Optimality conditions for interval-valued univex programming

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.

Another important type of generalized convex functions, called univex functions and preunivex functions, is introduced in [8]. Suppose \(\emptyset \neq X\subseteq R^{n}\), \(\eta :X \times X\rightarrow R^{n}\), \(\varPhi :R\rightarrow R\), and \(b=b(x,y):X \times X \rightarrow R^{+}\). A differentiable function \(F: X\rightarrow R\) is said to be univex at \(y\in X\) with respect to η, Φ, b if, for all \(x\in X\),

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.

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 \(\mathcal{I}\).

Given \(A=[a^{L},a^{U}]\) and \(B=[b^{L},b^{U}] \in \mathcal{I}\), the arithmetic operations and order are defined in [32] as follows:

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 \(x_{0}\).

Let \(x_{0} \in (a, b)\) and h be such that \(x_{0} + h \in (a, b)\). Then If \(F^{\prime }(x_{0})\in \mathcal{I}\) exists, then F is gH- differentiable at \(x_{0}\).

If \(F^{L}(x)\) and \(F^{U}(x)\) are differentiable functions at \(x\in (a, b)\), then \(F(x)\) is gH-differentiable at x, and

We say that an interval-valued function F is gH-differentiable at \(x=(x_{1},\ldots ,x_{n})\in X\) if all the partial gH-derivatives \(( \frac{\partial F}{\partial x_{1}})(x),\ldots , ( \frac{\partial F}{ \partial x_{n}})(x)\) exist on some neighborhood of x and are continuous at x. We write and we call \(\nabla F(x)\) the gradient of a gH-differentiable interval-valued function F at x.

Let \(\mathbb{H}(R^{n})\) denote the family of nonempty compact subsets of \(R^{n}\). For \(A,B\in \mathbb{H}(R^{n})\), the Hausdorff metric \(h(A,B)\) on \(\mathbb{H}(R^{n})\) is defined by where

The following basic result (which can be found in Lemma 3.1. of [19]) of the mathematical analysis is well known:

Suppose that \(\varPhi :R^{n}\rightarrow R^{n}\) is continuous and let \(X\in \mathbb{H}(R^{n})\). Then the mapping is uniformly continuous in h-metric.

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, \(\varPsi ([a^{L}, a^{U}])=[\varPhi (a^{L}),\varPhi (a^{U})]\).

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].

Suppose F is a weakly differentiable interval-valued function. Then F is weakly univex at \(y\in X\) with respect to η, Φ, b if and only if both \(F^{L}(x)\) and \(F^{U}(x)\) are univex at \(y\in X\), that is, for all \(x\in X\), where \(\eta =\eta (x,y):X \times X\rightarrow R^{n}\), \(\varPhi :R\rightarrow R\), and \(b = b(x,y):X\times X \times [0, 1]\rightarrow R^{+}\).

Let X be a nonempty open set in \(R^{n}\), \(\eta :X \times X\rightarrow R^{n}\), \(\varPsi :\mathcal{I}\rightarrow \mathcal{I}\), and \(b = b(x,y): X \times X \rightarrow R^{+}\).

The following example shows that an interval-valued univex function may not be an interval-valued weakly univex function.

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 \((P)\).

Suppose \(F(x)\), \(g_{1}(x),\ldots , g_{m}(x)\) are gH-differentiable interval-valued mappings defined on a nonempty open set \(X\subseteq R ^{n}\). Then, we consider the primal problem:

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.

In Theorem 18 of [20], the authors also gave a sufficient optimality condition for a feasible solution \(x^{\ast }\) to be an optimal solution. In this theorem, the equation was used, substituted for (9) of Theorem 4.1. We can prove that the previous equation is very restrictive. In fact, in case \(F(x)\) is a unary function, suppose \(\nabla F(x^{\ast })=[a,b]\) and \(y^{\ast t} \nabla g(x^{\ast })=[yc,yd]\). Then we have \([a,b]+[yc,yd]=[a+yc,b+yd]=0\), where \(a\leq b\) and \(yc\leq yd\). Therefore we have \(a=b\) and \(c=d\) since \(y\geq 0\). That is to say, \(\nabla F(x^{\ast })\) is a real number instead of an interval. In the following example, we can observe that \(x^{\ast }\) is an optimal solution of \((P)\), but \(x^{\ast }\) do not satisfies the previous equation. The following example also shows the advantages of our method over [6, 33, 36, 37].

The following example shows the advantages of our method over [22].

The following example also shows the advantages of our method over [10] and [23, 28].