Optimality conditions for interval-valued univex programming

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.

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.

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 $x\in X$,

$$\begin{aligned} b(x,y)\varPhi \bigl[F(x)-F(y)\bigr]\geq \eta ^{t}(x,y)\nabla F(y). \end{aligned}$$

(1)

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, 36,37,38,39] 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 $\mathcal{I}$.

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:

• (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 $x_{0}$.

Let $x_{0} \in (a, b)$ and h be such that $x_{0} + h \in (a, b)$. Then

$$\begin{aligned} F^{\prime }(x_{0})= \lim_{x\rightarrow 0} \bigl[F(x_{0}+h)\ominus _{gH}F(x _{0})\bigr]. \end{aligned}$$

(2)

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

$$\begin{aligned} F^{\prime }(x)= \bigl[\min \bigl\{ \bigl(F^{L}\bigr)^{\prime }(x), \bigl(F^{U}\bigr)^{\prime }(x)\bigr\} , \max \bigl\{ \bigl(F^{L}\bigr)^{\prime }(x),\bigl(F^{U} \bigr)^{\prime }(x)\bigr\} \bigr]. \end{aligned}$$

(3)

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

$$\begin{aligned} \nabla F(x)=\biggl(\biggl( \frac{\partial F}{\partial x_{1}}\biggr) (x),\biggl( \frac{\partial F}{\partial x_{2}}\biggr) (x),\ldots ,\biggl( \frac{\partial F}{\partial x_{n}}\biggr) (x) \biggr)^{t}, \end{aligned}$$

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

$$\begin{aligned} h(A,B)=\inf \bigl\{ \varepsilon \mid A\subseteq N(B,\varepsilon ),B\subseteq N(A, \varepsilon )\bigr\} , \end{aligned}$$

where

$$\begin{aligned} N(A,\varepsilon )=\bigl\{ x\in R^{n}\mid d(x,A)< \varepsilon \bigr\} , \quad d(x,A)= \inf_{a\in A} \Vert x-a \Vert . \end{aligned}$$

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

$$\begin{aligned} \varPsi : \mathbb{H}\bigl(X^{n}\bigr)\rightarrow \mathbb{H} \bigl(R^{n}\bigr), \qquad \varPsi (A)=\bigl\{ \phi (a)\mid a\in A\bigr\} \end{aligned}$$

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

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].

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 $x\in X$,

$$\begin{aligned}& b(x,y)\varPhi \bigl[F^{L}(x)-F^{L}(y)\bigr]\geq \eta ^{t}(x,y)\nabla F^{L}(y), \end{aligned}$$

(4)

$$\begin{aligned}& b(x,y)\varPhi \bigl[F^{U}(x)-F^{U}(y)\bigr]\geq \eta ^{t}(x,y)\nabla F^{U}(y), \end{aligned}$$

(5)

where $\eta =\eta (x,y):X \times X\rightarrow R^{n}$, $\varPhi :R\rightarrow R$, and $b = b(x,y):X\times X \times [0, 1]\rightarrow R^{+}$.

Remark 3.1

The concept of LU-invexity for interval-valued functions is introduced in [39], since it considers the endpoint functions; in this paper, we call them weakly invex. Every interval-valued weakly invex function is interval-valued weakly univex with respect to η, b, Φ, where

$$\begin{aligned}& \varPhi (x) = x,\quad b=1, \end{aligned}$$

but the converse is not true.

Example 3.1

Consider the function $F: (-\infty ,0)\rightarrow \mathcal{I}$ defined by

$$\begin{aligned}& F(x)=[1,2]x^{3}, \\& \eta (x,y)=\textstyle\begin{cases} x^{2}+xy+y^{2}, & x>y, \\ x-y, & x\leq y, \end{cases}\displaystyle \\& b(x,y)=\textstyle\begin{cases} \frac{y^{2}}{x-y}, & x>y, \\ 0, & x\leq y. \end{cases}\displaystyle \end{aligned}$$

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

Suppose F is a gH-differentiable interval-valued function. Then F is univex at $y\in X$ with respect to η, Ψ, b if for all $x\in X$,

$$\begin{aligned}& b(x,y)\varPsi \bigl[F(x)\ominus _{gH}F(y)\bigr]\succeq \eta ^{t}(x,y)\nabla F(y). \end{aligned}$$

(6)

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

Example 3.2

Suppose $F(x)=[-|x|,|x|]$, $x\in R$, $b=1$, and $\varPhi (a)=a$. Then $\varPsi [a,b]=[a,b]$ is induced by $\varPhi (a)=a$, and

$$\begin{aligned} \eta (x,y)=\textstyle\begin{cases} x-y, &x y\geq 0, \\ x+y, &x y< 0. \end{cases}\displaystyle \end{aligned}$$

Then $F(x)$ is gH-differentiable on R, and $F^{\prime }(y)=[-1, 1]$. We can prove that

$$\begin{aligned} b\varPsi \bigl[F(x)\ominus _{gH} F(y)\bigr]\succeq \eta ^{t}(x,y)\nabla F(y). \end{aligned}$$

Therefore $F(x)$ is univex with respect to η, b, Ψ, but $F(x)$ is not weakly univex since $F^{L}(x)$ is not univex with respect to η, b, Φ.

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

Since $F(x)$ is weakly univex at y, then real-valued functions $F^{L}$ and $F^{U}$ are univex at y, that is,

$$\begin{aligned}& b(x,y)\varPhi \bigl[F^{L}(x)-F^{L}(y)\bigr]\geq \eta ^{t}(x,y)\nabla F^{L}(y)\quad \text{and} \\& b(x,y)\varPhi \bigl[F^{U}(x)-F^{U}(y)\bigr]\geq \eta ^{t}(x,y)\nabla F^{U}(y) \end{aligned}$$

for all $x\in X$.

(i) Under the condition $\eta ^{t}(x,y)\nabla F^{L}(y) \leq \eta ^{t}(x,y) \nabla F^{U}(y)$, we have

$$\begin{aligned}& \eta ^{t}(x,y)\nabla F(y)=\bigl[\eta ^{t}(x,y)\nabla F^{L}(y), \eta ^{t}(x,y) \nabla F^{U}(y)\bigr]. \end{aligned}$$

If $F(x)\ominus _{gH} F(y)=[F^{L}(x)-F^{L}(y), F^{U}(x)-F^{U}(y)]$, then since Φ is increasing, we have

$$\begin{aligned}& b(x,y)\varPsi \bigl[F(x)\ominus _{gH}F(y)\bigr] \\& \quad =\bigl[b(x,y)\varPhi \bigl(F^{L}(x)-F^{L}(y)\bigr), b(x,y)\varPhi \bigl(F^{U}(x)-F^{U}(y)\bigr)\bigr] \\& \quad \succeq \bigl[\eta ^{t}(x,y)\nabla F^{L}(y), \eta ^{t}(x,y)\nabla F^{U}(y)\bigr] \\& \quad = \eta ^{t}(x,y)\nabla F(y). \end{aligned}$$

If $F(x)\ominus _{gH} F(y)=[F^{U}(x)-F^{U}(y), F^{L}(x)-F^{L}(y)]$, then

$$\begin{aligned}& b(x,y)\varPhi \bigl(F^{L}(x)-F^{L}(y)\bigr) \\& \quad \succeq b(x,y)\varPhi \bigl(F^{U}(x)-F^{U}(y)\bigr) \\& \quad \succeq \eta ^{t}(x,y)\nabla F^{U}(y) \\& \quad \succeq \eta ^{t}(x,y)\nabla F^{L}(y), \end{aligned}$$

and since Φ is increasing, we have

$$\begin{aligned}& b(x,y)\varPsi \bigl[F(x)\ominus _{gH}F(y)\bigr] \\& \quad =[b(x,y)\varPsi \bigl[F^{U}(x)-F^{U}(y), F^{L}(x)-F^{L}(y)\bigr] \\& \quad =\bigl[b(x,y)\varPhi \bigl(F^{U}(x)-F^{U}(y)\bigr), b(x,y)\varPhi \bigl(F^{L}(x)-F^{L}(y)\bigr)\bigr] \\& \quad \succeq \bigl[\eta ^{t}(x,y)\nabla F^{L}(y), \eta ^{t}(x,y)\nabla F^{U}(y)\bigr] \\& \quad = \eta ^{t}(x,y)\nabla F(y). \end{aligned}$$

(ii) Under the condition $\eta ^{t}(x,y)\nabla F^{L}(y) > \eta ^{t}(x,y) \nabla F^{U}(y)$, we have

$$\begin{aligned} \eta ^{t}(x,y)\nabla F(y)=\bigl[\eta ^{t}(x,y)\nabla F^{U}(y),\eta ^{t}(x,y) \nabla F^{L}(y)\bigr]. \end{aligned}$$

If $F(x)\ominus _{gH} F(y)=[F^{U}(x)-F^{U}(y), F^{L}(x)-F^{L}(y)]$, then since Φ is increasing, we have

$$\begin{aligned}& b(x,y)\varPsi \bigl[F(x)\ominus _{gH}F(y)\bigr] \\& \quad =\bigl[b(x,y)\varPhi \bigl(F^{U}(x)-F^{U}(y)\bigr), b(x,y)\varPhi \bigl(F^{L}(x)-F^{L}(y)\bigr)\bigr] \\& \quad \succeq \bigl[\eta ^{t}(x,y)\nabla F^{U}(y), \eta ^{t}(x,y)\nabla F^{L}(y)\bigr] \\& \quad = \eta ^{t}(x,y)\nabla F(y). \end{aligned}$$

If $F(x)\ominus _{gH} F(y)=[F^{L}(x)-F^{L}(y), F^{U}(x)-F^{U}(y)]$, then

$$\begin{aligned}& b(x,y)\varPhi \bigl(F^{U}(x)-F^{U}(y)\bigr) \\& \quad \succeq b(x,y)\varPhi \bigl(F^{L}(x)-F^{L}(y)\bigr) \\& \quad \succeq \eta ^{t}(x,y)\nabla F^{L}(y) \\& \quad \succeq \eta ^{t}(x,y)\nabla F^{U}(y). \end{aligned}$$

Since Φ is increasing, we have

$$\begin{aligned}& b(x,y)\varPsi \bigl[F(x)\ominus _{gH}F(y)\bigr] \\& \quad =b(x,y)\varPsi \bigl[F^{L}(x)-F^{L}(y), F^{U}(x)-F^{U}(y)\bigr] \\& \quad =\bigl[b(x,y)\varPhi \bigl(F^{L}(x)-F^{L}(y) \bigr),b(x,y)\varPhi \bigl(F^{U}(x)-F^{U}(y)\bigr)\bigr] \\& \quad \succeq \bigl[ \eta ^{t}(x,y)\nabla F^{U}(y), \eta ^{t}(x,y)\nabla F^{L}(y)\bigr] \\& \quad = \eta ^{t}(x,y)\nabla F(y). \end{aligned}$$

 □

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

Suppose $F(x)=[-2,1]x^{2}$, $x<0$. Then $F(x)$ is gH-differentiable and weakly differentiable. It is easy to check that $F(x)$ is weakly univex with respect to $\eta (x,y)=x-y$,

$$b(x,y)=\textstyle\begin{cases} 1, & x\leq y< 0, \\ \frac{-2y(x-y)}{-x^{2}+y^{2}}, & y< x< 0, \end{cases} $$

and $\varPhi (a)=|a|$. However, $F(x)$ is not univex with respect to the same $\eta (x,y)$, b, and Ψ, where Ψ is defined by the extension of $\varPhi (a)=|a|$.

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 $(P)$.

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:

$$\begin{aligned}& (P)\quad \min F(x) \\& \hphantom{(P)\quad} \text{s.t.}\quad g(x)\preceq 0. \end{aligned}$$

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

Let $x^{\ast }$ be P-feasible. Suppose that:

• (i) there exist η, $\varPsi _{0}$, $b_{0}$, $\varPsi _{i}$, $b_{i}$, $i=1, 2 , \ldots ,m$, such that

  $$\begin{aligned}& b_{0}(x,y)\varPsi _{0}\bigl[F(x)\ominus _{gH}F \bigl(x^{\ast }\bigr)\bigr]\succeq \eta ^{t}\bigl(x,x ^{\ast }\bigr)\nabla F\bigl(x^{\ast }\bigr) {} \end{aligned}$$

  (7)

  and

  $$\begin{aligned}& -b_{i}\bigl(x,x^{\ast }\bigr)\varPsi _{i} \bigl[g_{i}\bigl(x^{\ast }\bigr)\bigr]\succeq \eta ^{t}\bigl(x,x^{ \ast }\bigr)\nabla g_{i} \bigl(x^{\ast }\bigr) \end{aligned}$$

  (8)

  for all feasible x;

• (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)
  $$\begin{aligned}& y^{\ast }\geq 0. \end{aligned}$$

  (10)

  Further suppose that

  $$\begin{aligned}& \varPsi _{0}(\mu )\succeq 0 \quad \Rightarrow \quad \mu \succeq 0, \end{aligned}$$

  (11)
  $$\begin{aligned}& \mu \preceq 0 \quad \Rightarrow\quad \varPsi _{i}( \mu )\succeq 0, \end{aligned}$$

  (12)

  and

  $$\begin{aligned}& b_{0}\bigl(x,x^{\ast }\bigr)> 0,\qquad b_{i} \bigl(x,x^{\ast }\bigr)\geq 0, \end{aligned}$$

  (13)

  for all feasible x. Then $x^{\ast }$ is an optimal solution of $(P)$.

Proof

Let x be P-feasible. Then

$$\begin{aligned} g(x)\preceq 0. \end{aligned}$$

This, along with (12), yields

$$\begin{aligned} \varPsi _{i}\bigl[g_{i}(x)\bigr]\succeq 0. \end{aligned}$$

From (7)–(13) it follows that

$$\begin{aligned} b_{0}\bigl(x,x^{\ast }\bigr)\varPsi _{0}\bigl[F(x) \ominus _{gH}F\bigl(x^{\ast }\bigr)\bigr] \succeq &\eta ^{t}\bigl(x,x^{\ast }\bigr)\nabla F\bigl(x^{\ast }\bigr) \\ =&-\eta ^{t}\bigl(x,x^{\ast }\bigr) \sum _{i=1} ^{m}y_{i}\nabla g_{i} \bigl(x^{\ast }\bigr) \\ \succeq &\sum_{i=1} ^{m}b_{i} \bigl(x,x^{\ast }\bigr)y_{i}\varPsi _{i} \bigl[g_{i}(x^{ \ast }\bigr] \\ \succeq &0. \end{aligned}$$

From (13) it follows that

$$\begin{aligned} \varPsi _{0}\bigl[F(x)\ominus _{gH}F\bigl(x^{\ast } \bigr)\bigr]\succeq 0. \end{aligned}$$

By (11) we have

$$\begin{aligned} F(x)\ominus _{gH}F\bigl(x^{\ast }\bigr)\succeq 0. \end{aligned}$$

Thus

$$\begin{aligned} F(x)\succeq F\bigl(x^{\ast }\bigr). \end{aligned}$$

Therefore $x^{\ast }$ is an optimal solution of $(P)$. □

Remark 4.1

If we change the condition

$$\begin{aligned} \varPsi _{0}(\mu )\succeq 0 \quad \Rightarrow\quad \mu \succeq 0 \end{aligned}$$

of Theorem 4.1 by

$$\begin{aligned} \varPsi _{0}(\mu )\nprec 0 \quad \Rightarrow\quad \mu \nprec 0, \end{aligned}$$

(14)

then $x^{\ast }$ is a nondominated solution of $(P)$.

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

$$\begin{aligned} \nabla F\bigl(x^{\ast }\bigr)+y^{\ast t}\nabla g \bigl(x^{\ast }\bigr)=0 \end{aligned}$$

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].

Example 4.1

$$\begin{aligned}& \min F(x)=\biggl[\frac{1}{2},\frac{3}{2}\biggr]\sin ^{2}x_{1}+\biggl[\frac{1}{2},\frac{3}{2} \biggr] \sin ^{2}x_{2} \\& \textit{s.t.}\quad g(x)=\biggl[\frac{1}{2},\frac{3}{2}\biggr](\sin x_{1}-1)^{2}+\biggl[\frac{1}{2}, \frac{3}{2} \biggr](\sin x_{2}-1)^{2}\preceq \frac{1}{4}\biggl[ \frac{1}{2}, \frac{3}{2}\biggr], \\& \hphantom{\mbox{s.t.}\quad} x_{1},x_{2}\in \biggl(0,\frac{\pi }{2} \biggr). \end{aligned}$$

We can observe that $F(x)$ is weakly differentiable, H-differentiable, and gH-differentiable. Since the interval-valued function $F(x)$ is not convex, the method in [6, 33, 36, 37] cannot be used.

The function $F(x)$ is interval-valued univex with respect to

$$\begin{aligned}& \eta (x,y)=\textstyle\begin{cases} (\frac{\sin x_{1}-\sin y_{1}}{\cos y_{1}},\frac{\sin x_{2}-\sin y_{2}}{ \cos y_{2}})^{t}, &(x_{1},x_{2})\geq (y_{1},y_{2}), \\ 0 & \text{otherwise}, \end{cases}\displaystyle \\& b_{0}(x,y)=\textstyle\begin{cases} 1, &(x_{1},x_{2})\geq (y_{1},y_{2}), \\ 0 & \text{otherwise}, \end{cases}\displaystyle \end{aligned}$$

and Ψ is induced by $\varPhi (a)=2a$, $b_{1}(x,y)=b_{0}(x,y)$, and $\varPsi _{1}$ is induced by $\varPhi _{1}(a)=|a|$, where $x=(x_{1},x_{2})^{t}$ and $y=(y_{1},y_{2})^{t}$. The point $x^{\ast }=(\sin ^{-1}(1-\frac{1}{2 \sqrt{2}}),\sin ^{-1}(1-\frac{1}{2\sqrt{2}}))^{t}$ is a feasible solution. We can also see that $(F,g)$ satisfies the hypotheses of Theorem 4.1. Therefore $x^{\ast }=(\sin ^{-1}(1-\frac{1}{2\sqrt{2}}), \sin ^{-1}(1-\frac{1}{2\sqrt{2}}))^{t}$ is an optimal solution.

Theorem 4.2

Let $x^{\ast }$ be P-feasible. Suppose that:

• (i) there exist η, $\varPsi _{0}$, $b_{0}$, $\varPsi _{i}$, $b_{i}$, $i=1, 2 , \ldots ,m$, such that

  $$\begin{aligned}& b_{0}(x,y)\varPsi _{0}\bigl[F(x)\ominus _{gH}F\bigl(x^{\ast }\bigr)\bigr]\succeq \eta ^{t} \bigl(x,x ^{\ast }\bigr)\nabla F\bigl(x^{\ast }\bigr) \end{aligned}$$

  (15)

  and

  $$\begin{aligned}& -b_{i}\bigl(x,x^{\ast }\bigr)\varPsi _{i}\bigl[g_{i}\bigl(x^{\ast }\bigr)\bigr]\succeq \eta ^{t}\bigl(x,x^{ \ast }\bigr)\nabla g_{i} \bigl(x^{\ast }\bigr) \end{aligned}$$

  (16)

  for all feasible x;

• (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)
  $$\begin{aligned}& y^{\ast }\geq 0. \end{aligned}$$

  (18)

  Further, suppose that

  $$\begin{aligned}& \varPsi _{0}(\mu )\nprec 0 \quad \Rightarrow \quad \mu \nprec 0, \end{aligned}$$

  (19)
  $$\begin{aligned}& \mu \preceq 0 \quad \Rightarrow \quad \varPsi _{i}( \mu )\succeq 0, \end{aligned}$$

  (20)

  and

  $$\begin{aligned}& b_{0}\bigl(x,x^{\ast }\bigr)> 0,\qquad b_{i}\bigl(x,x^{\ast }\bigr)\geq 0 \end{aligned}$$

  (21)

  for all feasible x. Then $x^{\ast }$ is a nondominated solution of $(P)$.

Proof

Let x be P-feasible. Then

$$\begin{aligned}& \widetilde{g}(x)\preceq 0. \end{aligned}$$

From (20) we conclude

$$\begin{aligned}& \varPsi _{i}\bigl[g_{i}(x)\bigr]\succeq 0. \end{aligned}$$

From (15), (16) it follows that

$$\begin{aligned}& b_{0}(x,y)\bigl\{ \varPsi _{0}\bigl[F(x)\ominus _{gH}F\bigl(x^{\ast }\bigr)\bigr]\bigr\} ^{L}\geq \bigl\{ \eta ^{t}\bigl(x,x^{\ast }\bigr)\nabla F\bigl(x^{\ast } \bigr)\bigr\} ^{L}, \\& b_{0}(x,y)\bigl\{ \varPsi _{0}\bigl[F(x)\ominus _{g}F\bigl(x^{\ast }\bigr)\bigr]\bigr\} ^{U}\geq \bigl\{ \eta ^{t}\bigl(x,x^{\ast }\bigr)\nabla F\bigl(x^{\ast } \bigr)\bigr\} ^{U}, \end{aligned}$$

and

$$\begin{aligned}& b_{i}\bigl(x,x^{\ast }\bigr)\bigl\{ \varPsi _{i} \bigl[g_{i}\bigl(x^{\ast }\bigr)\bigr]\bigr\} ^{L}\leq \bigl\{ -\eta ^{t}\bigl(x,x ^{\ast }\bigr)\nabla g_{i} \bigl(x^{\ast }\bigr)\bigr\} ^{L}, \\& b_{i}\bigl(x,x^{\ast }\bigr)\bigl\{ \varPsi _{i} \bigl[g_{i}\bigl(x^{\ast }\bigr)\bigr]\bigr\} ^{U}\leq \bigl\{ -\eta ^{t}\bigl(x,x ^{\ast }\bigr)\nabla g_{i} \bigl(x^{\ast }\bigr)\bigr\} ^{U}. \end{aligned}$$

Since

$$\begin{aligned} \eta ^{t}\bigl(x,x^{\ast }\bigr)\nabla F\bigl(x^{\ast } \bigr) =&\eta ^{t}\bigl(x,x^{\ast }\bigr)\bigl[\bigl\{ \nabla F \bigl(x^{\ast }\bigr)\bigr\} ^{L},\bigl\{ \nabla F \bigl(x^{\ast }\bigr)\bigr\} ^{U}\bigr]] \\ =&\textstyle\begin{cases} [\eta ^{t}(x,x^{\ast })\{\nabla F(x^{\ast })\}^{L},\eta ^{t}(x,x^{ \ast })\{\nabla F(x^{\ast })\}^{U}], &\eta ^{t}(x,x^{\ast })\geq 0, \\ [\eta ^{t}(x,x^{\ast })\{\nabla F(x^{\ast })\}^{U},\eta ^{t}(x,x^{ \ast })\{\nabla F(x^{\ast })\}^{L}], &\eta ^{t}(x,x^{\ast })< 0, \end{cases}\displaystyle \end{aligned}$$

and

$$\begin{aligned} -\eta ^{t}\bigl(x,x^{\ast }\bigr)\nabla g_{i} \bigl(x^{\ast }\bigr) =&-\eta ^{t}\bigl(x,x^{\ast }\bigr) \bigl[ \bigl\{ \nabla g_{i}\bigl(x^{\ast }\bigr)\bigr\} ^{L},\bigl\{ \nabla g_{i}\bigl(x^{\ast }\bigr)\bigr\} ^{U}\bigr]] \\ =&\textstyle\begin{cases} [\eta ^{t}(x,x^{\ast })\{-\nabla g_{i}(x^{\ast })\}^{U},\eta ^{t}(x,x ^{\ast })\{-\nabla g_{i}(x^{\ast })\}^{L}], &\eta ^{t}(x,x^{\ast }) \geq 0, \\ [\eta ^{t}(x,x^{\ast })\{-\nabla g_{i}(x^{\ast })\}^{L},\eta ^{t}(x,x ^{\ast })\{-\nabla g_{i}(x^{\ast })\}^{U}], &\eta ^{t}(x,x^{\ast })< 0, \end{cases}\displaystyle \end{aligned}$$

we consider the following two cases.

Case (i)

$$\begin{aligned}& \bigl\{ \eta ^{t}\bigl(x,x^{\ast }\bigr)\nabla F \bigl(x^{\ast }\bigr)\bigr\} ^{L}=\eta ^{t} \bigl(x,x^{\ast }\bigr) \bigl\{ \nabla F\bigl(x^{\ast }\bigr)\bigr\} ^{L} \end{aligned}$$

and

$$\begin{aligned}& \bigl\{ -\eta ^{t}\bigl(x,x^{\ast }\bigr)\nabla g_{i} \bigl(x^{\ast }\bigr)\bigr\} ^{L}=\eta ^{t}\bigl(x,x ^{\ast }\bigr)\bigl\{ -\nabla g_{i}\bigl(x^{\ast }\bigr) \bigr\} ^{U} \end{aligned}$$

yield

$$\begin{aligned}& \bigl\{ \eta ^{t}\bigl(x,x^{\ast }\bigr)\nabla F \bigl(x^{\ast }\bigr)\bigr\} ^{U}=\eta ^{t} \bigl(x,x^{\ast }\bigr) \bigl\{ \nabla F\bigl(x^{\ast }\bigr)\bigr\} ^{U} \end{aligned}$$

and

$$\begin{aligned}& \bigl\{ -\eta ^{t}\bigl(x,x^{\ast }\bigr)\nabla g_{i} \bigl(x^{\ast }\bigr)\bigr\} ^{U}=\eta ^{t}\bigl(x,x ^{\ast }\bigr)\bigl\{ -\nabla g_{i}\bigl(x^{\ast }\bigr) \bigr\} ^{L}. \end{aligned}$$

Thus

$$\begin{aligned} b_{0}(x,y)\bigl\{ \varPsi _{0}\bigl[F(x)\ominus _{gH}F\bigl(x^{\ast }\bigr)\bigr]\bigr\} ^{L} \geq & \bigl\{ \eta ^{t}\bigl(x,x^{\ast }\bigr)\nabla F \bigl(x^{\ast }\bigr)\bigr\} ^{L} \\ =&\eta ^{t}\bigl(x,x^{\ast }\bigr)\bigl\{ \nabla F \bigl(x^{\ast }\bigr)\bigr\} ^{L} \\ =&\eta ^{t}\bigl(x,x^{\ast }\bigr)\bigl\{ -y^{\ast t} \nabla g\bigl(x^{\ast }\bigr)\bigr\} ^{L} \\ \geq &\sum_{i=1} ^{m}b_{i} \bigl(x,x^{\ast }\bigr)y_{i}\bigl\{ \varPsi _{i} \bigl[g_{i}\bigl(x^{ \ast }\bigr)\bigr]\bigr\} ^{L} \\ \geq &0. \end{aligned}$$

From (21) it follows that

$$\begin{aligned}& \varPsi _{0}\bigl[F(x)\ominus _{gH}F\bigl(x^{\ast } \bigr)\bigr]\succeq 0. \end{aligned}$$

Then

$$\begin{aligned}& F(x)\ominus _{gH}F\bigl(x^{\ast }\bigr)\nprec 0, \end{aligned}$$

and thus

$$\begin{aligned}& F(x)\nprec F\bigl(x^{\ast }\bigr). \end{aligned}$$

Therefore $x^{\ast }$ is a nondominated solution of $(P)$.

Case (ii)

$$\begin{aligned}& \bigl\{ \eta ^{t}\bigl(x,x^{\ast }\bigr)\nabla F \bigl(x^{\ast }\bigr)\bigr\} ^{L}=\eta ^{t} \bigl(x,x^{\ast }\bigr) \bigl\{ \nabla F\bigl(x^{\ast }\bigr)\bigr\} ^{U} \end{aligned}$$

and

$$\begin{aligned}& \bigl\{ -\eta ^{t}\bigl(x,x^{\ast }\bigr)\nabla g_{i} \bigl(x^{\ast }\bigr)\bigr\} ^{L}=\eta ^{t}\bigl(x,x ^{\ast }\bigr)\bigl\{ -\nabla g_{i}\bigl(x^{\ast }\bigr) \bigr\} ^{L} \end{aligned}$$

yield

$$\begin{aligned}& \bigl\{ \eta ^{t}\bigl(x,x^{\ast }\bigr)\nabla F \bigl(x^{\ast }\bigr)\bigr\} ^{U}=\eta ^{t} \bigl(x,x^{\ast }\bigr) \bigl\{ \nabla F\bigl(x^{\ast }\bigr)\bigr\} ^{L} \end{aligned}$$

and

$$\begin{aligned}& \bigl\{ -\eta ^{t}\bigl(x,x^{\ast }\bigr)\nabla g_{i} \bigl(x^{\ast }\bigr)\bigr\} ^{U}=\eta ^{t}\bigl(x,x ^{\ast }\bigr)\bigl\{ -\nabla g_{i}\bigl(x^{\ast }\bigr) \bigr\} ^{U}. \end{aligned}$$

Thus

$$\begin{aligned} b_{0}(x,y)\bigl\{ \varPsi _{0}\bigl[F(x)\ominus _{gH}F\bigl(x^{\ast }\bigr)\bigr]\bigr\} ^{U} \geq & \bigl\{ \eta ^{t}\bigl(x,x^{\ast }\bigr)\nabla F \bigl(x^{\ast }\bigr)\bigr\} ^{U} \\ =&\eta ^{t}\bigl(x,x^{\ast }\bigr)\bigl\{ \nabla F \bigl(x^{\ast }\bigr)\bigr\} ^{L} \\ =&\eta ^{t}\bigl(x,x^{\ast }\bigr)\bigl\{ -y^{\ast t} \nabla g\bigl(x^{\ast }\bigr)\bigr\} ^{L} \\ \geq &\sum_{i=1} ^{m}b_{i} \bigl(x,x^{\ast }\bigr)y_{i}\bigl\{ \varPsi _{i} \bigl[g_{i}\bigl(x^{ \ast }\bigr)\bigr]\bigr\} ^{L} \\ \geq &0, \end{aligned}$$

From (21) it follows that

$$\begin{aligned}& \varPsi _{0}\bigl[F(x)\ominus _{gH}F\bigl(x^{\ast } \bigr)\bigr]\nprec 0. \end{aligned}$$

Then

$$\begin{aligned}& F(x)\ominus _{gH}F\bigl(x^{\ast }\bigr)\nprec 0, \end{aligned}$$

and thus

$$\begin{aligned}& F(x)\nprec F\bigl(x^{\ast }\bigr). \end{aligned}$$

Therefore $x^{\ast }$ is a nondominated solution of $(P)$. □

Theorem 4.3

Let $x^{\ast }$ be P-feasible. Suppose that:

• (i) there exist η, $\varPsi _{0}$, $b_{0}$, $\varPsi _{i}$, $b_{i}$, $i=1, 2 , \ldots ,m$, such that

  $$\begin{aligned}& b_{0}(x,y)\varPsi _{0}\bigl[F(x)\ominus _{gH}F\bigl(x^{\ast }\bigr)\bigr]\succeq \eta ^{t} \bigl(x,x ^{\ast }\bigr)\nabla F\bigl(x^{\ast }\bigr) \end{aligned}$$

  (22)

  and

  $$\begin{aligned}& -b_{i}\bigl(x,x^{\ast }\bigr)\varPsi _{i}\bigl[g_{i}\bigl(x^{\ast }\bigr)\bigr]\succeq \eta ^{t}\bigl(x,x^{ \ast }\bigr)\nabla g_{i} \bigl(x^{\ast }\bigr) \end{aligned}$$

  (23)

  for all feasible x;

• (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)
  $$\begin{aligned}& y^{\ast }\geq 0. \end{aligned}$$

  (25)

  Further, suppose that

  $$\begin{aligned}& \varPsi _{0}(\mu )\nprec 0 \quad \Rightarrow\quad \mu \nprec 0, \end{aligned}$$

  (26)
  $$\begin{aligned}& \mu \preceq 0 \quad \Rightarrow\quad \varPsi _{i}( \mu )\succeq 0, \end{aligned}$$

  (27)

  and

  $$\begin{aligned}& b_{0}\bigl(x,x^{\ast }\bigr)> 0,\qquad b_{i}\bigl(x,x^{\ast }\bigr)\geq 0 \end{aligned}$$

  (28)

  for all feasible x. Then $x^{\ast }$ is a nondominated solution of $(P)$.

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

Example 4.2

$$\begin{aligned}& \min F(x)=[-1,1] \vert x \vert \\& \textit{s.t.}\quad g(x)=x-1\leq 0. \end{aligned}$$

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.

Note that the objective function $F(x)$ is gH-differentiable on R and that $F^{\prime }(y)=[-1,1]$. Let

$$\begin{aligned} b_{0}(x,y)=\textstyle\begin{cases} 1, &x< y< 0 \text{ or } 0< x< y, \\ 0 &\text{otherwise}. \end{cases}\displaystyle \end{aligned}$$

the function $\varPsi _{0}[a,b]=[a,b]$ is induced by $\varPhi _{0}(a)=a$, and

$$\begin{aligned}& \eta (x,y)=\textstyle\begin{cases} x-y, &x< y< 0 \text{ or } 0< x< y, \\ 0 &\text{otherwise}. \end{cases}\displaystyle \end{aligned}$$

Let $b_{1}=1$, and let $\varPsi _{1}$ be induced by $\varPhi _{1}(a)=|a|$. The point $x^{\ast }=1$ is a feasible solution. We can see that $(F,g)$ satisfies the hypotheses of Theorem 4.2. Therefore $x^{\ast }=1$ is a nondominated solution.

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

Example 4.3

$$\begin{aligned}& \min F(x)=[-2,1]x^{2},\quad x< 0, \\& \textit{s.t.}\quad g(x)=x+1\leq 0. \end{aligned}$$

Then $F(x)$ is gH-differentiable and weakly differentiable. Since $F(x)$ is not LU-convex, the methods of [10] cannot be used, and since $F^{L}(x)+F^{U}(x)=-x^{2}$ is not convex, the methods of [23, 28] cannot be used.

Let

$$\begin{aligned}& b_{0}(x,y)=\textstyle\begin{cases} 1, & x\leq y< 0, \\ \frac{-2y(x-y)}{-x^{2}+y^{2}}, & y< x< 0, \end{cases}\displaystyle \quad \mbox{and} \quad \varPsi _{0}[a,b]=\textstyle\begin{cases} [a,b], & [a,b]\preceq 0, \\ \varPsi ([a,b]), &[a,b]\npreceq 0, \end{cases}\displaystyle \end{aligned}$$

where $\varPsi ([a,b])$ induced by $\varPhi (a)=|a|$, and

$$\begin{aligned} \eta (x,y)=\textstyle\begin{cases} x-y, &x< y< 0 \text{ or } 0< x< y, \\ 0 &\text{otherwise}. \end{cases}\displaystyle \end{aligned}$$

Let $b_{1}(x,y)=1$ and $\varPhi _{1}(a)=\varPhi (a)=|a|$. The point $x^{\ast }=-1$ is a feasible solution. We can see that $(F,g)$ satisfies the hypotheses of Theorem 4.3, and therefore $x^{\ast }=-1$ is a nondominated solution.

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.