On Interval-Valued Pseudolinear Functions and Interval-Valued Pseudolinear Optimization Problems

Some basic characterizations of an interval-valued 
pseudolinear function are derived. By means of the properties of interval-valued 
pseudolinearity, a class of interval-valued pseudolinear programs is considered, 
and the solution set of the interval-valued pseudolinear optimization problem 
is characterized.


Introduction
In recent years, many approaches for interval-valued optimization problems have been explored in considerable details; see examples in [1][2][3].Several authors have been interested in the optimality conditions and duality results for the interval-valued optimization problems.Wu has extended the concept of convexity for real-valued functions to LUconvexity for interval-valued functions, and then he has established the Karush-Kuhn-Tucker (KKT) optimality conditions [4] and duality theory [5] for an optimization problem with an interval-valued objective function under the assumption of LU-convexity.Sun and Wang [6] have derived the Fritz John type and Kuhn-Tucker type necessary and sufficient optimality conditions for nondifferentiable interval-valued programming.Jayswal et al. [7] have discussed Mond-Weir and Wolfe type duality theorems for the interval-valued programming problems under the conditions of generalized convexity.Chalco-Cano et al. [8] have obtained KKT optimality conditions for interval-valued programming problems by using a more general concept of gH-derivative.Bhurjee and Panda [9,10] have introduced the parametric form of interval-valued functions and studied the solution of convex interval-valued programming problems.Jana and Panda [11] have studied the preferable efficient solutions of the problem of interval-valued vector optimization.Zhang et al. [12] have extended the concepts of preinvexity and invexity to intervalvalued functions and derived the KKT optimality conditions for LU-preinvex and invex optimization problems with an interval-valued objective function.
Interval-valued linear optimization [13,14] is a class of important and simple interval-valued optimization problems.Recently, Hladík [15] has proposed a method for testing basis stability of interval-valued linear optimization problems; Hladík's studies have shown that if some basis stability criterion holds true, then the problems become much more easy to solve.In [16], Hladík has discussed lower and upper bound approximation for the best case optimal value of the interval-valued linear optimization problems and proposed an algorithm for computing the best case optimal value.Li et al. [17,18] and Luo et al. [19] have established some necessary and sufficient conditions for checking weak and strong optimality of given feasible solutions for the intervalvalued linear optimization problems.
In this paper, we consider and give some characterizations for a class of generalized interval-valued linear function, which is interval-valued pseudolinear function.Then, by means of the basic properties of interval-valued pseudolinearity, a class of interval-valued pseudolinear programs is considered, and the solution set of the interval-valued pseudolinear optimization problem is characterized.It can be shown that the interval-valued linear optimization problems

Preliminaries
In this section, we recall some basic concepts with regard to interval-valued functions.
Let   be -dimensional Euclidean space, and let   + be its nonnegative orthant.Let us denote by I the class of all closed intervals in . = [  ,   ] ∈ I denotes a closed interval, where   and   mean the lower and upper bounds of , respectively.The closed interval  also can be expressed in terms of a parameter in several ways.Throughout this paper, we consider specific parametric representations of the interval as } be in I, and the interval operations can be performed with respect to parameters [9,10] as follows: . ., .The interval-valued function in parametric form is as follows.

The partial derivatives of F C 𝑘
V at x * may be calculated as follows: If  () (x * ) is continuous in , then The gradient of F C  V at x * is an interval vector: In this paper, we denote the point y + (1 − )x,  ∈ [0, 1], by xy.For the two interval-valued functions of for all  1 ,  2 ∈ [0, 1]  and  1 may or may not be equal to  2 .However, we say that for all  1 =  2 ∈ [0, 1]  ; that is to say,  1 and  2 are the same in both sides.
Definition 2 (see [9]).Suppose that  ⊆   is a convex set and for given which means for all   ,   ∈ [0, 1]  ;   may or may not be equal to which means for all  ∈ [0, 1]  ;  is same in both sides.
It can be shown that F C  V is convex [9] with respect to ⪯  if and only if  () (x) is a convex function on  for every .Similar result does not hold for convexity with respect to ⪯.
For a real-valued differentiable function, which is said to be a pseudolinear function [21][22][23], if it is both pseudoconvex and pseudoconcave, similar to the case of real-valued functions, we can also define the following interval-valued pseudolinear functions.

Definition 7. A differentiable interval-valued function F C 𝑘
V :  → I, on the convex set  ⊆   for given C  V ∈ I  , is  with respect to ⪯  on  if it is both pseudoconvex and pseudoconcave on .
This class of pseudolinear interval-valued functions includes many useful functions.For example, the following function is a pseudolinear interval-valued function.

Characterizations of Pseudolinear Interval-Valued Functions
In this section, we provide some characterizations of the pseudolinear interval-valued function.
Theorem 9. Let a differentiable interval-valued function F C  V :  → I, on the convex set  ⊆   for given C  V ∈ I  .Then, the following statements (i)-(iii) are equivalent.

(i) F C 𝑘
V is pseudolinear over .(ii) For any x and y in , (y − x)

x). (iii) There exists an interval-valued function P C 𝑘
V defined on  ×  such that P C  V (x, y)≻  0 for given C  V ∈ I  and for any x and y in .
Proof.(i)⇒(ii) Suppose that F C  V is pseudolinear intervalvalued function for given C  V ∈ I  on .It can be shown that F C  V (y)=  F C  V (x) implies (y − x)  ∇F C  V (x)=  0.