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Vector critical points and generalized quasiefficient solutions in nonsmooth multiobjective programming
Journal of Inequalities and Applications volumeÂ 2017, ArticleÂ number:Â 184 (2017)
Abstract
In this work, several extended approximately invex vectorvalued functions of higher order involving a generalized Jacobian are introduced, and some examples are presented to illustrate their existences. The notions of higherorder (weak) quasiefficiency with respect to a function are proposed for a multiobjective programming. Under the introduced generalization of higherorder approximate invexities assumptions, we prove that the solutions of generalized vector variationallike inequalities in terms of the generalized Jacobian are the generalized quasiefficient solutions of nonsmooth multiobjective programming problems. Moreover, the equivalent conditions are presented, namely, a vector critical point is a weakly quasiefficient solution of higher order with respect to a function.
1 Introduction
Convexity and its generalizations played a critical role in multiobjective programming problems. In many generalizations, approximate convexity and invexity are two significant generalized versions of convexity, which tried to weaken the convexity hypotheses thus to study the relations between vector variationallike inequalities and multiobjective programming problems. Invexity was firstly put forward by Hanson [1]. Then OsunaGÃ³mez et al. [2] introduced the notions of generalized invexity for differentiable functions in a finitedimensional contex. And this generalized invexity has been extended to locally Lipschitz functions using the generalized Jacobian (see [3, 4]). BenIsrael and Mond [5] presented pseudoinvex functions which generalized pseudoconvex functions in the same manner as invex functions generalized convex functions. Mishra et al. [6] and Ngai et al. [7] introduced the concept of approximately convex functions. Inspired and motivated by this ongoing research work, we present the concept of approximately invex function of higher order.
The notion of an efficient solution in multiobjective programming is widely used. Considering the complexity of optimization problems, several variants of the efficient solutions have been introduced (see [8â€“11]). Recently, researchers have shown great interests in quasiefficiency of multiobjective programming (see [12, 13]). In this work, we give the notion of a quasiefficient solution of higher order for a class of nonsmooth multiobjective programming problems (NMPs) with respect to a function.
The vector variational inequality was initially introduced by Giannessi [14]. Since then vector variational inequalities, which were used as an efficient tool to study multiobjective programming, have attracted much attention and have been extended to generalized vector variationallike inequalities (GVVI). Recently, a great quantity of work focused on the study of relations between (GVVI) and multiobjective programming under different convexity assumptions (see [15â€“17]). Motivated by the previous contributions, in this note, our purpose is to obtain the relations between (GVVI) and (NMP) under approximate invexity of higher order.
The rest of this work is organized as follows. In SectionÂ 2, we recall some basic definitions and preliminary results. Besides, the notions of approximately invex function of higher order with respect to vectorvalued functions and (weakly) quasiefficient solution of higher order for (NMP) with respect to a vectorvalued function are introduced, and examples are provided to illustrate their existence. In SectionÂ 3, the relations between (GVVI) and (NMP) are established under the approximate invexity of higherorder assumptions. In SectionÂ 4, we study the relations between vector critical points and weakly quasiefficient solutions of higher order for (NMP) with respect to a vectorvalued function.
2 Notations and preliminaries
Throughout the current paper, unless otherwise stated, \(\mathbb{R}\), \(\mathbb{R}^{n}\), \(\mathbb{R}^{n}_{+}\) stand for the set of all real numbers, the ndimensional Euclidean space and the nonnegative orthant of \(\mathbb{R}^{n}\), respectively. For any \(x, y\in\mathbb {R}^{n}\), the inner product of x and y is denoted \(x^{T}y\), where the superscript T represents the transpose of a vector. Let \(X\subseteq \mathbb{R}^{n}\) be a nonempty subset and \(m\geq1\) be a positive integer, the symbols \(\operatorname{co}(X)\) and \(\operatorname{int}(X)\) represent the convex hull of X and the interior of X, respectively. We employ the following conventions for vectors in \(\mathbb{R}^{n}\):
For the sake of convenience, we firstly recall some notations that will be used in the sequel. We always suppose that \(f: X\rightarrow \mathbb{R}^{p}\) , \(\eta: X\times X\rightarrow\mathbb{R}^{n}\) and \(\psi: X\times X\rightarrow\mathbb{R}^{n}\) are vectorvalued functions in the rest of this paper.
Definition 2.1
see [18]
The function \(f: X\rightarrow\mathbb{R}^{p}\) is said to be locally Lipschitz on X, if for every \(x\in X\) there exist a neighborhood \(U_{x}\subseteq X\) of x and a constant \(L>0\) such that, for all y, \(z\in U_{x}\),
Rademacherâ€™s theorem (see CorollaryÂ 4.12 in [19]) indicates that a function f satisfying the Lipschitz condition (2.1) is FrÃ©chet differentiable. Based on this fact, Clarke [18] presented the following concept of the generalized Jacobian of f at some point.
Definition 2.2
see [18]
Let \(x_{0}\in X\) and \(Jf(x)\) represent the usual Jacobian matrix of f at x whenever f is FrÃ©chet differentiable at x. The generalized Jacobian of f at \(x_{0}\) is
Let a scalar function \(\varphi: X\rightarrow\mathbb{R}\) be locally Lipschitz at \(x_{0}\), then the upper Clarke directional derivative of Ï† at \(x_{0}\) in the direction \(v\in\mathbb{R}^{n}\) is given by
and the Clarke subdifferential of Ï† at \(x_{0}\), denoted by \(\partial\varphi(x_{0})\), is defined as follows:
For a vectorvalued function \(f=(f_{1},\ldots,f_{p})^{T}: X\rightarrow\mathbb {R}^{p}\), its Clarke subdifferential is the cartesian product of Clarke subdifferentials of the components \(f_{i}: X\rightarrow\mathbb{R}\), \(i=1,2,\ldots,p\) of f, that is, \(\partial f(x)=\partial f_{1}(x)\times \cdots\times\partial_{p}f(x)\). It has been shown in [18] that, for a scalar function \(\varphi: X\rightarrow\mathbb{R}\), \(\partial _{J}\varphi(x)=\partial\varphi(x)\), but for the vectorvalued function f, \(\partial_{J} f(x)\) is contained and is different from \(\partial f(x)\).
Definition 2.3
see [20]
The subset \(\emptyset\neq X\subseteq \mathbb{R}^{n}\) is said to be invex with respect to \(\eta: X\times X\rightarrow\mathbb{R}^{n}\), if for every \(x, y\in X\), \(\lambda\in[0, 1]\), we have
From now on, we always assume that the subset \(X\subseteq\mathbb {R}^{n}\) is a nonempty invex set with respect to some Î· unless otherwise specified.
The generalized invexity of differentiable functions in a finitedimensional space (see [2]) has been extended to locally Lipschitz functions using the generalized Jacobian as follows (see [3, 4]).
Definition 2.4
see [4]
Let \(x_{0}\in X\) and \(\eta: X\times X\rightarrow\mathbb{R}^{n}\). The function \(f: X\rightarrow\mathbb{R}^{p}\) is said to be invex at \(x_{0}\) with respect to Î·, if for all \(x\in X\),
f is said to be invex with respect to Î· on X, if for every \(x\in X\), f is invex at x with respect to Î·.
Definition 2.5
see [4]
Let \(x_{0}\in X\) and \(\eta: X\times X\rightarrow\mathbb{R}^{n}\). The function \(f: X\rightarrow\mathbb{R}^{p}\) is said to be pseudoinvex at \(x_{0}\) with respect to Î·, if for all \(x\in X\),
or equivalently,
f is said to be pseudoinvex with respect to Î· on X, if for every \(x\in X\), f is pseudoinvex at x with respect to Î·.
In the generalized convexity of functions, the study of approximately convex functions (see [6, 7, 12, 21]) is a hot spot. Mishra and Upadhyay [21] introduced the following concept of vectorvalued approximately convex functions.
The function \(f:X\rightarrow\mathbb{R}^{p}\) is said to be approximately convex at \(x_{0}\in X\), if for all \(\alpha\in \operatorname{int}(\mathbb{R}^{p}_{+})\) such that
Motivated by above definitions, we give the notions of approximate invexity of order m with respect to Î· and Ïˆ, strictly approximate invexity of order m with respect to Î· and Ïˆ and approximate pseudoinvexity of type I of order m with respect to Î· and Ïˆ as follows.
Definition 2.6
Let \(x_{0}\in X\) and \(m\geq1\) be a positive integer. The function \(f: X\rightarrow\mathbb{R}^{p}\) is said to be approximately invex of order m at \(x_{0}\) with respect to Î· and Ïˆ, if there exist \(\eta: X\times X\rightarrow\mathbb{R}^{n}\), \(\psi: X\times X\rightarrow\mathbb {R}^{n}\) and \(\alpha\in \operatorname{int}(\mathbb{R}^{p}_{+})\) such that, for all \(x\in X \),
f is said to be approximately invex of order m with respect to Î· and Ïˆ on X, if for every \(x\in X\), f is approximately invex of order m at x with respect to Î· and Ïˆ.
Remark 2.1
Replacing \(A\in\partial_{J} f(x_{0})\) by \(\xi\in\partial f(x_{0})\), setting \(\eta(x,x_{0})=xx_{0}\), \(\psi(x,x_{0})=xx_{0}\) and \(m=1\) in DefinitionÂ 2.6, then we arrive at the notion of approximately convex function, defined by Mishra and Upadhyay [21].
Remark 2.2
A function which is invex at \(x_{0}\) with respect to Î· is also approximately invex of order m at \(x_{0}\) with respect to Î· and Ïˆ, but in the contrary case, it does not hold. The following example is given to illustrate this fact.
Example 2.1
Consider the vectorvalued function \(f:\mathbb{R}\rightarrow\mathbb {R}^{2}\), defined by \(f(x)=(x,\varphi(x))^{T}\), where
It can easily be seen that
We firstly prove that f is not invex with respect to \(\eta:\mathbb {R}\times\mathbb{R}\rightarrow\mathbb{R}\) on \(\mathbb{R}\). Indeed, choose \(\bar{x}=0\) and \(x_{0}=1\), then there exists no \(u=\eta(\bar {x},x_{0})\in\mathbb{R}\) such that \(f(\bar{x})f(x_{0})\geqq Au\) for every \(A=(1, a)^{T}\in\partial_{J} f(x_{0})\). In other words, if \((1,0)^{T}\geqq (u,au)^{T}, \forall a\in[1,0]\), this implies \(1\geqq u\) and \(0\geqq au\). For \(a=1\), from above inequality \(0\geqq au\) we obtain \(u\geqq 0\), which contradicts \(1\geqq u\). Finally, we show that f is approximately invex of order m at \(x_{0}=1\) with respect to \(\eta:\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}\) and \(\psi:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}\). Take \(\eta(x,x_{0})=1\), \(\psi(x,x_{0})=xx_{0}1=x2\), \(m=1\) and \(\alpha =(\alpha_{1},\alpha_{2})^{T}=(1,2)^{T}\). From the above we have \(A=(1, a)^{T}\in\partial_{J} f(1)\), \(1\leq a\leq0\). then we can see the following inequality holds true invariably:
Definition 2.7
Let \(x_{0}\in X\) and \(m\geq1\) be a positive integer. The function \(f: X\rightarrow\mathbb{R}^{p}\) is said to be strictly approximately invex of order m at \(x_{0}\) with respect to Î· and Ïˆ, if there exist \(\eta: X\times X\rightarrow\mathbb{R}^{n}\), \(\psi: X\times X\rightarrow \mathbb{R}^{n}\) and \(\alpha\in \operatorname{int}(\mathbb{R}^{p}_{+})\) such that, for all \(x\in X \),
f is said to be strictly approximately invex of order m with respect to Î· and Ïˆ on X, if for every \(x\in X\), f is strictly approximately invex of order m at x with respect to Î· and Ïˆ.
Definition 2.8
Let \(x_{0}\in X\) and \(m\geq1\) be a positive integer. The function \(f: X\rightarrow\mathbb{R}^{p}\) is said to be approximately pseudoinvex type I of order m at \(x_{0}\) with respect to Î· and Ïˆ, if there exist \(\eta: X\times X\rightarrow\mathbb{R}^{n}\), \(\psi: X\times X\rightarrow\mathbb{R}^{n}\) and \(\alpha\in \operatorname{int}(\mathbb{R}^{p}_{+})\) such that, for all \(x\in X \),
or, equivalently,
f is said to be approximately pseudoinvex type I of order m with respect to Î· and Ïˆ on X, if for every \(x\in X\), f is approximately pseudoinvex type I of order m at x with respect to Î· and Ïˆ.
The following example illustrates the existence of approximate invexity of order m with respect to Î· and Ïˆ and of an approximately pseudoinvex type I function of order m with respect to Î· and Ïˆ.
Example 2.2
Let \(X=\mathbb{R}\), \(\alpha=(\alpha_{1},\alpha_{2})^{T}>0\) and \(m\geq1\) be a positive integer. Consider the following functions: \(f: X\rightarrow\mathbb{R}^{2}\), \(f': X\rightarrow\mathbb{R}^{2}\), \(\eta: X\times X\rightarrow\mathbb{R}\) and \(\psi: X\times X\rightarrow \mathbb{R}^{2}\) defined by
For any positive integer \(m\geq1\), it is easy to verify that f is approximately invex of order m at \(x_{0}=1\) with respect to Î· and Ïˆ. In fact, we can easily obtain \(A=(1, 1)^{T}\in\partial _{J}f(x_{0})\). By direct calculation, we derive
Obviously,
Furthermore, because of \(\alpha=(\alpha_{1},\alpha_{2})^{T}>0\), we can arrive at
That is,
So, we have verified that f is approximatelyly invex of order m at \(x_{0}=1\) with respect to Î· and Ïˆ.
Now, let us prove that \(f'(x)=(2x, \max\{x,0,x1\})^{T}\) is approximately pseudoinvex type I of order m at \(x_{0}=0\) with respect to \(\eta(x, x_{0})=xx_{0}\) and \(\psi(x,x_{0})=(\frac{x^{2}}{\sqrt {1+x_{0}^{2}}},0)^{T}\). Actually, it is not difficult to get \(A=(2, a)^{T}\in\partial_{J}f'(x_{0})=\{(2, a)^{T}:1\leq a\leq0\}\). We suppose that
then we arrive at
This fulfills the condition of an approximately pseudoinvex type I of order m function at \(x_{0}=0\) with respect to Î· and Ïˆ.
Remark 2.3
It is obvious that if \(f:X \rightarrow\mathbb{R}^{p}\) is pseudoinvex at \(x_{0}\in X\) with respect to Î·, then it is also approximately pseudoinvex type I of order m at \(x_{0}\) with respect to Î· and Ïˆ. But the converse does not hold. For example, consider \(\varphi :X\rightarrow\mathbb{R}\), given by
Taking \(\eta(x,x_{0})=xx_{0}\), \(\psi(x,x_{0})=xx_{0}\) and \(m=1\), then Ï† is approximately pseudoinvex type I of order m at \(x_{0}=0\) with respect to Î· and Ïˆ. As for any \(\alpha>0\), there exists \(\delta=\min(\pi,\alpha)>0\) such that \(A\eta(x,x_{0})\geq0\) for some \(A\in \partial_{J}\varphi(x_{0})\) implies
However, Ï† is not pseudoinvex at \(x_{0}\) with respect to Î·. Indeed, for every \(\delta>0\), there exists \(x\in B(x_{0},\delta)\cap X\) such that \(A\eta(x,x_{0})\geq0\), \(A\in\partial_{J}\varphi(x_{0})\) does not imply \(\varphi(x)\geq\varphi(x_{0})\) (see RemarkÂ 3 in [22]).
We consider the following nonsmooth multiobjective programming problem (NMP):
where \(f_{i}:X \rightarrow\mathbb{R}\), \(i\in P=\{1,2,\ldots,p\}\) are nondifferentiable functions.
In multiobjective programming problems, efficient and weakly efficient solutions are widely used. Considering the complexity of the optimization problem in reality and in order to find the optimal solution of multiobjective optimization problem in a smaller range, the notion of quasiefficient and weakly quasiefficient are introduced as follows (see [12, 21, 22]).
Definition 2.9
A point \(x_{0}\in X\) is said to be an efficient solution to the (NMP), if there exists no \(x\in X\) such that
Definition 2.10
A point \(x_{0}\in X\) is said to be a weakly efficient solution to the (NMP), if there exists no \(x\in X\) such that
Definition 2.11
Let \(x_{0}\in X\).

(i)
A point \(x_{0}\) is said to be a quasiefficient solution to the (NMP), if there exists \(\alpha\in \operatorname{int}(\mathbb{R}^{p}_{+})\) such that, for any \(x\in X\), the following cannot hold:
$$ f(x)\leqslant f(x_{0})\alpha\xx_{0}\. $$ 
(ii)
A point \(x_{0}\) is said to be a weakly quasiefficient solution to the (NMP), if there exists \(\alpha\in \operatorname{int}(\mathbb{R}^{p}_{+})\) such that, for any \(x\in X\), the following cannot hold:
$$ f(x)< f(x_{0})\alpha\xx_{0}\. $$
Now, we present the concepts of (weakly) quasiefficient solution of order m with respect to a function Ïˆ for the problem (NMP).
Definition 2.12
Let \(m\geq1\) be a positive integer. A point \(x_{0}\in X\) is called a quasiefficient solution of order m for (NMP) with respect to Ïˆ, if there exist a function \(\psi: X\times X\rightarrow\mathbb{R}^{n}\) and \(\alpha\in \operatorname{int}(\mathbb{R}^{p}_{+})\) such that, for any \(x\in X\), the following cannot hold:
Definition 2.13
Let \(m\geq1\) be a positive integer. A point \(x_{0}\in X\) is called a weakly quasiefficient solution of order m for (NMP) with respect to Ïˆ, if there exist a function \(\psi: X\times X\rightarrow\mathbb {R}^{n}\) and \(\alpha\in \operatorname{int}(\mathbb{R}^{p}_{+})\) such that, for any \(x\in X\), the following cannot hold:
Remark 2.4
It is clear that efficient solution implies quasiefficient solution of order m with respect to Ïˆ to the (NMP), but the converse may not be true. To illustrate this fact, we consider the following multiobjective programming problem:
where \(f: \mathbb{R}_{+}\rightarrow\mathbb{R}^{2}\). Then \(x_{0}=0\) is a quasiefficient solution of order m for (NMP) with respect to \(\psi (x,x_{0})=(x,x_{0})^{T}\) for \(\alpha=(1,1)^{T}\) and \(m=2\), but not an efficient solution.
Remark 2.5
A quasiefficient solution of order m for (NMP) with respect to Ïˆ is not to be a quasiefficient solution in the sense of DefinitionÂ 2.11. For example, let \(X=\{x\in\mathbb{R}:0\leq x\leq1\}\) and \(f:X\rightarrow\mathbb{R}^{2}\) be defined as \(f(x)=(x^{4},\sin^{4} x)^{T}\), then \(x_{0}=0\) is not a quasiefficient solution in the sense of DefinitionÂ 2.11, because, for any \(\alpha=(\alpha_{1},\alpha_{2})^{T}\in \operatorname{int}(\mathbb{R}^{2}_{+})\), there exists an x satisfying \(x\geq\alpha _{1}^{\frac{1}{3}}\), \(\frac{\sin^{4}x}{x}\geq\alpha_{2}\) such that \(f(x)\leqslant f(x_{0})\alpha\xx_{0}\\); however, \(x_{0}=0\) is a quasiefficient solution of order \(m=4\) for (NMP) with respect to \(\psi (x,x_{0})=(x\sin^{4}x_{0},x_{0})^{T}\) for \(\alpha=(1,1)^{T}\).
Associated with the problem (NMP), we consider the following generalized (weakly) vector variationallike inequalities problems:
 (GVVI):

Find a point \(x_{0}\in X\) such that there exists no \(x\in X\) such that
$$ A\eta(x,x_{0})\leqslant0, \quad \forall A\in\partial_{J} f(x_{0}). $$  (GWVVI):

Find a point \(x_{0}\in X\) such that there exists no \(x\in X\) such that
$$ A\eta(x,x_{0})< 0, \quad \forall A\in\partial_{J} f(x_{0}). $$
3 Relations between (GVVI), (GWVVI) and (NMP)
In this section, by using the tools of nonsmooth analysis, we shall disclose that the solutions of generalized vector variationallike inequalities (GVVI) or (GWVVI) are the generalized quasiefficient solutions under the extended invexity (defined in SectionÂ 2).
Theorem 3.1
Let \(f: X\rightarrow\mathbb{R}^{p}\) be approximately invex of order m at \(x_{0}\in X\) with respect to Î· and Ïˆ. If \(x_{0}\) solves (GVVI), then \(x_{0}\) is a quasiefficient solution of order m for (NMP) with respect to the same Ïˆ.
Proof
Suppose that \(x_{0}\) is not a quasiefficient solution of order m for (NMP) with respect to Ïˆ, then there exist \(\hat{x}\in X\) and \(\alpha\in \operatorname{int}(\mathbb{R}^{p}_{+})\) such that
Since f is approximately invex of order m at \(x_{0}\) with respect to Î· and Ïˆ on X, it follows from DefinitionÂ 2.6 that
From inequalities (3.1) and (3.2), we see that there exists \(\hat{x}\in X\) such that
which is inconsistent with the fact that \(x_{0}\) solves (GVVI).â€ƒâ–¡
Theorem 3.2
Let \(f: X \rightarrow\mathbb{R}^{p}\) be approximately invex of order m at \(x_{0}\in X\) with respect to Î· and Ïˆ. If \(x_{0}\) solves (GWVVI), then \(x_{0}\) is a weakly quasiefficient solution of order m for (NMP) with respect to the same Ïˆ.
Proof
Assume that \(x_{0}\) is not a weakly quasiefficient solution of order m for (NMP) with respect to Ïˆ, then there exist \(\alpha\in \operatorname{int}(\mathbb{R}^{p}_{+})\) and \(\hat{x}\in X\) such that
Because f is approximately invex of order m at \(x_{0}\) with respect to Î· and Ïˆ on X, therefore, in particular for \(\alpha\in \operatorname{int}(\mathbb{R}^{p}_{+})\) and xÌ‚, we have
Furthermore, we arrive at
which contradicts the hypothesis that \(x_{0}\) solves (GWVVI).â€ƒâ–¡
Theorem 3.3
Let \(f: X \rightarrow\mathbb{R}^{p}\) be approximately pseudoinvex type I of order m at \(x_{0}\in X\) with respect to Î· and Ïˆ. If \(x_{0}\) solves (GWVVI), then \(x_{0}\) is a weakly quasiefficient solution of order m for (NMP) with respect to the same Ïˆ.
Proof
Suppose \(x_{0}\) solves (GWVVI) but is not a weakly quasiefficient solution of order m for (NMP) with respect to Ïˆ, then there exist \(\alpha\in \operatorname{int}(\mathbb{R}^{p}_{+})\) and \(\hat{x}\in X\), satisfying
Noticing that f is approximately pseudoinvex type I of order m at \(x_{0}\) with respect to Î· and Ïˆ on X, it follows from DefinitionÂ 2.8 and inequality (3.3) that there exist \(\alpha\in \operatorname{int}(\mathbb{R}^{p}_{+})\) and xÌ‚ such that
This is obviously not in agreement with the hypothesis that \(x_{0}\) solves (GWVVI).â€ƒâ–¡
4 Characterization of generalized quasiefficient solutions by vector critical points
This section is devoted to investigating the relations between vector critical points and weakly quasiefficient solution of order m for (NMP) with respect to Ïˆ under generalized invexity (introduced in SectionÂ 2) hypotheses imposed on the involved functions.
Definition 4.1
see [4]
A feasible solution \(x_{0}\in X\) is said to be a vector critical point of (NMP), if there exists a vetor \(\xi\in\mathbb{R}^{p}\) with \(\xi\geqslant0\) such that \(\xi^{T} A=0\) for some \(A\in\partial_{J} f(x_{0})\).
Lemma 4.1
see [23] (Gordanâ€™s theorem)
Let A be a \(p\times n\) matrix. Then exactly one of the following two systems has a solution:

System 1:
\(Ax<0\) for some \(x\in\mathbb{R}^{n}\).

System 2:
\(A^{T}y=0\), \(y\geqslant0\) for some nonzero \(y\in\mathbb{R}^{p}\).
Theorem 4.1
Let \(x_{0}\in X\) be a vector critical point of (NMP) and \(f: X\rightarrow\mathbb{R}^{p}\) be approximately pseudoinvex type I of order m at \(x_{0}\) with respect to Î· and Ïˆ, then \(x_{0}\) is a weakly quasiefficient solution of order m for (NMP) with respect to Ïˆ.
Proof
Let \(x_{0}\) be a vector critical point of (NMP), it follows from DefinitionÂ 4.1 that there exist a vector \(\xi\in\mathbb{R}^{p}\) with \(\xi\geqslant0\) and \(A\in\partial_{J}f(x_{0})\) such that
By contradiction, suppose that \(x_{0}\) is not a weakly quasiefficient solution of order m for (NMP) with respect to Ïˆ, then for any \(\alpha\in \operatorname{int}(\mathbb{R}^{p}_{+})\) there exists an \(\hat{x}\in X\) satisfying
Noticing that f is approximately pseudoinvex type I of order m at \(x_{0}\) with respect to Î· and Ïˆ on X, it follows from DefinitionÂ 2.8 and inequality (4.1) that
Using Gordanâ€™s theorem, the system
has no solution for Î¾, which contradicts the fact that \(x_{0}\) is a vector critical point of (NMP).â€ƒâ–¡
Theorem 4.2
Any vector critical point is a weakly quasiefficient solution of order m for (NMP) with respect to Ïˆ, if and only if \(f: X\rightarrow \mathbb{R}^{p}\) is approximately pseudoinvex type I of order m at that point with respect to Î· and Ïˆ.
Proof
The sufficient condition is obtained by TheoremÂ 4.1. In the following we only need to prove the necessary condition. Let \(x_{0}\) be a weakly quasiefficient solution of order m for (NMP) with respect to Ïˆ, then there exists \(\alpha\in \operatorname{int}(\mathbb {R}^{p}_{+})\) such that, for any \(x\in X\), the following cannot hold:
Noticing that \(x_{0}\) is a vector critical point, then there exist \(\xi \in\mathbb{R}^{p}\) with \(\xi\geqslant0\) and \(A\in\partial_{J} f(x_{0})\) such that
Using Gordanâ€™s theorem, there exists \(A\in\partial_{J}f(x_{0})\) such that the system
has no solution \(\mu\in\mathbb{R}^{p}\). Thus, the system
has no solution \(\mu\in\mathbb{R}^{p}\). Therefore, (4.2) and (4.3) are equivalent. Hence, if \(x_{0}\) is a weakly quasiefficient solution of order m for (NMP) with respect to Ïˆ, that is, for any \(\alpha\in \operatorname{int}(\mathbb{R}^{p}_{+})\), there exists no \(x\in X\) such that
then \(x_{0}\) solves (GWVVI), that is, there exists no \(x\in X\) with \(\eta(x,x_{0})\in\mathbb{R}^{p}\) satisfying
This satisfies the condition of the approximately pseudoinvexity of type I of order m of f at \(x_{0}\).â€ƒâ–¡
5 Conclusions
In the current work, we present several extended approximately invex vectorvalued functions of higher order involving a generalized Jacobian. Furthermore, the notions of higherorder (weak) quasiefficiency with respect to a function for a multiobjective programming are also introduced, and some examples are given to illustrate their existence. Under generalization of higherorder approximate invexities assumptions, it proves that the solutions of generalized vector variationallike inequalities in terms of the generalized Jacobian are the generalized quasiefficient solutions to nonsmooth multiobjective programming problems (i.e. Theorems 3.13.3). In addition, we also focused on examining the equivalent conditions. By employing the Gordan theorem [23], the equivalent conditions are obtained, that is, a vector critical point is a weakly quasiefficient solution of higher order with respect to a function (TheoremÂ 4.1 and TheoremÂ 4.2).
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This research was supported by the Natural Science Foundation of China under Grant Nos. 61650104,Â 11361001.
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Wang, Z., Li, R. & Yu, G. Vector critical points and generalized quasiefficient solutions in nonsmooth multiobjective programming. J Inequal Appl 2017, 184 (2017). https://doi.org/10.1186/s1366001714562
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DOI: https://doi.org/10.1186/s1366001714562