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Secondorder optimality conditions for intervalvalued functions
Journal of Inequalities and Applications volume 2023, Article number: 139 (2023)
Abstract
This work is included in the search of optimality conditions for solutions to the scalar interval optimization problem, both constrained and unconstrained, by means of secondorder optimality conditions. As it is known, these conditions allow us to reject some candidates to minima that arise from the firstorder conditions. We will define new concepts such as secondorder gHderivative for intervalvalued functions, 2critical points, and 2KKTcritical points. We obtain and present new types of intervalvalued functions, such as 2pseudoinvex, characterized by the property that all their secondorder stationary points are global minima. We extend the optimality criteria to the semiinfinite programming problem and obtain duality theorems. These results represent an improvement in the treatment of optimization problems with intervalvalued functions.
1 Introduction
Convexity has always been a desirable property of a function because it ensures that local optima are global. Pseudoconvexity is also important as this property ensures that stationary points are optimal. This property is shared by invex functions (see Hanson [12]) and makes it possible to design numerical algorithms that find solutions to optimization problems. These functions have been applied to fields such as fractional programming or programming with intervalvalued or fuzzy objective functions (see [1, 22, 23]). In the case of constrained problems, invexity is replaced by KTinvexity [19] to ensure that the KKT stationary points are optimal. Examples of invex functions are \(\theta (x)=\ln(x)\) for \(x>0\) and \(\theta (x)=x^{2}+3\sin^{2}(x)\).
It is also well known that to obtain the optima, firstorder conditions are not enough, and we must resort to secondorder conditions to rule out some of the candidates. An example of this is illustrated in Ginchev and Ivanov [8]:
Example 1
We consider:
The only stationary point \(x=0\) is a global minimum, but this problem cannot be solved with the classical sufficient conditions since our goal is secondorder pseudoconvex but not pseudoconvex.
Authors such as Antczak [4], Luc [18], and Mishra [20] related invexity with secondorder derivatives.
More closely related to our study, Ginchev and Ivanov [8] introduced the parabolic local minimum notion and proved that every parabolic local minimum satisfies the necessary and sufficient conditions for a global minimum with real functions.
In [13] the author utilized secondorder Fréchet differentiable functions in \(R^{n}\) and defined the secondorder KTpseudoconvexity to prove that each KT point of secondorder is a global minimum. Later, Ivanov in [14] generalized the firstorder sufficient optimality conditions from invex functions to secondorder invex ones.
Uncertainty in the coefficients of a programming problem can be trapped through one of these methods:

Assume that they are random variables and have some known distribution function, but sometimes it is difficult to know their shape.

Using robust optimization, which always considers the worstcase scenario of the model but does not consider that a good scenario could occur and therefore may not be appropriate to use.

Using interval analysis through the calculation of the lower and upper extremes of the estimated parameters of historical data.
Since the creation of interval analysis in the 1960s by Ramon Moore [21] to control the imprecision of data, there have been significant advances in the way of working with intervals and in concepts as important as the differentiability of intervalvalued functions, see [5, 35]. Differential calculus is essential for finding solutions by the gradient method or for the Karush–Kuhn–Tucker optimality conditions. In [29] Roy, Panda, and Qiu propose a solution search scheme for an interval optimization problem based on the gradient.
Singh et al., in two papers [33, 34], derived KKT optimality conditions for multiobjective optimization problems in which both objective and constraints are assumed to be intervalvalued functions. So did Jayswal et al. [16, 17] and Ahmad et al. [2, 3] generalizing firstdegree optimality conditions to intervalvalued functions in both the differentiable and nondifferentiable cases. But these conditions do not involve secondorder conditions as we do in this paper.
Also, Daidai [7] presented a generalization of the first and secondorder approximations to optimality conditions for strongly convex functions. The author solved the Euler equation with secondorder approximation data by Newton’s method.
Within the field of intervalvalued functions, in 2015, Osuna et al. [22] using the concept of gHdifferentiability characterized pseudoinvex functions as those in which the stationary and optimal points of unconstrained multiobjective programming problems coincide. In 2017, Osuna et al. [24] generalized the above results to the multiobjective intervalvalued programming problems with constraints. They proved that if this problem is KTpseudoinvex, then every vector interval Karush–Kuhn–Tucker solution is a strictly weakly efficient solution. In 2022, Osuna et al. [26] proved that if \(x^{*}\) is an optimum of a gHdifferentiable intervalvalued function F on \(\mathbb{R}\), then \(0 \in F'(x^{*})\), i.e., 0 belongs to this interval. This simple property opens the way to design algorithms for the search of solutions.
In 2018, RuizGarzón et al. [31] obtained firstorder optimality conditions for the scalar and vector optimization problem on Riemann manifolds but not secondorder conditions, which will be discussed in this article. In 2020, RuizGarzón et al. [30] used Lipschitz functions, and in [32] secondorder optimality conditions were given but not for the intervalvalued functions.
Semiinfinite programming originated in the 1920s in a work by Haar [11] and was named after Charnes et al. [6]. In this field, the systematization work carried out by Goberna and López is noteworthy [9, 10]. In this line of work it is worth mentioning the recent paper by Tung and Tam [37], Upadhyay, Gosh, Mishra, and Treanţă [38, 39] where they study the optimality and duality conditions for multiobjective semiinfinite programming on Hadamard manifolds, obtaining firstorder results but not secondorder results like the ones we deal with in this paper nor with intervalvalued functions.
Motivated by the previous work, our objective is focused on extending the secondorder optimality conditions obtained with real functions to intervalvalued functions. Therefore, we present necessary and sufficient optimality conditions for both unconstrained and constrained scalar interval optimization problems, looking for the function types for which the secondorder critical points and the global minimum points coincide.
To the best of our knowledge, there is no paper to study the secondorder optimality conditions for interval optimization problem. This paper is to make in this direction. Our contributions are the following:

Proposing a new concept of 2invexity, 2pseudoinvexity intervalvalued functions, and 2critical points.

Characterizing the 2PIX functions as those in which 2critical points and minima coincide.

Defining the concept of 2Karush–Kuhn–Tucker stationary point to constrained scalar interval optimization problem.

Proposing when the constrained scalar interval optimization problem can be considered to be 2KKTpseudoinvex.

Analyzing the environmental conditions so that the 2KKT points and the minima coincide.

Establishing weak and strong duality theorems of a dual Mond–Weir type problem.

Extend the optimality criteria from the finite case to the semiinfinite case.
Interval problems capture the uncertainty that classical optimization is not able to capture. The advantages of the interval approach over classical multiobjective optimization is that conventional optimization techniques cannot be applied directly to the interval problem because when working with intervals the lower end point must always be smaller than the upper end point. Furthermore, solving an interval problem based separately on the lower and upper end points of the interval, and not jointly, means that the topological structure of the interval space is not exploited.
One of the fundamental tools for the study of problems with fuzzyvalued functions is their connection with interval problems based on the level sets of a fuzzy interval, which are real intervals. Differentiability concepts developed for fuzzy optimization are often based on differentiability in interval optimization. For example, OsunaGómez et al. (2022) in [25] defined a levelwise gHdifferentiable fuzzy function when \(n \geq 1\), using the results obtained for intervalvalued functions. The derivative in interval space was quasilinear, while in real functions it is linear. An interval function can be gHdifferentiable even though the extremes are not. Therefore, the results established in the article can be used to solve fuzzy optimization problems.
Summary.
Next, we will show the different sections of this article. In Sect. 2, we recall interval arithmetic operations and define the new concept of secondorder gHderivative. In Sect. 3, we define the concepts of 2invex and 2pseudoinvex functions, and 2critical points. We study the relationships between them. Subsequently, we extend some of these concepts to the constrained case. We define the 2KKT stationary points and the 2KKTpseudoinvex problem to identify the conditions under which stationary points and minima coincide. Our challenge is to extend and generalize the classical results in the literature obtained with Fréchet differentiable functions given by Ginchev and Ivanov [8], Ivanov [13, 14], and RuizGarzón [32] to intervalvalued functions. In Sect. 4, we study the weak and strong duality theorems, and in Sect. 5 we extending the optimality criteria to the semiinfinite programming problem.
2 Preliminaries
2.1 Notations for intervals
We denote by \(\mathcal{K}_{C}\) the family of all bounded closed intervals in \(\mathbb{R}\). Let \(A=[a^{L},a^{U}]\) and \(B=[b^{L}, b^{U}]\) be two closed intervals. By definition, we have the sum of two intervals and the product of a scalar by an interval as follows:

(a)
\(A+B=[a^{L}+b^{L}, a^{U}+b^{U}]\) and \(\lambda A= \begin{cases} [ \lambda a^{L},\lambda a^{U} ], & \lambda \geq 0 \\ [\lambda a^{U}, \lambda a^{L} ], & \lambda <0 \end{cases} \), where \(\lambda \in \mathbb{R}\).

(b)
Subtraction of two intervals can also be set broadly. We have that gHdifference between the two intervals [5] is as follows:
$$ A\ominus _{gH} B= \bigl[\mbox{min } \bigl\{ a^{L}b^{L}, a^{U}b^{U} \bigr\} , \mbox{max } \bigl\{ a^{L}b^{L}, a^{U}b^{U} \bigr\} \bigr]. $$
We need to establish an order between the intervals.
Definition 1
Let \(A=[a^{L}, a^{U}]\) and \(B=[b^{L}, b^{U}]\) be two closed intervals in \(\mathbb{R}\). We write:
 •:

\(A\underline{\preceq} B \Leftrightarrow a^{L} \leq b^{L} \mbox{ and } a^{U} \leq b^{U}\);
 •:

\(A\preceq B \Leftrightarrow A\underline{\preceq} B\) and \(A\neq B, \mbox{ i.e.,} a^{L} \leq b^{L}\) and \(a^{U} \leq b^{U}\), with a strict inequality;
 •:

\(A\prec B \Leftrightarrow a^{L} < b^{L} \mbox{ and } a^{U}< b^{U}\).
The function \(f:(a,b) \rightarrow \mathcal{K}_{C} \) is called an intervalvalued function, i.e., \(f(x)\) is a closed interval in \(\mathbb{R}\) for each \(x \in \mathbb{R}\). We will denote \(f(x)=[f^{L}(x),f^{U}(x)]\), where \(f^{L}\) and \(f^{U}\) are realvalued functions and satisfy \(f^{L}(x) \leq f^{U}(x)\) for every \(x\in \mathbb{R}\).
Definition 2
([35], Definition 10)
Let \(u \in (a,b)\) and h be such that \(\bar{x}+h \in (a,b)\). Then the gHderivative of a function \(f: (a,b) \to \mathcal{K}_{C}\) is defined as
if the limit exists. The interval \(f'_{gH}(\bar{x}) \in \mathcal{K}_{C}\) satisfying (1) is called the generalized Hukuhara derivative of f (gHderivative for short) at x̄.
The gHderivative can be easily calculated as follows.
Lemma 1
[5, 28] Let \(D \subseteq \mathbb{R}\) be a nonempty open set and let \(f: D \to \mathcal{K}_{C}\) be a continuous function. Then f is gHderivative at \(\bar{x}\in D\) if and only if \(f^{L}\) and \(f^{U}\) have derivative at x̄. Furthermore, we have
where \(f^{\prime \,L}(\bar{x})\) and \(f^{\prime \,U}(\bar{x})\) are the derivatives of \(f^{L}\) and \(f^{U}\) at x̄, respectively.
Example 2
Suppose that \(S=\{ x\in \mathbb{R}, x \geq 1 \}\) and \(f: D\subseteq S \to \mathcal{K}_{C} \) is defined by
We have that
Thus, Stefanini et al. [36] proposed the following definition of secondorder derivative, which plays a central role in our study.
Definition 3
Let \(f:(a,b) \to \mathcal{K}_{C}\) be gHdifferentiable on (a,b) and \(\bar{x} \in (a,b)\) and h be such that \(\bar{x}+h \in (a,b)\). The secondorder gHderivative of a function \(f(x)\) at x̄ is defined as
if the limit exists. The interval \(f''_{gH}(\bar{x}) \in \mathcal{K}_{C}\) satisfying (2) is called the secondorder generalized Hukuhara derivative of f (secondorder gHderivative for short) at x̄.
With all the tools defined above, we are finally in a position where we can start to look for the necessary and sufficient secondorder conditions in different contexts.
3 Applications to scalar interval optimization problems
In this section, we characterize the functions whose critical points are global optimums in the context of scalar interval optimization problems.
3.1 Unconstrained case
We start by considering the unconstrained scalar interval optimization problem:
where \(f: (a,b)\subseteq \mathbb{R} \rightarrow \mathcal{K}_{C}\) is a differentiable function.
In [15] Jayswal et al. introduced invexity, pseudoinvexity, and quasiinvexity function concepts for intervalvalued functions.
Definition 4
A differentiable \(f: (a,b)\subseteq \mathbb{R} \rightarrow \mathcal{K}_{C}\) function is said to be an invex (IX) at \(\bar{x} \in (a,b)\) with respect to \(\eta :\mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R} \) if there exists \(\eta (x,\bar{x})\in \mathbb{R}\) nonidentically zero, such that \(\forall x \in (a,b)\),
Definition 5
Let \(f: (a,b)\subseteq \mathbb{R} \rightarrow \mathcal{K}_{C}\) be a differentiable function. Then f is said to be pseudoinvex at \(\bar{x} \in (a,b)\) with respect to \(\eta :\mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R} \) if there exists \(\eta (x,\bar{x}) \in \mathbb{R}\) such that \(\forall x \in (a,b)\),
Definition 6
Let \(f: (a,b)\subseteq \mathbb{R} \rightarrow \mathcal{K}_{C}\) be a differentiable function. Then f is said to be quasiinvex at \(\bar{x} \in (a,b)\) with respect to \(\eta :\mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R} \) if there exists \(\eta (x,\bar{x}) \in \mathbb{R}\) such that \(\forall x \in (a,b)\),
Example 3
[15] Let us consider that \(f: \mathbb{R}+ \rightarrow \mathcal{K}_{C} \), defined by \(f(x)=[3,4]x+[1,5]x^{3}\), is pseudoinvex and quasiinvex with respect to \(\eta (x,y)=x^{2}y^{2}\) but not an invex function.
We are interested in characterizing intervalvalued functions where the stationary and optimal points coincide.
Thus, we will now propose a generalization of the concept of 2invexity given by Ivanov [14] for Fréchet differentiable functions in dimensional finite Euclidean space to intervalvalued functions.
Definition 7
A secondorder differentiable \(f: (a,b)\subseteq \mathbb{R} \rightarrow \mathcal{K}_{C}\) function is said to be a 2invex (2IX) at \(\bar{x} \in (a,b)\) with respect to \(\eta ,\xi :\mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R} \) if there exist \(\eta (x,\bar{x})\in \mathbb{R}\) nonidentically zero, \(\xi (x,\bar{x}) \in \mathbb{R}\) such that \(\forall x \in (a,b)\),
Remark 1
Note that each invex function is also a 2invex function.
Example 4
Let us consider that \(f: \mathbb{R}_{+} \rightarrow \mathcal{K}_{C} \), defined by \(f(x)=[3,4]x+[1,5]x^{3}\), is not an invex function with respect to \(\eta (x,y)=x^{2}y^{2}\) but is a 2invex function.
We can define a new concept of secondorder stationary point for the intervalvalued functions.
Definition 8
Suppose that the function \(f:(a,b)\subseteq \mathbb{R}\rightarrow \mathcal{K}_{C}\) is secondorder differentiable at any \(\bar{x}\in (a,b)\). A feasible point x̄ for SIOP is said to be a 2critical point (2CP) if
Thus, we can propose and prove the following theorem that characterizes the concept of 2invex functions.
Theorem 2
Let \(f:(a,b)\subseteq \mathbb{R}\rightarrow \mathcal{K}_{C}\) be a secondorder differentiable function at any \(\bar{x}\in (a,b)\). The function f is 2invex at \(\bar{x} \in (a,b)\) with respect to \(\eta ,\xi :\mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R} \) if and only if each 2CP is a global minimum of f on \((a,b)\).
Proof
We will argue by contradiction. Suppose that f is 2invex at \(\bar{x}\in (a,b)\) with respect to \(\eta , \xi :\mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R} \) and x̄ is a 2CP but it is not a global minimum. Thus it follows that \(0 \in f'_{gH}(\bar{x})\). Furthermore, \(f^{\prime\prime \,L}_{gH}(\bar{x}) \geq 0\), and there is \(x\in (a,b)\) with \(f(x)\prec f(\bar{x})\).
By the 2invexity of f with respect to η and ξ, there exist \(\eta (x,\bar{x})\in \mathbb{R}\) and \(\xi (x,\bar{x}) \in \mathbb{R}\) such that
and then \(f^{\prime\prime \,U}_{gH}(\bar{x})<0 \), and therefore \(f^{\prime\prime \,L}_{gH}(\bar{x})<0\), which is a contradiction.
Now, we will prove the sufficient condition for 2invexity.
Suppose that each 2CP x̄ is a global minimum, but f is not 2invex at \(\bar{x} \in (a,b)\) with respect to \(\eta ,\xi :\mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R} \). Then there exist \(\eta (x,\bar{x})\in \mathbb{R}\) nonidentically zero and \(\xi (x,\bar{x}) \in \mathbb{R}\) such that
Then we choose \(\xi (x,\bar{x})=0\) and \(\eta (x,\bar{x})=tf'_{gH}(\bar{x})\), where t is arbitrary positive real. Thus, it follows that
which contradicts with the base assumption made of x̄ being a global minimum. □
This result extends Theorem 2.6 given by Ivanov [14] and Theorem 2 given by RuizGarzón et al. [32] of scalar functions to intervalvalued functions.
Example 5
Let us consider \(f: \mathbb{R}\rightarrow \mathcal{K}_{C} \) defined by \(f(x)=[x^{4},e^{x^{2}}]\). We can calculate:
The point \(\bar{x}=0\) for SIOP is a 2critical point (2CP) because
and a global minimum of f, then, by Theorem 2, the function f is 2invex at x̄.
We will now define the new concept of the 2pseudoinvex function for the intervalvalued functions.
Definition 9
Let \(f:(a,b)\subseteq \mathbb{R}\rightarrow \mathcal{K}_{C}\) be a secondorder differentiable function at any \(\bar{x}\in (a,b)\). A differentiable f function is said to be a 2pseudoinvex (2PIX) at \(\bar{x} \in (a,b)\) with respect to \(\eta ,\xi :\mathbb{R}\times \mathbb{R} \rightarrow \mathbb{R} \) if there exists \(\eta (x,\bar{x}) \in \mathbb{R}\) nonidentically zero such that
We will now discuss when the 2PIX and 2IX functions coincide.
Theorem 3
Suppose that:

(a)
\(f:(a,b)\subseteq \mathbb{R} \rightarrow \mathcal{K}_{C}\) is secondorder gHdifferentiable at every \(\bar{x}\in (a,b)\);

(b)
\(x\in (a,b)\) such that if \(f(x)\prec f(\bar{x})\) then \(0\in f'_{gH}(\bar{x})\).
If f is a 2pseudoinvex function at \(\bar{x} \in S_{1}\), then f is also 2invex at \(\bar{x} \in S_{1}\).
Proof
Let \(\bar{x},x\in (a,b)\) be two points such that \(f(x)\prec f(\bar{x})\).
If \(f'_{gH}(\bar{x})(\eta (x,\bar{x}))\prec [0, 0]\), then the inequality
is ensured with \(\xi (x,\bar{x})=0\).
If \(0 \in f'_{gH}(\bar{x})\), then inequality (8) holds since f is 2pseudoinvex with respect to \(\eta ,\xi \).
Inequality (8) implies the 2invexity of f since if f is not a 2invex function then there exist \(\eta (x,\bar{x}),\xi (x,\bar{x}) \in \mathbb{R}\) such that
a contradiction with all x̄ being a minimum. □
We can move forward and get the following result.
Corollary 4
Suppose that:

(a)
\(f:(a,b)\subseteq \mathbb{R} \rightarrow \mathcal{K}_{C}\) is secondorder gHdifferentiable at every \(\bar{x}\in (a,b)\);

(b)
\(x\in (a,b)\) such that if \(f(x)\prec f(\bar{x})\) then \(0\in f'_{gH}(\bar{x})\).
The function f is 2pseudoinvex at \(\bar{x} \in S_{1}\) if and only if each 2CP is a global minimum of f on \(S_{1}\).
Proof
On the one hand, from Theorem 3, the 2pseudoinvexity implies the 2invexity. On the other hand, the 2invexity implies the 2pseudoinvexity. Together with Theorem 2, we obtain our assertion. □
The previous corollary extends the results obtained by Ivanov, Theorems 2.12 and 2.14 in [14], by RuizGarzón et al. [32], Corollary 1, from an environment of convexity and scalar functions to a more general environment of invexity and intervalvalued functions. The 2PIX functions are characterized by the fact that the minimum points and the 2CP points coincide.
3.2 Constrained case
In this section, we consider the constrained scalar interval optimization problem of the form:
where \(f:(a,b)\subseteq \mathbb{R} \rightarrow \mathcal{K}_{C}\), \(g_{j}:(a,b)\subseteq \mathbb{R}\rightarrow \mathbb{R}, j=1,2, \ldots ,m\), is a set of differentiable functions. Hence, let us consider
and let \(I(x)\) be the set of active constraints.
Similar to the unconstrained case, our aim is to find the kind of functions for which the Karush–Kuhn–Tucker points and the optimums coincide.
Definition 10
Suppose that the functions \(f:(a,b)\subseteq \mathbb{R} \rightarrow \mathcal{K}_{C}\), \(g_{j}:(a,b)\subseteq \mathbb{R}\rightarrow \mathbb{R}, j=1,2, \ldots ,m\), are secondorder differentiable at any \(\bar{x}\in S_{1}\). A feasible point x̄ for CIOP is said to be a 2Karush–Kuhn–Tucker stationary point (in short, 2KKT point) if there exist nonnegative multipliers \(\lambda =(\lambda ^{L}, \lambda ^{U}), \mu _{1},\ldots , \mu _{m}\) with \((\lambda , \mu ) \neq (0,0)\) such that
where \(L=\lambda f +\sum_{j=1}^{n} \mu _{j}g_{j}\) is the Lagrange function.
These conditions are an extension of the classical KKT conditions by two further conditions.
In this section, as in the previous section, our aim to analyze the conditions under which the KKT points and minima coincide.
In the following, our focus is to extend the kind of KTinvex functions created by Martin [19] and other later ones introduced by Osuna et al. [27] to generalized invexity intervalvalued functions. To do so, let us set the following definitions.
Definition 11
The CIOP problem is said to be 2KKTpseudoinvex (2KKTPIX) with respect to \(\eta :\mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R} \) and for all feasible points \(x,\bar{x}\) for CIOP, if \(f(x) \ominus _{gH} f(\bar{x})\prec [0, 0]\) then
where \(I(\bar{x})=\{j=1,\dots ,m:g_{j}(\bar{x})=0\}\).
Remark 2
In [24] OsunaGómez et al. defined a multiobjective interval optimization problem (MIVOP) as KTpseudoinvexI problem if expressions (14) and (16) hold.
We now can obtain the sufficient condition for global optimality.
Theorem 5
Suppose that the functions \(f:(a,b)\subseteq \mathbb{R} \rightarrow \mathcal{K}_{C}\), \(g_{j}:(a,b)\subseteq \mathbb{R}\rightarrow \mathbb{R}, j=1,2, \ldots ,m\), are secondorder differentiable at any \(\bar{x}\in S_{1}\). Furthermore, assume that the constrained scalar interval optimization problem CIOP is 2KKT pseudoinvex problem with respect to η. Then each 2KKT point is a global minimum.
Proof
Suppose that the constrained scalar interval optimization problem CIOP is a 2KKT pseudoinvex problem with respect to η and that \(\bar{x}\in S_{1}\) is a 2KKT point, and we need to prove that x̄ is a global minimum. By reductio ad absurdum, let us assume the opposite, and thus, that there is \(x\in S_{1}\) with \(f(x)\prec f(\bar{x})\). By 2KKTPIX we have that \(f'_{gH}(\bar{x})(\eta (x,\bar{x}))\underline{\preceq} [0, 0]\) and \(g'_{j}(\bar{x})(\eta (y,\bar{x})) \leq 0, j\in I(\bar{x})\).
Since x is a 2KKT stationary point, there exist \(\lambda >0\) and \(\mu _{j} \geq 0, j \in I(\bar{x})\) such that expressions (9)–(13) hold. Then we conclude from \(L'_{\bar{x}}(\eta (x,\bar{x}))=0\) that
such that \(\mu _{j}>0\).
From the 2KKTPIX with respect to η of CIOP, we have that \(f''_{gH}({\bar{x}})(\xi (x,\bar{x})) \prec [0, 0]\) and \(g''_{j}(\bar{x})(\eta (x,\bar{x})) \leq 0\) for all \(j\in I(\bar{x})\) with \(\mu _{j}\geq 0\). Thus, on the one hand, we get \(L''_{\bar{x}} (\eta (x,\bar{x}))\prec [0, 0]\), and on the other hand, we get expression (13), a contradiction. □
Remark 3
OsunaGómez et al. [24] proved that if MIVOP is a KTpseudoinvexI problem, then every interval KKT solution is a strictly weakly efficient solution.
We now can obtain the sufficient condition for global optimality.
Theorem 6
Suppose that:

(a)
The functions \(f:(a,b)\subseteq \mathbb{R} \rightarrow \mathcal{K}_{C}\), \(g_{j}:(a,b)\subseteq \mathbb{R}\rightarrow \mathbb{R}, j=1,2, \ldots ,m\), are secondorder differentiable at any \(\bar{x}\in S_{1}\);

(b)
The functions \(f:(a,b)\subseteq \mathbb{R} \rightarrow \mathcal{K}_{C}\), \(g_{j}:(a,b)\subseteq \mathbb{R}\rightarrow \mathbb{R}, j=1,2, \ldots ,m\), are quasiinvex differentiable at \(\bar{x}\in S_{1}\) with respect to η nonidentically zero.
If each 2KKT point is a global minimum, then the problem CIOP is 2KKT pseudoinvex.
Proof
Suppose that each 2KKT stationary point is a global minimum, we will prove that CIOP is 2KKT pseudoinvex. Given two \(x,\bar{x}\in S_{1}\) points with
According to the quasiinvexity of f at \(\bar{x}\in S_{1}\) with respect to η, the expression \(f'_{gH}(\bar{x}) (\eta (x,\bar{x})) \underline{\preceq} [0, 0]\) holds.
If \(0\in f'_{gH}(\bar{x}) \), we can prove that \(f''_{gH}({\bar{x}})(\xi (x,\bar{x})) \prec [0, 0]\).
By reductio ad absurdum, suppose that \(f''_{gH}({\bar{x}})(\eta (x,\bar{x})) \succeq [0, 0]\), then x̄ is a 2KKT point, which implies, by the hypothesis, that x̄ is a global minimum, which is in contradiction with expression (18).
We need to prove that \(g'_{j}(\bar{x})(\eta (x,\bar{x}))\leq 0\) and the expression
But it follows directly from the assumption \(x\in S_{1}, j\in I(\bar{x})\) and the quasiinvexity of \(g_{j}\) at \(\bar{x}\in S_{1}\) with respect to η.
In conclusion, all this shows that equations (14)–(17) hold, and then the problem CIOP is 2KKT pseudoinvex with respect to η. □
And we obtain the following corollary.
Corollary 7
Suppose that:

(a)
The functions \(f:(a,b)\subseteq \mathbb{R} \rightarrow \mathcal{K}_{C}\), \(g_{j}:(a,b)\subseteq \mathbb{R}\rightarrow \mathbb{R}, j=1,2, \ldots ,m\), are secondorder differentiable at any \(\bar{x}\in S_{1}\).

(b)
The functions \(f:(a,b)\subseteq \mathbb{R} \rightarrow \mathcal{K}_{C}\), \(g_{j}:(a,b)\subseteq \mathbb{R}\rightarrow \mathbb{R}, j=1,2, \ldots ,m\), are quasiinvex differentiable at \(\bar{x}\in S_{1}\) with respect to η nonidentically zero.
Then, each 2KKT point is a global minimum if and only if the problem CIOP is 2KKT pseudoinvex with respect to η.
Then, in quasiinvexity environments, if the problem is 2KKTPIX, the minima coincide with 2KKTpoints. This result generalizes Ivanov’s [13] Theorems 3.1 and 3.2, and RuizGarzón et al., Corollary 2, [32] of scalar functions to intervalvalued functions.
We now illustrate Corollary 7 with an example.
Example 6
The functions \(f:\mathbb{R} \rightarrow \mathcal{K}_{C}\), \(g:\mathbb{R} \rightarrow \mathbb{R}\) are differentiable and secondorder gHdirectionally differentiable at \(\bar{x}=0\) with respect to \(\eta (x,\bar{x})= (x^{2}\bar{x}^{2}) \in \mathbb{R}\).
Let the constrained scalar interval optimization problem CIOP1 be defined as follows:
We can calculate
We will see that \(\bar{x}=0\) is a 2KKTpoint and a global minimum to the constrained scalar interval optimization problem CIOP1.
where \(L=\lambda f +\sum_{j=1}^{n} \mu _{j}g_{j}\) is the Lagrange function.
Now, according to conditions (19) and (23) of the theorem, we have to solve the following simultaneous equations:
We obtain \(\bar{x}=0\), \(\lambda ^{L}=\lambda ^{U}=1\), \(\mu _{1}=0\).
Since f and g are quasiinvex at \(\bar{x}=0\), according to the previous Corollary 7, the constrained scalar interval optimization problem CIOP is then 2KKTPIX.
4 Duality
We consider the secondorder interval Mond–Weir dual problem of the form:
where \(L''_{u}(\eta (x,u))=\lambda f''_{gH}(u)(\eta (x,u))+\sum_{j\in I(u)} \mu _{j}g''_{j}(u)( \eta (x,u))\). Hence, let us consider \(S_{2}\) the feasible set of \(SOIDP\).
Our objective is to establish weak and strong duality theorems.
Theorem 8
(Weak duality)
Let \(x\in S_{1}\) and \((u, \lambda , \mu ) \in S_{2}\). Suppose that:

(a)
The functions \(f:(a,b)\subseteq \mathbb{R} \rightarrow \mathcal{K}_{C}\), \(g_{j}:(a,b)\subseteq \mathbb{R}\rightarrow \mathbb{R}, j=1,2, \ldots ,m\), are secondorder differentiable at any \(u\in S_{2}\).

(b)
Let SOIDP be 2KKT pseudoinvex problem with respect to η nonidentically zero.
If u is a 2KKT point of SOIDP, then
Proof
By contradiction, \(f(x) \underline{\preceq} \widehat{f}(u)=f(u)\) since u is a 2KKTstationary point
From (26) and (27) it follows that
From the 2KKTpseudoinvexity of \((CIOP)\) it follows that
From (31) and (32) we obtain that
Contradiction with (28). □
Theorem 9
(Strong duality)
Let \(\bar{x}\in S_{1}\) be a 2KKT point of CIOP. Let us assume that the functions \(f:(a,b)\subseteq \mathbb{R} \rightarrow \mathcal{K}_{C}\), \(g_{j}:(a,b)\subseteq \mathbb{R}\rightarrow \mathbb{R}, j=1,2, \ldots ,m\), are secondorder differentiable at any \(\bar{x}\in S_{1}\). Then there exist \(\lambda =(\lambda ^{L}, \lambda ^{U}) \in \mathbb{R}^{2}_{+} \setminus \{0\}, \mu _{j}\in \mathbb{R}_{+}, \textit{with} ( \lambda , \mu ) \neq (0,0)\) such that \((\bar{x}, \lambda , \mu ) \in S_{2}\) and
Further, let the assumptions of Theorem 8hold, then \((\bar{x}, \lambda , \mu )\) is a solution of SOIDP.
Proof
As x̄ is a 2KKT point, there exist \(\lambda =(\lambda ^{L}, \lambda ^{U}) \in \mathbb{R}^{2}_{+} \setminus \{0\}, \mu _{j}\in \mathbb{R}_{+} \mbox{ with } ( \lambda , \mu ) \neq (0,0)\) such that
where \(L=\lambda f +\sum_{j=1}^{n} \mu _{j}g_{j}\) is the Lagrange function.
And therefore, \((\bar{x}, \lambda , \mu ) \in S_{2}\), hence \(f(\bar{x})= \widehat{f}(\bar{x})\).
On the other hand, if \((\bar{x}, \lambda , \mu )\) is not a solution of SOIDP, then there exists \((u, \lambda , \mu ) \in S_{2}\) such that
Contradiction with Theorem 8. □
Example 7
Let us consider the constrained scalar interval optimization problem CIOP1 as defined in Example 6. We denote the feasible set of CIOP1 by \(S_{1}\).
The Mond–Weir dual problem related to CIOP1, denoted by SOIDP1, may be formulated as follows:
where
Hence, let us consider \(S_{2}\) the feasible set of \(SOIDP1\). The point \(\bar{x}=0\in S_{1}\) is a 2KKT point of CIOP1 because conditions (19)–(23) hold, as we have seen in Example 6. Thus, we see that all the assumptions for strong duality of Mond–Weir dual problem are satisfied. Hence, there exist \(\lambda ^{L}=\lambda ^{U}=1\), \(\mu _{1}=0\) such that \((\bar{x}, \lambda , \mu )\) is a feasible point of SOIDP1 and \(f(\bar{x})=\widehat{f}(\bar{x})\).
Now, from Example 6, we see SOIDP1 is a 2KKTPIX problem, then \((\bar{x}, \lambda , \mu )\) is a solution of SOIDP1.
5 Extension: semiinfinite case
Let us consider the semiinfinite interval optimization problem (SIOP) defined as follows:
where \(f:(a,b)\subseteq \mathbb{R} \rightarrow \mathcal{K}_{C}\), \(g_{t}:(a,b)\subseteq \mathbb{R}\rightarrow \mathbb{R}, t\in T\) is a set of differentiable functions. We denote the feasible solution set of (SIOP):
The index set T is an arbitrary nonempty set, not necessarily finite and \(T(\bar{x})=\{t\in T  g_{t}(\bar{x})=0 \}\). The set of active constraint multipliers at \(\bar{x} \in S\) is
As in the finite case, we can define the following.
Definition 12
Suppose that the functions \(f:(a,b)\subseteq \mathbb{R} \rightarrow \mathcal{K}_{C}\), \(g_{t}:(a,b)\subseteq \mathbb{R}\rightarrow \mathbb{R}, t\in T\), are secondorder differentiable at any \(\bar{x}\in S_{2}\). A feasible point x̄ for SIOP is said to be a 2Karush–Kuhn–Tucker stationary point (in short, 2KKT point) if there exist nonnegative multipliers \(\lambda =(\lambda ^{L}, \lambda ^{U}), \mu \in \Lambda (\bar{x})\) with \((\lambda , \mu ) \neq (0,0)\) such that
where \(L=\lambda f +\sum_{t \in T} \mu _{t}g_{t}\) is the Lagrange function.
Definition 13
The SIOP problem is said to be 2KKTpseudoinvex (2KKTPIX) with respect to \(\eta :\mathbb{R}\times \mathbb{R}\rightarrow \mathbb{R} \) and for all feasible points \(x,\bar{x}\) for SIOP, if \(f(x) \ominus _{gH} f(\bar{x})\prec [0, 0]\) then
where \(T(\bar{x})=\{t\in T  g_{t}(\bar{x})=0 \}\).
And we obtain the following corollary.
Corollary 10
Suppose that:

(a)
The functions \(f:(a,b)\subseteq \mathbb{R} \rightarrow \mathcal{K}_{C}\), \(g_{t}:(a,b)\subseteq \mathbb{R}\rightarrow \mathbb{R}, t \in T\), are secondorder differentiable at any \(\bar{x}\in S_{2}\);

(b)
The functions \(f:(a,b)\subseteq \mathbb{R} \rightarrow \mathcal{K}_{C}\), \(g_{t}:(a,b)\subseteq \mathbb{R}\rightarrow \mathbb{R}, t \in T\), are quasiinvex differentiable at \(\bar{x}\in S_{2}\) with respect to η nonidentically zero.
Then each 2KKT point is a global minimum if and only if the problem SIOP is 2KKT pseudoinvex with respect to η.
Proof
The proof of this result follows the lines of the theorems obtained for the finite case. □
We now illustrate the previous corollary 10 with an example.
Example 8
The functions \(f:\mathbb{R} \rightarrow \mathcal{K}_{C}\), \(g:\mathbb{R} \rightarrow \mathbb{R}\) are differentiable and secondorder gHdirectionally differentiable at \(\bar{x}=1\) with respect to \(\eta (x,\bar{x})= (x^{2}\bar{x}^{2}) \in \mathbb{R}\).
Let the semiinfinite scalar interval optimization problem SIOP be defined as follows:
then \(g_{t}(x) \leq 0, \forall t\in T \Leftrightarrow x\in [0,2], S_{2}=[0,2]\).
We can calculate:
We will see that \(\bar{x}=0\) is a 2KKTpoint and a global minimum to the semiinfinite scalar interval optimization problem SIOP.
where \(L=\lambda f +\sum_{t\in T} \mu _{t}g_{t}\) is the Lagrange function.
Now, according to conditions (45) and (49) of the theorem, we have to solve the following simultaneous equations:
We obtain \(\bar{x}=0\), \(\lambda ^{L}=\lambda ^{U}=1\), \(\mu _{1}=0\).
Since f and g are quasiinvex at \(\bar{x}=0\), according to the previous Corollary 10, the semiinfinite scalar interval optimization problem SIOP is then 2KKTPIX. Thus, we have extended the optimality conditions for the finite optimization problem to the semiinfinite case.
6 Conclusions
Throughout this paper, we have obtained secondorder optimality conditions for the scalar interval optimization problems in both the constrained and unconstrained cases. This has been done by extending the notions from the literature of Ginchev and Ivanov [8], Ivanov [13, 14], and RuizGarzón [32] of scalar functions to intervalvalued functions. We have analyzed the precise conditions for the critical or stationary points to coincide with the minimums of the interval optimization problem.
To do so, our work has proposed and made use of:

An adequate secondorder gHdifferential definition of an intervalvalued function.

An extension of concepts such as 2invexity, 2pseudoinvexity, and 2critical points to the case of intervalvalued functions.

Identifying the 2invex with the 2pseudoinvex intervalvalued functions and characterizing them according to the equivalence of 2critical points and minimums.

An adequate 2KKT stationary point and 2KKTpseudoinvex problem for the constrained interval optimization problem has allowed us to identify the conditions under which stationary points and minima coincide.

A dual problem model of the Mond–Weir type, deriving the weak and strong duality theorems related to the primal problem.

A generalization of the optimality conditions for the semiinfinite scalar interval optimization problem SIOP.
One possible way forward may be to implement the theoretical conditions under which such coincidences occur in numerical software to find the minima of the interval optimization problem.
Availability of data and materials
Not applicable.
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The authors wish to thank the referees for their several valuable suggestions, which have considerably improved the presentation of the article, and the Instituto de Desarrollo Social y Sostenible (INDESS) for the facilities provided for the preparation of this work.
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RuizGarzón, G., OsunaGómez, R., RufiánLizana, A. et al. Secondorder optimality conditions for intervalvalued functions. J Inequal Appl 2023, 139 (2023). https://doi.org/10.1186/s13660023030545
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DOI: https://doi.org/10.1186/s13660023030545