- Research
- Open Access
Optimality conditions for interval-valued univex programming
- Lifeng Li^{1, 2}Email author,
- Jianke Zhang^{1, 2} and
- Chang Zhou^{1}
https://doi.org/10.1186/s13660-019-2002-1
© The Author(s) 2019
- Received: 4 September 2018
- Accepted: 18 February 2019
- Published: 27 February 2019
Abstract
We introduce concepts of interval-valued univex mappings, consider optimization conditions for interval-valued univex functions for the constrained interval-valued minimization problem, and show examples for the illustration purposes.
Keywords
- Interval-valued univex mappings
- Interval-valued univex programming
- Optimality
1 Introduction
Convexity and generalized convexity are important in mathematical programming. Invex functions, introduced by Hanson [17], are important generalized convex functions and are successfully used in optimization and equilibrium problems. For example, necessary and sufficient conditions are obtained for K-invex functions in [14]. The concept of G-invex functions is introduced by Antczak [3]. Optimality and duality for differentiable G-multiobjective problems are considered in [4, 5]. Noor [26] considered invex equilibrium problems in the context of invexity. As an extension and refinement of Noor [26], Farajzadeh [15] gave some results for invex Ky Fan inequalities in topological vector spaces.
Later, some generalized optimality conditions of primal and dual problems were considered by Hanson and Mond [18]. Combing with generalized type I and univex functions, optimality conditions and duality for several mathematical programming problems were considered by many researchers [1, 16, 29], and more and more scholars pay attention to type I and univex functions [24, 25, 34, 35].
The authors of [2, 6, 9, 12, 27, 30, 33, 36–39] have studied generalized convex interval-valued mappings and their connection with interval-valued optimization. For example, Steuer [33] proposed three algorithms, called the F-cone algorithm, E-cone algorithm, and emanating algorithms, to solve the linear programming problems with interval-valued objective functions. To prove strong duality theorems, Wu [37] derived KKT optimality conditions in the interval-valued problems under convexity hypotheses. Wu [36] also obtained KKT conditions in an optimization problem with an interval-valued objective function using H-derivatives and the concept of weakly differentiable functions. Since the H-derivative suffers certain disadvantages, Chalco-Cano et al. [10] gave KKT-type optimality conditions, which were obtained using the gH-derivatives of interval-valued functions. Also, they studied the relationship between the approach presented with other known approaches given by Wu [36]. However, these methods cannot solve a kind of optimization problems with interval-valued objective functions that are not LU-convex but univex. Antczak [6] used the classical exact \(l_{1}\) penalty function method for solving nondifferentiable interval-valued optimization problems under convexity hypotheses. Optimality conditions in invex optimization problems with an interval-valued objective function were discussed by Zhang et al. [39]. Using gH-differentiability, Li et al. [21] introduced interval-valued invex mappings and gave the optimality conditions for interval-valued objective functions under invexity. By using the weak derivative of fuzzy functions, Li et al. [22] defined fuzzy weakly univex functions and considered optimization conditions for fuzzy minimization problem.
Followed by [21] and [22], in this paper, we introduce the concept of interval-valued univex mappings, consider optimization conditions for interval-valued univex functions for the constrained interval-valued minimization problem, and show examples for illustration purposes. The present paper can be seen as promotion and expansion of [20]. The method presented in this paper is different from that in [6]. Our method cannot solve Example 3.1 of [6] because the objective function is not gH-differentiable. Example 4.1 shows that the methods given by [6, 33, 36, 37] cannot solve a kind of optimization problems for interval-valued univex mappings. Example 4.2 shows that the methods given by Li et al. [22] cannot solve a kind of fuzzy optimization problems for interval-valued univex mappings. Finally, Example 4.3 shows that the method given in [10] cannot solve a kind of optimization problems for interval-valued univex mappings. In Sect. 3, we introduce the concept of interval-valued univex mappings and discuss some their properties. Section 4 deals with optimality conditions for the constrained interval-valued minimization problem under the assumption of interval-valued univexity.
2 Preliminaries
In this paper, a closed interval in R is denoted by \(A=[a^{L}, a ^{U}]\). Every \(a\in R\) is considered as a particular closed interval \(a=[a,a]\). The set of closed intervals is denoted by \(\mathcal{I}\).
- (1)
\(A+B=[a^{L}+b^{L}, a^{U}+b^{U}] \) and \(-A=\{-a:a\in A\}=[-a^{U},-a ^{L}]\);
- (2)
\(A\ominus _{gH} B=[\min (a^{L}-b^{L},a^{U}-b^{U}),\max (a^{L}-b ^{L},a^{U}-b^{U})]\);
- (3)
\(A\preceq B\Leftrightarrow a^{L}\leq b^{L}\) and \(a^{U}\leq b^{U}\); \(A\prec B \Leftrightarrow A\preceq B\) and \(A\neq B\).
For \(X\subseteq R^{n}\), a mapping \(F:X\rightarrow \mathcal{I}\) is called an interval-valued function. Then \(F(x)=[F^{L}(x),F^{U}(x)]\), where \(F^{L}(x)\) and \(F^{U}(x)\) are two real-valued functions defined on \(R^{n}\) and satisfying \(F^{L}(x)\leq F^{U}(x)\) for every \(x\in X\). If \(F^{L}(x)\) and \(F^{U}(x)\) are continuous, then \(F(x)\) is said to be continuous.
It is well known that the derivative and subderivative of a function is important in the study of generalized convexity and mathematical programming. For example, a classic subdifferential is introduced by Azimov and Gasimov [7]. Some theorems connecting operations on the weak subdifferential in the nonsmooth and nonconvex analysis are provided in [13]. The derivative and subderivative of interval-valued functions are extensions of real-valued functions. Due to different arithmetics of intervals, several definitions about derivatives of interval-valued functions are introduced by the authors, such as weakly differentiable functions [36], H-differentiable functions (based on the Hukuhara difference of two closed intervals [36]), gH-differentiable functions (based on the operation \(\ominus _{gH}\) of two closed intervals [11, 31]), and subdifferentiable functions (based on the difference \(A- B=[a^{L}-b^{U}, a^{U}-b^{L}]\) of two closed intervals [6]). In this paper, we always use weakly differentiable and gH-differentiable functions, which are defined as follows.
Let X be an open set in \(R^{n}\), and let \(F(x)=[F^{L}(x),F^{U}(x)]\). Then \(F(x)\) is called weakly differentiable at \(x_{0}\) if \(F^{L}(x)\) and \(F^{U}(x)\) are differentiable at \(x_{0}\).
The following basic result (which can be found in Lemma 3.1. of [19]) of the mathematical analysis is well known:
We say that \(\varPsi :\mathcal{I}\rightarrow \mathcal{I}\) is increasing if \(A\preceq B\) implies \(\varPsi (A)\preceq \varPsi (B)\). From the above result we can prove the following result:
If function \(\varPhi : R\rightarrow R\) is increasing, then \(\varPsi : \mathcal{I}\rightarrow \mathcal{I}\) is increasing. Moreover, \(\varPsi ([a^{L}, a^{U}])=[\varPhi (a^{L}),\varPhi (a^{U})]\).
3 Interval-valued univex functions
In this section, we define interval-valued univex functions as a generalization of univex functions [8] and discuss some their properties.
Let X be an invex set in \(R^{n}\) (the concept of an invex set can be found in [8]), and let F be an interval-valued function. The following definition is a particular case of fuzzy weakly univex functions, which has been introduced in [22].
Remark 3.1
Example 3.1
Let \(\varPhi :R\rightarrow R\) be defined by \(\varPhi (V)=3V\), \(F^{L}(x)=2x ^{3}\), and \(F^{U}(x)=x^{3}\); then \(\nabla F^{L}(x)=6x^{2}\) and \(\nabla F^{U}(x)=3 x^{2}\). Then F is interval-valued weakly univex but not interval-valued weakly invex, since for \(x=-2\) and \(y=-1\), \(F^{U}(x)-F^{U}(y)< \eta ^{t}(x,y)\nabla F^{U}(y)\).
Let X be a nonempty open set in \(R^{n}\), \(\eta :X \times X\rightarrow R^{n}\), \(\varPsi :\mathcal{I}\rightarrow \mathcal{I}\), and \(b = b(x,y): X \times X \rightarrow R^{+}\).
Definition 3.1
The following example shows that an interval-valued univex function may not be an interval-valued weakly univex function.
Example 3.2
Theorem 3.1
Suppose \(F(x)\) is gH-differentiable. If \(F(x)\) is an interval-valued weakly univex function with respect to η, b, Φ and Φ is increasing, then \(F(x)\) is an interval-valued univex function with respect to the same η, b, and Ψ, where Ψ is an extension of Φ.
Proof
Remark 3.2
If Φ is nonincreasing, then Theorem 3.1 may not be true (as shown in the following Example 3.3).
Example 3.3
4 Optimality criteria for interval-valued univex mappings
In this section, for gH-differentiable interval-valued univex functions, we establish sufficient optimality conditions for a feasible solution \(x^{\ast }\) to be an optimal solution or a nondominated solution for \((P)\).
Let \(P:=\{x\in X :g(x)\preceq 0\}\) denote the feasible set of \((P)\).
Since ⪯ is a partial order, the optimal solution may not exist for some interval-valued optimization problems. Therefore, authors always consider the concept of a nondominated solution in this situation. We reconsider an optimal solution and nondominated solution as follows.
Definition 4.1
- (i)
\(x^{\ast }\in P\) is an optimal solution of \((P)\Leftrightarrow F(x^{\ast })\preceq F(x)\) for all \(x\in P\). In this case, \(F(x^{\ast })\) is called the optimal objective value of F.
- (ii)
\(x^{\ast }\in P\) is a nondominated solution of \((P)\Leftrightarrow \) there exists no \(x_{0}\in P\) such that \(F(x_{0})\prec F(x^{\ast })\). In this case, \(F(x^{\ast })\) is called the nondominated objective value of F.
Theorem 4.1
- (i)there exist η, \(\varPsi _{0}\), \(b_{0}\), \(\varPsi _{i}\), \(b_{i}\), \(i=1, 2 , \ldots ,m \), such thatand$$\begin{aligned}& b_{0}(x,y)\varPsi _{0}\bigl[F(x)\ominus _{gH}F \bigl(x^{\ast }\bigr)\bigr]\succeq \eta ^{t}\bigl(x,x ^{\ast }\bigr)\nabla F\bigl(x^{\ast }\bigr) {} \end{aligned}$$(7)for all feasible x;$$\begin{aligned}& -b_{i}\bigl(x,x^{\ast }\bigr)\varPsi _{i} \bigl[g_{i}\bigl(x^{\ast }\bigr)\bigr]\succeq \eta ^{t}\bigl(x,x^{ \ast }\bigr)\nabla g_{i} \bigl(x^{\ast }\bigr) \end{aligned}$$(8)
- (ii)there exists \(y^{\ast }\in R^{m}\) such that$$\begin{aligned}& \nabla F\bigl(x^{\ast }\bigr)=-y^{\ast t}\nabla g \bigl(x^{\ast }\bigr), \end{aligned}$$(9)Further suppose that$$\begin{aligned}& y^{\ast }\geq 0. \end{aligned}$$(10)$$\begin{aligned}& \varPsi _{0}(\mu )\succeq 0 \quad \Rightarrow \quad \mu \succeq 0, \end{aligned}$$(11)and$$\begin{aligned}& \mu \preceq 0 \quad \Rightarrow\quad \varPsi _{i}( \mu )\succeq 0, \end{aligned}$$(12)for all feasible x. Then \(x^{\ast }\) is an optimal solution of \((P)\).$$\begin{aligned}& b_{0}\bigl(x,x^{\ast }\bigr)> 0,\qquad b_{i} \bigl(x,x^{\ast }\bigr)\geq 0, \end{aligned}$$(13)
Proof
Remark 4.1
Example 4.1
Theorem 4.2
- (i)there exist η, \(\varPsi _{0}\), \(b_{0}\), \(\varPsi _{i}\), \(b_{i}\), \(i=1, 2 , \ldots ,m \), such thatand$$\begin{aligned}& b_{0}(x,y)\varPsi _{0}\bigl[F(x)\ominus _{gH}F\bigl(x^{\ast }\bigr)\bigr]\succeq \eta ^{t} \bigl(x,x ^{\ast }\bigr)\nabla F\bigl(x^{\ast }\bigr) \end{aligned}$$(15)for all feasible x;$$\begin{aligned}& -b_{i}\bigl(x,x^{\ast }\bigr)\varPsi _{i}\bigl[g_{i}\bigl(x^{\ast }\bigr)\bigr]\succeq \eta ^{t}\bigl(x,x^{ \ast }\bigr)\nabla g_{i} \bigl(x^{\ast }\bigr) \end{aligned}$$(16)
- (ii)there exists \(y^{\ast }\in R^{m}\) such that$$\begin{aligned}& \bigl\{ \nabla F\bigl(x^{\ast }\bigr)\bigr\} ^{L}= \bigl\{ -y^{\ast t}\nabla g\bigl(x^{\ast }\bigr)\bigr\} ^{L}, \end{aligned}$$(17)Further, suppose that$$\begin{aligned}& y^{\ast }\geq 0. \end{aligned}$$(18)$$\begin{aligned}& \varPsi _{0}(\mu )\nprec 0 \quad \Rightarrow \quad \mu \nprec 0, \end{aligned}$$(19)and$$\begin{aligned}& \mu \preceq 0 \quad \Rightarrow \quad \varPsi _{i}( \mu )\succeq 0, \end{aligned}$$(20)for all feasible x. Then \(x^{\ast }\) is a nondominated solution of \((P)\).$$\begin{aligned}& b_{0}\bigl(x,x^{\ast }\bigr)> 0,\qquad b_{i}\bigl(x,x^{\ast }\bigr)\geq 0 \end{aligned}$$(21)
Proof
Theorem 4.3
- (i)there exist η, \(\varPsi _{0}\), \(b_{0}\), \(\varPsi _{i}\), \(b_{i}\), \(i=1, 2 , \ldots ,m \), such thatand$$\begin{aligned}& b_{0}(x,y)\varPsi _{0}\bigl[F(x)\ominus _{gH}F\bigl(x^{\ast }\bigr)\bigr]\succeq \eta ^{t} \bigl(x,x ^{\ast }\bigr)\nabla F\bigl(x^{\ast }\bigr) \end{aligned}$$(22)for all feasible x;$$\begin{aligned}& -b_{i}\bigl(x,x^{\ast }\bigr)\varPsi _{i}\bigl[g_{i}\bigl(x^{\ast }\bigr)\bigr]\succeq \eta ^{t}\bigl(x,x^{ \ast }\bigr)\nabla g_{i} \bigl(x^{\ast }\bigr) \end{aligned}$$(23)
- (ii)there exists \(y^{\ast }\in R^{m}\) such that$$\begin{aligned}& \bigl\{ \nabla F\bigl(x^{\ast }\bigr)\bigr\} ^{U}= \bigl\{ -y^{\ast t}\nabla g\bigl(x^{\ast }\bigr)\bigr\} ^{U}, \end{aligned}$$(24)Further, suppose that$$\begin{aligned}& y^{\ast }\geq 0. \end{aligned}$$(25)$$\begin{aligned}& \varPsi _{0}(\mu )\nprec 0 \quad \Rightarrow\quad \mu \nprec 0, \end{aligned}$$(26)and$$\begin{aligned}& \mu \preceq 0 \quad \Rightarrow\quad \varPsi _{i}( \mu )\succeq 0, \end{aligned}$$(27)for all feasible x. Then \(x^{\ast }\) is a nondominated solution of \((P)\).$$\begin{aligned}& b_{0}\bigl(x,x^{\ast }\bigr)> 0,\qquad b_{i}\bigl(x,x^{\ast }\bigr)\geq 0 \end{aligned}$$(28)
The following example shows the advantages of our method over [22].
Example 4.2
Since \(F^{L}(x)=-|x|\) and \(F^{U} (x)=|x|\) is not differentiable at \(x=0\), \(F(x)\) is not weakly differentiable at \(x=0\). Therefore the method in [22] cannot be used.
The following example also shows the advantages of our method over [10] and [23, 28].
Example 4.3
5 Conclusion
The objective of this paper is to introduce the concept of gH-differentiable interval-valued univex mappings and discuss the relationship between interval-valued univex mappings and interval-valued weakly univex mappings. We derive sufficient optimality conditions for constrained interval-valued minimization problem under interval-valued univex mappings. In future work, we hope to give sufficient optimality conditions for a nondifferentiable interval-valued optimization problem under univexity hypotheses.
Declarations
Funding
This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 11401469, 11701446) and the Natural Science Foundation of Shaanxi Province (2018JM1055) and Shaanxi key disciplines of special funds projects.
Authors’ contributions
All authors have equal contributions. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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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