On semidifferentiable interval-valued programming problems

*Correspondence: mskklai@outlook.com 1College of Economics, Shenzhen University, Shenzhen, 518060, China Full list of author information is available at the end of the article Abstract In this paper, we consider the semidifferentiable case of an interval-valued minimization problem and establish sufficient optimality conditions and Wolfe type as well as Mond–Weir type duality theorems under semilocal E-preinvex functions. Furthermore, we present saddle-point optimality criteria to relate an optimal solution of the semidifferentiable interval-valued programming problem and a saddle point of the Lagrangian function.


Introduction
The technique of solving optimization problems has wide applications in many research areas. Optimization problems having real coefficients are known as the deterministic optimization problems; however, having random variables with known distributions, they are classified as the stochastic optimization problems, see for instance [1,2]. The specifications of the distributions are more subjective, as many authors invoke the Gaussian distributions for various parameters in the stochastic theory, so it is hard to tackle the large area of real-life problems by these specifications. For resolving such difficulties, intervalvalued optimization problems, where coefficients must be chosen as closed intervals, are preferred for studying uncertainty in this optimization problem.
Paper [3] dealt with two types of solutions for an interval-valued optimization problem and established the Karush-Kuhn-Tucker optimality conditions. In addition to that, many solution concepts in the multiobjective view of interval-valued programming problems were proposed in [4]. Further, [5] presented the concept of a nondominated solution for vector optimization problems and established weak and strong duality results for intervalvalued programming problems in the presence of an interval-valued Lagrangian function and its dual. For more details on solution concepts of interval-valued programming, one can see [6][7][8][9] and the references therein.
The sufficient optimality conditions and duality theorems for Mond-Weir type as well as Wolfe type dual models under generalized invexity assumptions for interval-valued programming problems have been established in [10]. Paper [11] presented the concepts of invexity and preinvexity for interval-valued functions and studied the KKT optimality conditions for interval-valued programming using Hukuhara differentiability. The Mond-Weir type duality theorems and saddle-point optimality conditions for interval-valued programming problems have been derived in [12]. Recently, [13] extended the invexity assumptions for interval-valued functions with the help of generalized Hukuhara differentiability and presented the Kuhn-Tucker optimality conditions. Further, [14] discussed some properties of the univex mappings for interval-valued functions and established sufficient optimality conditions for the nondominated solution.
On the other hand, [15] introduced the concept of semidifferentiable functions and discussed locally star-shaped functions and generalizations of convex functions using semidifferentiability. Further, [16] extended the concept of semidifferentiability to E-ηsemidifferentiability and, using this, introduced (generalized) semilocal E-preinvex functions.
This paper is prepared as follows: in Sect. 2, we give some basic ideas related to interval analysis and semilocal E-preinvex functions. In Sect. 3, we establish sufficient optimality conditions for interval-valued programming using E-η-semidifferentiable and semilocal E-preinvex functions. We give an example to verify our result. In Sect. 4, we propose Wolfe type and Mond-Weir type dual models involving E-η-semidifferentiable functions. Further, we derive weak, strong, and strict converse duality results for the described models. Finally, in the last section, we present relations between an optimal solution of the interval-valued programming problem and a saddle point of the Lagrangian function in case of E-η-semidifferentiability.

Preliminaries
Suppose that J is the set of all closed and bounded intervals in R.
where c L (d L ) and c U (d U ) are respectively the lower and upper bounds of C(D) with c L ≤ c U and d L ≤ d U , we have Now, for any real number μ, we have For further details on interval analysis, one can see [17].
Suppose that R n denotes an n-dimensional Euclidean space, E : R n → R n and η : R n × R n → R n are two fixed mappings.
Remark 2.1 When E becomes an identity map, then the notion of E-η-semidifferentiability is η-semidifferentiability, and for E as an identity map with η(x,x) = x -x, the same is converted into a semidifferentiable function (see [15]).
Lemma 2.1 (see [16]) (i) Suppose that f is (strictly) semilocal E-preinvex and E-ηsemidifferentiable atx ∈ X ⊂ R n , where X is a local E-invex set with respect to η. Then (ii) Suppose that f is pseudo(quasi)-semilocal E-preinvex and E-η-semidifferentiable at x ∈ X ⊂ R n , where X is a local E-invex set with respect to η. Then

Optimality conditions for interval-valued programming problem
Consider the following interval-valued minimization problem: . . , l} be a feasible set of (IVP).

Definition 3.1 ([19])
Letx be a feasible solution of problem (IVP). We say thatx is an LU optimal solution of the problem if there exists no x ∈ P such that Now, we define semilocal E-preinvexity for interval-valued functions as follows.

Lemma 3.1 Suppose that F is an E-η-semidifferentiable interval-valued function. Then F is semilocal E-preinvex with respect to η atx if both real-valued functions F L and F U are semilocal E-preinvex with respect to the same η atx.
Motivated by [19] and [20], we state the Karush-Kuhn-Tucker type necessary conditions for interval-valued programming problems in terms of E-η-semidifferentiable functions.
Suppose thatx is an LU optimal solution to (IVP) and the suitable constraint qualification is satisfied, and all functions F L , F U , and g j are E-ηsemidifferentiable atx. Then there exist scalars w L , w U (> 0) ∈ R, and τ j (≥ 0) ∈ R, j = 1, 2, . . . , l, such that Now, we present some sufficient optimality conditions for (IVP).
Proof Suppose thatx is not an LU optimal solution to (IVP), then there exists a point For w L , w U > 0, we can write Since F is semilocal E-preinvex with respect to η atx, then and Multiplying (3.4) by w L and (3.5) by w U and adding them, we get Using (3.3), the above inequality becomes From (ii) with the feasibility of x to (IVP), we find that Since l j=1 τ j g j is semilocal E-preinvex with respect to η atx, then Using (3.7), the above inequality becomes Adding (3.6) and (3.8), we obtain a contradiction to assumption (i). Hence,x is an LU optimal solution to (IVP).
Example 3.1 Consider the interval-valued programming problem: It is easy to see that F L , F U , and and the map η : R × R → R is defined as follows: The feasible set of the problem is P = {x : -3x + 2 ≤ 0}. Clearly,x = 1 is feasible. Choose another point x = 3 ∈ P. Now, If we choose w L = 1, w U = 1, and τ 1 = 0, then Moreover, functions F and τ 1 g 1 are semilocal E-preinvex atx = 1. Therefore,x = 1 is an LU optimal solution to the given problem. Thus, Theorem 3.2 is verified.
Proof On the contrary, suppose thatx is not an LU optimal solution to (IVP), then there exists a point x ∈ P such that F(x ) ≺ F(x).
Since w L , w U > 0, we can write The above inequality together with the pseudo-semilocal E-preinvexity of w L F L + w U F U with respect to η atx gives Again, from the feasibility of x to (IVP) and by (ii), we have With above inequality, use the fact that l j=1 τ j g j is quasi-semilocal E-preinvex with respect to η atx, then (3.10) Adding (3.9) and (3.10), we get a contradiction to assumption (i). Thus,x is an LU optimal solution to (IVP).

Wolfe type duality
Consider the following Wolfe type dual model:  Proof On the contrary, suppose that F(x) ≺ F(z) + l j=1 τ j g j (z). That is, Using the fact that w L , w U > 0 and w L + w U = 1 with the feasibility ofx to (IVP), the above inequalities become Since F is semilocal E-preinvex with respect to η at z, then Since w L , w U > 0, then the above inequalities become and From the semilocal E-preinvexity of l j=1 τ j g j with respect to η at z, we have Adding (4.4), (4.5), and (4.6), we get On using inequality (4.3), the above inequality becomes a contradiction to dual constraint (4.1) with (z, w L , w U , τ ) feasible to (IVWD). This completes the proof. Proof Asx is an LU optimal solution to (IVP) and suitable constraint qualification holds atx, so by Theorem 3.1 there exist scalarsw L > 0,w U > 0,τ j ≥ 0, j = 1, 2, . . . , l, such that and l j=1τ j g j (x) = 0 show that (x,w L ,w U ,τ ) is a feasible solution to (IVWD) and the analogous objective values are the same. Assume that (x,w L ,w U ,τ ) is not an LU optimal solution to (IVWD), this means there is a feasible solution (z,w L ,w U ,τ ) to (IVWD) such that which is a contradiction to weak duality Theorem 4.1. Thus, (x,w L ,w U ,τ ) is an LU optimal solution to (IVWD).
Proof Suppose, on the contrary, thatx =z. Since F is strictly semilocal E-preinvex with respect to η atz, i.e., Multiplying the above inequalities byw L andw U , respectively, and adding them, we get Adding (4.8) and (4.9), we get The above inequality with the feasibility of (z,w L ,w U ,τ ) to (IVWD) (i.e., with dual constraint (4.1)) becomes a contradiction to inequality (4.7). Hence,x =z.

Mond-Weir type duality
Consider the following Mond-Weir type dual model:  Proof On the contrary, suppose that F(x) ≺ F(z).
This means For w L > 0 and w U > 0, we can write which together with the pseudo-semilocal E-preinvexity of w L F L + w U F U with respect to η at z gives Again, from the feasibility ofx and (z, w L , w U , τ ) to (IVP) and (IVMWD), respectively, we have l j=1 τ j g j (x) ≤ l j=1 τ j g j (z).
With the above inequality, use the fact that l j=1 τ j g j is quasi-semilocal E-preinvex with respect to η at z, then l j=1 τ j (dg j ) + z; η E(x), z ≤ 0. (4.14) Adding (4.13) and (4.14), we get a contradiction to dual constraint (4.10) for (z, w L , w U , τ ) feasible to (IVMWD). Hence, F(x) F(z). Proof Asx is an LU optimal solution to (IVP) and suitable constraint qualification holds atx, so by Theorem 3.1 there exist scalarsw L > 0,w U > 0, andτ j ≥ 0, j = 1, 2, . . . , l, such thatw and l j=1τ j g j (x) = 0 show that (x,w L ,w U ,τ ) is a feasible solution to (IVMWD) and the analogous objective values are the same. Assume that (x,w L ,w U ,τ ) is not an LU optimal solution to (IVMWD), this means there is a feasible solution (z,w L ,w U ,τ ) to (IVMWD) such that which is a contradiction to weak duality Theorem 4.4. Thus, (x,w L ,w U ,τ ) is an LU optimal solution to (IVMWD).
Theorem 4.6 (Strict converse duality) Letx and (z,w L ,w U ,τ ) be the feasible solutions to (IVP) and (IVMWD), respectively, with E(z) =z. Assume thatw L F L +w U F U is strictly pseudo-semilocal E-preinvex and l j=1τ j g j is quasi-semilocal E-preinvex, and all functions are E-η-semidifferentiable atz with Thenx =z.
The above inequality, with the fact that l j=1τ j g j is quasi-semilocal E-preinvex with respect to η atz, gives l j=1τ j (dg j ) + z; η E(x),z ≤ 0.
Since (z,w L ,w U ,τ ) is feasible to (IVMWD), then from dual constraint (4.10) and the above inequality, we obtain w L dF L + z; η E(x),z +w U dF U + z; η E(x),z ≥ 0.
With the above inequality using the strict pseudo-semilocal E-preinvexity ofw L F L +w U F U with respect to η atz, we get a contradiction to inequality (4.15). Hence,x =z.

Lagrangian function and saddle-point criteria
Consider the following Lagrangian function for interval-valued optimization problem (IVP): where x ∈ X, w L ≥ 0, w U ≥ 0, τ ∈ R l + .
Definition 5.1 ([20]) Suppose thatw L ≥ 0 andw U ≥ 0 are fixed. A point (x,w L ,w U ,τ ) ∈ X × R + × R + × R l + is called a saddle point of the real-valued function L(x, w L , w U , τ ) if the following condition holds: Theorem 5.1 ([20]) Suppose thatw L > 0 andw U > 0 are fixed and (x,w L ,w U ,τ ) is a saddle point of L(x, w L , w U , τ ). Thenx is an LU optimal solution to (IVP).
Proof As F is semilocal E-preinvex with respect to η atx, then Multiplying the above inequalities byw L andw U , respectively, and adding them, we get By the semilocal E-preinvexity of l j=1τ j g j with respect to η atx, we get l j=1τ j g j (x) -l j=1τ j g j (x) ≥ l j=1τ j (dg j ) + x; η E(x),x .
On adding the above two inequalities w L F L (x) +w U F U (x) + l j=1τ j g j (x) -w L F L (x) +w U F U (x) + l j=1τ j g j (z) The above inequality together with given assumption (i) gives i.e., L x,w L ,w U ,τ ≤ L x,w L ,w U ,τ . Then (x,w L ,w U ,τ ) is a saddle-point of L(x, w L , w U , τ ).
Proof Assumption (i), together with the semilocal E-preinvexity ofw L F L +w U F U + l j=1τ j g j with respect to η atx, yields w L F L (x) +w U F U (x) + l j=1τ j g j (x) ≤w L F L (x) +w U F U (x) + l j=1τ j g j (x). Now, the rest of the proof is the same as the proof of Theorem 5.2.

Conclusions
We have considered interval-valued programming problems (IVP) for E-η-semidifferentiable functions. We established sufficient optimality conditions for (IVP) and illustrated the result with the help of an example. We formulated the Wolfe type and Mond-Weir type dual models for (IVP) and established the usual duality results for the described models. Further, we presented the saddle-point optimality criteria to establish the relation between an optimal solution of semidifferentiable (IVP) and a saddle point of the Lagrangian function. In the future, the results obtained in this paper can be extended to multiobjective case and generalized type-I functions.