Vector critical points and generalized quasiefficient solutions in nonsmooth multiobjective programming
 Zhen Wang^{1},
 Ru Li^{1} and
 Guolin Yu^{1}Email author
https://doi.org/10.1186/s1366001714562
© The Author(s) 2017
Received: 16 June 2017
Accepted: 18 July 2017
Published: 9 August 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.
Keywords
MSC
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 OsunaGómez 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 [8–11]). 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 [15–17]). 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 Section 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 Section 3, the relations between (GVVI) and (NMP) are established under the approximate invexity of higherorder assumptions. In Section 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
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]
Rademacher’s theorem (see Corollary 4.12 in [19]) indicates that a function f satisfying the Lipschitz condition (2.1) is Fréchet 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]
Definition 2.3
see [20]
From now on, we always assume that the subset \(X\subseteq\mathbb {R}^{n}\) is a nonempty invex set with respect to some η 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]
Definition 2.5
see [4]
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.
Motivated by above definitions, we give the notions of approximate invexity of order m with respect to η and ψ, strictly approximate invexity of order m with respect to η and ψ and approximate pseudoinvexity of type I of order m with respect to η and ψ as follows.
Definition 2.6
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 Definition 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 η is also approximately invex of order m at \(x_{0}\) with respect to η and ψ, but in the contrary case, it does not hold. The following example is given to illustrate this fact.
Example 2.1
Definition 2.7
Definition 2.8
The following example illustrates the existence of approximate invexity of order m with respect to η and ψ and of an approximately pseudoinvex type I function of order m with respect to η and ψ.
Example 2.2
Remark 2.3
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
Definition 2.10
Definition 2.11
 (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 ψ for the problem (NMP).
Definition 2.12
Definition 2.13
Remark 2.4
Remark 2.5
A quasiefficient solution of order m for (NMP) with respect to ψ is not to be a quasiefficient solution in the sense of Definition 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 Definition 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 \(\alpha=(1,1)^{T}\).
 (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 Section 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 η and ψ. If \(x_{0}\) solves (GVVI), then \(x_{0}\) is a quasiefficient solution of order m for (NMP) with respect to the same ψ.
Proof
Theorem 3.2
Let \(f: X \rightarrow\mathbb{R}^{p}\) be approximately invex of order m at \(x_{0}\in X\) with respect to η and ψ. If \(x_{0}\) solves (GWVVI), then \(x_{0}\) is a weakly quasiefficient solution of order m for (NMP) with respect to the same ψ.
Proof
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 η and ψ. If \(x_{0}\) solves (GWVVI), then \(x_{0}\) is a weakly quasiefficient solution of order m for (NMP) with respect to the same ψ.
Proof
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 ψ under generalized invexity (introduced in Section 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] (Gordan’s theorem)
 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 η and ψ, then \(x_{0}\) is a weakly quasiefficient solution of order m for (NMP) with respect to ψ.
Proof
Theorem 4.2
Any vector critical point is a weakly quasiefficient solution of order m for (NMP) with respect to ψ, if and only if \(f: X\rightarrow \mathbb{R}^{p}\) is approximately pseudoinvex type I of order m at that point with respect to η and ψ.
Proof
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 (Theorem 4.1 and Theorem 4.2).
Declarations
Acknowledgements
This research was supported by the Natural Science Foundation of China under Grant Nos. 61650104, 11361001.
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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