)= (f_{1}(x),f_{2}(x),\ldots,f_{p}(x) )\) , \(f_{i}:X \rightarrow \mathbb{R}\) and \(i\in P=\{1,2,\ldots,p\}\) .
In multi-objective programming problems, objectives often conflict with each other. In this regard, the concepts of efficient and weak efficient solutions are widely used.
Definition 2.7
Definition 2.8
Next, we are about to introduce the following Stampacchia and Minty vector variational-like inequalities, respectively, with also their weak formulations, which will be used to ensure the efficient solutions of the problem (MP). Let \(\rho_{i}:X\times X \rightarrow \mathbb{R}\), \(i\in P\).
Remark 2.2
Let \(\alpha \in \mathbb{R}_{++}\) and \(\rho_{i}=0\) in above (SVVI) (respectively (WSVVI)), then this problem reduces to the vector variational inequality (respectively weak) introduced by Farajzadeh and Lee [9].
Example 2.3
3 The relationships between vector variational-like inequalities and multi-objective programming problems
In this section, we shall examine the relationships between Stampacchia and Minty vector variational-like inequalities and multi-objective programming problems in terms of \((\alpha,\rho,\eta)\)-invex functions, which are formulated in Section 2.
Theorem 3.1
Let α, \(\rho_{i}: X\times X\rightarrow \mathbb{R}\), \(\eta: X\times X\rightarrow \mathbb{R}^{n}\) and for each \(i\in P\), \(f_{i}\) be \((\alpha,\rho_{i},\eta)\)-invex function at \(\bar{x}\in X\). If x̄ solves (SVVI), then x̄ is an efficient solution of (MP).
Proof
Theorem 3.2
Let α, \(\rho_{i}: X\times X\rightarrow \mathbb{R}\), \(\eta: X\times X\rightarrow \mathbb{R}^{n}\) and for each \(i\in P\), \(f_{i}\) be an \((\alpha,\rho_{i},\eta)\)-pseudoinvex function at \(\bar{x}\in X\). If x̄ solves (WSVVI), then x̄ is a weak efficient solution of (MP).
Proof
Theorem 3.3
Let α, \(\rho_{i}: X\times X\rightarrow \mathbb{R}\), \(\eta: X\times X\rightarrow \mathbb{R}^{n}\) and for each \(i\in P\), \(f_{i}\) be an \((\alpha,\rho_{i},\eta)\)-invex function on X. If \(\bar{x}\in X\) is an efficient solution of (MP), then x̄ solves (MVVI).
Proof
Theorem 3.4
Let α, \(\rho_{i}: X\times X\rightarrow \mathbb{R}\), \(\eta: X\times X\rightarrow \mathbb{R}^{n}\) and for each \(i\in P\), \(f_{i}\) be an \((\alpha,\rho_{i},\eta)\)-invex function on X. If \(\bar{x}\in X\) solves (WSVVI), then x̄ solves (WMVVI).
Proof
Theorem 3.5
Let α, \(\rho_{i}: X\times X\rightarrow \mathbb{R}\), \(\eta: X\times X\rightarrow \mathbb{R}^{n}\) and for each \(i\in P\), \(f_{i}\) be a strictly \((\alpha,\rho_{i},\eta)\)-invex function on X. If \(\bar{x}\in X\) is a weak efficient solution of (MP), then x̄ solves (MVVI).
Proof
Theorem 3.6
Let α, \(\rho_{i}: X\times X\rightarrow \mathbb{R}\), \(\eta: X\times X\rightarrow \mathbb{R}^{n}\) and for each \(i\in P\), \(f_{i}\) be an \((\alpha,\rho_{i},\eta)\)-invex function on X. If \(\bar{x}\in X\) is a weak efficient solution of (MP), then x̄ solves (WMVVI).
Proof
4 The solution existence of vector variational-like inequalities
This section is devoted to a discussion of the existence of solutions for Stampacchia and Minty vector variational-like inequalities under the assumption of \((\alpha,\rho,\eta)\)-monotonicity. Now, we assume that Y is a topological vector space, \(X\subseteq\mathbb{R}^{n}\) is a convex set, and \(f_{i}:X \rightarrow \mathbb{R}\), \(i\in P\).
Definition 4.1
see [21]
Lemma 4.1
see [21] (KKM-Fan theorem)
Theorem 4.1
- (i)
for each \(i\in P\), \(-f_{i}\) is \((\alpha,\rho_{i},\eta)\)-monotone on X;
- (ii)
\(\alpha(x,y) \in \mathbb{R}_{++}\), \(x\neq y\), \(x,y\in X\). \(\rho_{i}(x,y)\in \mathbb{R}_{+}\), \(x,y\in X\);
- (iii)
α and \(\rho_{i}\) are affine functions with respect to their second argument such that \(\alpha(x,x)=0\) and \(\rho_{i}(x,x)=0\), \(\forall x\in X\);
- (iv)the set-valued map \(\Gamma: X\rightarrow2^{X}\) defined byis closed valued;$$\begin{aligned} \Gamma(x) =&\bigl\{ y\in X:\bigl(\alpha(x,y)f_{1}^{\prime}\bigl(y,\eta(x,y) \bigr)+\rho_{1}(x,y),\ldots, \\ & \alpha(x,y)f_{p}^{\prime} \bigl(y,\eta(x,y)\bigr)+\rho_{p}(x,y)\bigr)\nleqslant 0, \forall x\in X\bigr\} , \end{aligned}$$
- (v)
there exist a nonempty compact set \(M\subset X\) and a nonempty compact convex set \(N\subset X\) such that for each \(y\in X\backslash M\), there exists \(x\in N\) such that \(y \notin \Gamma(x)\).
Then (SVVI) is solvable on X.
Proof
We end this paper by presenting the following existence theorem for Minty vector variational-like inequality (MVVI). We omit its proof because it can be proven along similar lines of Theorem 4.1.
Theorem 4.2
- (i)
for each \(i\in P\), \(f_{i}\) is \((\alpha,\rho_{i},\eta)\)-monotone on X;
- (ii)
α and \(\rho_{i}\) are affine functions with respect to their first argument such that \(\alpha(x,x)=0\) and \(\rho_{i}(x,x)=0\), \(\forall x\in X\);
- (iii)the set-valued map \(\Gamma: X\rightarrow2^{X}\) defined byis closed valued;$$\begin{aligned} \Gamma(x) =&\bigl\{ y\in X: \bigl(\alpha(y, x)f_{1}^{\prime}\bigl(x, \eta(y, x)\bigr)+\rho_{1}(y, x), \ldots, \\ &\alpha(y, x)f_{p}^{\prime} \bigl(x, \eta(y, x)\bigr)+\rho_{p}(y, x)\bigr)\nleqslant 0, \forall x\in X \bigr\} , \end{aligned}$$
- (iv)
there exist a nonempty compact set \(M\subset X\) and a nonempty compact convex set \(N\subset X\) such that for each \(y\in X\backslash M\), there exists \(x\in N\) such that \(y \notin \Gamma(x)\).
Then (MVVI) is solvable in X.
5 Conclusion and remarks
In this paper, we have introduced a kind of generalized invex functions, termed \((\alpha,\rho,\eta)\)-invex functions, and Stampacchia and Minty vector variational-like inequalities associated with the introduced extended convexities. Moreover, we have also demonstrated the relationships between the discussed vector variational-like inequalities and nonsmooth multi-objective programming problems involving \((\alpha,\rho,\eta)\)-invex functions. Finally, the existence results for Stampacchia and Minty vector variational-like inequalities are proposed by utilizing the KKM-Fan theorem. The results of this note extend some earlier results of Farajzadeh and Lee [9] to a more general class of functions.
Declarations
Acknowledgements
This research was supported by the Natural Science Foundation of China under Grant No. 11361001; Natural Science Foundation of Ningxia under Grant No. NZ14101.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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