Weak sharp minima for interval-valued functions and its primal-dual characterizations using generalized Hukuhara subdifferentiability

This article introduces the concept of weak sharp minima (WSM) for convex interval-valued functions (IVFs). To identify a set of WSM of a convex IVF, we provide its primal and dual characterizations. The primal characterization is given in terms of $gH$-directional derivatives. On the other hand, to derive dual characterizations, we propose the notions of the support function of a subset of $I(\mathbb{R})^{n}$ and $gH$-subdifferentiability for convex IVFs. Further, we develop the required $gH$-subdifferential calculus for convex IVFs. Thereafter, by using the proposed $gH$-subdifferential calculus, we provide dual characterizations for the set of WSM of convex IVFs.


Introduction
Due to the presence of uncertainty, deterministic optimization fails to represent many real-life optimization problems. In such cases, we need to proceed with the tools of uncertain optimization. If the uncertainty is given by a random variable, then these optimization problems come under the umbrella of stochastic optimization. On the other, if the uncertainty is given by a membership function, then these optimization problems are solved with the techniques of fuzzy optimization. Also, it is seen that the uncertainty of many practical problems is expressed using closed and bounded intervals. Thus, interval optimization is an indispensable way to deal with the uncertainty present in many real-life problems.
In 1966, Moore [28] introduced interval analysis to investigate interval-valued functions (IVFs). In [28], Moore extensively gave arithmetic of intervals. Subsequently, there was a need to improve this arithmetic [18], especially the subtraction. Due to this, Hukuhara [18] presented a notion of the difference between intervals, which is known as Hukuhara difference (H-difference). Stefanini and Bede [34] proposed an extended version of H-difference, known as generalized Hukuhara difference (gH-difference), which has been comprehensively adopted in interval analysis.
We know that the solution concepts of optimization problems depend widely on ordering the range set of the objective function. Unlike real numbers, the set of intervals is not linearly ordered. Thus, to introduce a solution concept for optimization problems under interval uncertainties, many partial ordering relations of intervals were proposed in the literature (see [3,21,24,31,39,40], and the references therein). With the help of the existing ordering concepts of a pair of intervals, many theories and methods have been developed regarding solutions of optimization problems with IVFs or of interval optimization problems (IOPs) [20,22,29,32,37,41]. Inuiguchi et al. [20] proposed treatment of optima for IVFs by minimax regret criteria. Chanas and Kutcha [8] gave a solution concept based on a preference relation of intervals. A robust efficient solution for interval linear porgramming was given in [19]. Chen et al. [9] reported a solution concept by midpoint deterministic approach. Wu [38] defined type I and type II LU -optimal solution concepts similar to Pareto optimality. A survey on the different ordering of intervals and related optimality concepts can be found in [14,23] and from their references.
The major developments on IOPs started after a rich calculus of IVFs was ready to be used. Hukuhara [18] laid the foundation to develop the calculus of IVFs by introducing the concept of H-differentiability of IVFs. However, this definition of H-differentiability is restrictive [7]. To overcome the deficiencies of H-differentiability, Stefanini and Bede [34] proposed the notion of gH-differentiability for IVFs. Later, by using gH-differentiability, many concepts on calculus have been developed, for instance, see [7,13,26]. In 2007, Wu [38] proposed two solution concepts by considering two partial ordering concepts on the set of all closed intervals and derived KKT optimality conditions for IOPs using H-derivative. Subsequently, Wu investigated KKT optimality conditions for multi-objective IOPs [41]. In 2012, Bhurjee et al. [3] developed a methodology to study the existence of the solution of general IOPs by expressing IVFs in the parametric form. Chalco-cano et al. [6] derived KKT optimality conditions for IOPs using gH-derivative and explained the advantages of using gH-derivative instead of H-derivative. In 2016, Singh et al. [33] proposed the concept of Pareto optimal solution for the interval-valued multi-objective programming problems. Many other researchers have also proposed optimality conditions and solution concepts for IOPs, see for instance [1,12,15,16,35] and the references therein.

Motivation and work done
To analyse optimization problems with multiple minima points, Burke and Ferris [5] introduced the notion of WSM in conventional optimization. It is known that WSM plays an important role in the sensitivity analysis and convergence analysis of conventional optimization problems [5,11,42]. It is also seen that many algorithms exhibit finite termination at WSM [5,11,27,36,42]. Motivated by these properties and wide applications of WSM in conventional optimization, in this article, we attempt to propose and mathematically characterize the notion of WSM for convex IVFs. To give characterizations of WSM for convex IVFs, we defined gH-subdifferentiability for convex IVFs and support function of a subset of I(R) n . Some required fundamental characteristics of gH-subdifferential set are proposed. Few related results on the support function of a nonempty subset of I(R) n are also derived.

Delineation
The article is presented in the following manner. In Section 2, basic terminologies and definitions on intervals and IVFs are provided. In Section 3, we propose the concept of the support function of a subset of I(R) n ; alongside, few necessary results on extended support function are also given. Next, we derive the idea of gH-subdifferentiability for convex IVFs, in Section 4 that are required in the subsequent sections. The concept of WSM for convex IVFs is presented in Section 5; further, we give primal and dual characterizations of WSM. Lastly, the conclusion and future scopes are given in Section 6.

Preliminaries and Terminologies
In this article, the following notations are used throughout.
• R denotes the set of real numbers • R + denotes the set of nonnegative real numbers • · denotes the Euclidean norm and ·, · denotes the standard inner product on R n • I(R) represents the set of all closed and bounded intervals • Bold capital letters refer to the elements of I(R) x ≤ 1} denotes the closed unit ball in R n .

Fundamental Operations on Intervals
Arithmetic operations of two intervals A = [a, a] and B = b, b are defined by where λ is a real constant. The norm [28] of an interval A = [a, a] in I(R) is defined by A I(R) = max{|a|, |a|}.
The norm of an interval vector A = (A 1 , A 2 , . . . , A n ) ∈ I(R) n is given by (see [28]) It is to note that a real number p can be represented by the interval [p, p].
Definition 1 (gH-difference of intervals [34]). Let A and B be two elements in I(R). The gH-difference between A and B, denoted by A ⊖ gH B, is defined by the interval C such that It is to be noted that for A = [a, a] and B = b, b , and A ⊖ gH A = 0.
Definition 3 (Special product ). For any x = (x 1 , x 2 , . . . , x n ) ∈ R n and a vector of intervals A = (A 1 , A 2 , . . . , A n ) ∈ I(R) n with A i = [a i , a i ] for each i = 1, 2, . . . , n, the special product between x and A, denoted by x ⊤ ⊙ A, is given by Remark 1 It is to notice that if all the components of A are degenerate intervals, i.e., A ∈ R n , then the special product x ⊤ ⊙ A reduces to the standard inner product of x ∈ R n with A.

Calculus of IVFs
Throughout this subsection, F is an IVF defined on a nonempty subset X of R n .
Definition 5 (gH-continuity [12]). Let F be an IVF and letx be a point of X and h ∈ R n such thatx + h ∈ X. The function F is said to be gH-continuous atx if Definition 6 (gH-derivative [34]). The gH-derivative of an IVF F : R → I(R) atx ∈ R is defined by d , provided the limit exists.
Remark 2 (See [34]). Let F = [F , F ] be an IVF on X, where F and F are real-valued functions defined on X. Then, the gH-derivative of F atx ∈ X exists if the derivatives of F and F atx exist and Definition 7 (gH-partial derivative [12]). Letx = (x 1 ,x 2 , . . . ,x n ) ⊤ be a point of X. For a given i ∈ {1, 2, . . . , n}, we define a function G i by If gH-derivative of G i exists atx i , then we say that F has the ith gH-partial derivative at x. We denote the ith gH-partial derivative of F atx by Definition 8 (gH-gradient [12]). The gH-gradient of F at a pointx ∈ X, denoted by ∇F(x) ∈ I(R) n , is defined by Lemma 1 Let A, B, and C are in I(R). Then, for any real number r, (i) [r, r] A and A B ⊖ gH C =⇒ C ⊕ [r, r] B and WSM for IVFs and its characterizations using gH-subdifferentiability Proof See Appendix A.
Definition 9 (Convex IVF [38]). Let X be a nonempty convex subset of R n . An IVF F : X → I(R) is said to be convex on X if for any x 1 and x 2 in X, Lemma 2 (See [38]). Let X be a nonempty convex subset of R n , and F = [F , F ] be an IVF on X, where F and F are real-valued functions defined on X. Then, F is convex on X if and only if F and F are convex on X.
Definition 10 (gH-directional derivative [13]). Let F be an IVF on X. Letx ∈ X and d ∈ R n . If the limit exists, then the limit is said to be gH-directional derivative of F atx in the direction d, and it is denoted by F D (x)(d).
Definition 11 (gH-differentiability [12]). An IVF F is said to be gH-differentiable atx ∈ X if there exist two IVFs E(F(x); h) and Lx : R n → I(R) such that and Lx is such a function that satisfies (i) Lx(x + y) = Lx(x) ⊕ Lx(y) for all x, y ∈ X, and (ii) Lx(cx) = c ⊙ Lx(x) for all c ∈ R and x ∈ X.
Theorem 1 (See [12]). Let F : X → I(R) be gH-differential atx. Then, Lx exists for every h ∈ R n and Remark 3 (See [12]). Let F : X → I(R) be gH-differentiable atx ∈ X. Then, there exists a nonzero λ and δ > 0 such that for all h ∈ R n with |λ| h < δ, where Lx is an IVF, defined in Definition 11 of gH-differentiability.
Then, F has gH-directional derivative atx for every direction h ∈ R n and where Lx is as defined in Definition 11.
Proof Since F is gH-differentiable atx, by Remark 3, we have Hence, by Definition 10, we conclude that F has gH-directional derivative atx and Definition 13 (Effective domain of IVF ). The effective domain of an extended IVF F : X → I(R) is the collection of all such points at which F is finite. It is denoted by dom(F), i.e., Theorem 2 Let X be a nonempty convex subset of R n and F : Proof Similar to the proof of Theorem 3.1 in [13].
Definition 14 (gH-Lipschitz continuous IVF [13]). An IVF F is said to be gH-Lipschitz continuous on X if there exists M > 0 such that The constant M is called a Lipschitz constant. Lemma 4 (See [25]). Let F 1 and F 2 be two proper extended IVFs defined on S, which is a nonempty subset of X. Then, Definition 16 (Lower limit and gH-lower semicontinuity of an extended IVF [25]). The lower limit of an extended IVF F atx ∈ X, denoted by lim inf is an open ball with radius δ centered atx. F is called gH-lower semicontinuous Remark 5 By Note 5 of [25], we see that F is gH-lsc atx ∈ X if and only if F and F both are lsc atx.
Lemma 5 Let F : R n → I(R) be a proper convex IVF. Then, for all x, y ∈ dom(F), we have Proof By Definition 10 of gH-directional derivative, we have then according to Definition 2 and norm on I(R) n , the corresponding sequence {G ki } in I(R) converges to G i ∈ I(R) for each i = 1, 2, . . . , n. Also, by Definition 17, the sequences g ki and g ki in R converge to g i and g i in R, respectively, for each i = 1, 2, . . . , n.

Results from convex analysis
Apart from the results of interval analysis, we use the following results from classical convex analysis throughout the article.
Definition 18 (Projection [30]). Let A be a nonempty closed set in R n . Then, the projection of a point x ∈ R n onto the set A is denoted by P (x | A), and is defined by Definition 19 (Polar cone [30]). Let A be a nonempty set in R n . Then, the polar cone of the set A is Definition 20 (Tangent cone [30]). Let A be a nonempty closed convex set in R n . Then, the tangent cone to the set A at x ∈ A is defined by Definition 21 (Normal cone [30]). The normal cone to a nonempty set A in R n at x is polar of the tangent cone at x to the A, i.e., where the distance function is given by

Support function in I(R) n
In this section, we attempt to extend the conventional notion of support functions for subsets of I(R) n . The derived concepts of support function are used later in Section 5 to derive dual characterizations of WSM for convex IVFs.
Definition 22 (Support function of a subset of I(R) n ). Let S be a nonempty subset of I(R) n . Then, the support function of S at x ∈ R n , denoted by ψ * S (x), is defined by Since D is arbitrary, we get WSM for IVFs and its characterizations using gH-subdifferentiability 9 Theorem 3 Let K be a nonempty closed convex cone in X ⊆ R n . Let P and Q be two nonempty subsets of I(R) n . Then, Also, by hypothesis, we have ψ * Note that as λ → +∞, λx ⊤ z → +∞, and therefore x ⊤ ⊙ ( A ⊕ λz) → +∞, which implies Therefore, from (3) and (4), we have for all x ∈ X. Proof of the converse part follows from (2). This completes the proof.
Lemma 9 Let P be nonempty a subset of R n and Q be a nonempty closed convex subset of I(R) n . Then, for any x ∈ R n , Therefore, for any p ∈ P and x ∈ R n , we have We now consider the following two possible cases.
To show that p ∈ Q, we have to show that p = Q m as well.
Note that Thus, from equation (8) and Lemma 6, we have p ∈ S ′ , i.e., p = Q m . Hence, p ∈ Q. Since p is arbitrary, P ⊆ Q.
• Case 2. Let n j=1 x j q ij ≤ n j=1 x j q ij . By following similar steps as in Case 1, in this case also, we get P ⊆ Q.
Proof of the converse part follows from Lemma 8.
Lemma 10 For x ∈ R n and A = (A 1 , A 2 , . . . , A n ) ∈ S ⊆ R n , we have

Lemma 11
The support function of a nonempty set S ⊆ I(R) n is finite everywhere if and only if S is bounded.
Proof Suppose that S is bounded, i.e., we have M > 0 such that A I(R) n ≤ M for all A = (A 1 , A 2 , . . . , A n ) ∈ S with A i = [a i , a i ] for each i = 1, 2, . . . , n. By Lemma 10 and A I(R) n ≤ M , for any x ∈ R n , we have Since A ∈ S is arbitrary chosen, therefore Hence, ψ * S (x) is finite everywhere. Conversely, let ψ * S (x) is finite for every x ∈ R n . Therefore, there exists an M > 0 such that ψ * S (x) M, which implies that for any x ∈ R n and A ∈ S, we have x i a i ≤ M , then by Remark 1, we have x, a ≤ M, where a = (a 1 , a 2 , . . . , a n ) ∈ R n .
If a = 0, choose x = a a , then (9) gives Since A ∈ S was arbitrary chosen, therefore we have A M for all A ∈ S. Hence, S is bounded.

gH-subdifferentiability of convex IVFs
In this section we develop gH-subdifferential calculus for convex IVFs that are used later to find dual characterization of WSM for convex IVFs.
Definition 23 (gH-subdifferentiability). Let F : X ⊆ R n → I(R) be a proper convex IVF and x ∈ dom(F). Then, gH-subdifferential of F atx, denoted by ∂F(x) is defined by The elements of (10) are known as gH-subgradients of F atx. Further, if ∂F(x) = ∅, we say that F is gH-subdifferentiable atx.
Example 1 Consider F : R → I(R) be a convex IVF such that F(x) = |x| ⊙ A, where 0 A. Let us check gH-subdifferentiability of F at 0.
• Case 1. x ≤ 0. In this case, for all x ∈ R, (11) gives, • Case 2. x > 0. In this case, for all x ∈ R, (11) gives, Hence, from Case 1 and Case 2, we have Fig. 1 The IVF F of Example 1 In Fig. 1, the IVF F, with A = 1 4 , 1 , is drawn by the gray shaded region between two red dashed lines, and its possible two gH-subgradients G 1 and G 2 at 0 are shown by blue and green shaded regions, respectively.
Lemma 12 Let X be a nonempty convex subset of R n and F : X → I(R) be a proper convex IVF. Then, for anyx ∈ dom(F) and h ∈ R n such thatx + h ∈ X, the gH-subdifferential set of F atx is Proof Suppose G ∈ ∂F(x). Then, by Definition 23, we have By taking x =x + λh with λ > 0 and h ∈ R n in (12), we get Next, if we take any G ∈ I(R) n such that h ⊤ ⊙ G F D (x)(h) for all h ∈ R n . Then, by a similar reasoning as above it can be seen that G ∈ ∂F(x). Proof We first prove the closedness of ∂F(x). Let G k be a sequence in ∂F(x), which converges to G ∈ I(R) n , where G k = (G k1 , G k2 , . . . , G kn ) and G = (G 1 , G 2 , . . . , G n ). Since G k ∈ ∂F(x), for all h ∈ R n such thatx + h ∈ X, we have Since the sequence G k converges to G, in view of Remark 6, the sequences {g ki } and {g ki } converge to g i and g i , respectively, for each i = 1, 2, . . . , n. Thus, Therefore, in view of (13) and (14), we have Thus, Therefore, G ∈ ∂F(x), and hence ∂F(x) is closed.
Theorem 5 Let X be a nonempty convex subset of R n and let F : X → I(R) be a gHdifferentiable convex IVF atx ∈ X. Then, Proof Let G ∈ ∂F(x). Since F is gH-differentiable atx, with the help of Lemma 3 and Lemma 12, we get Replacing h by −h in (15), we obtain Thus, (15) and (16), simultaneously give Therefore, for each i ∈ {1, 2, . . . , n} , by choosing h = e i in (17), Lemma 13 Let X be a nonempty convex subset of R n and F : X → I(R) be a proper convex where F , F : X → R are extended real-valued functions. Then, the subdifferential set of F atx ∈ int(dom(F)) can be obtained by the subdifferential sets of F and F atx and vice-versa.
Proof Since F is proper convex, with the help of Lemma 2, we note that F and F are also convex. Therefore, by the property of real-valued proper convex functions, the subdifferential sets of F and F atx ∈ int(dom(F)) are nonempty (see [2]). Let g = (g 1 , g 2 , . . . , g n ) ∈ ∂F (x) and g = (g 1 , g 2 , . . . , g n ) ∈ ∂F (x). Then, by Definition 23 of gH-subdifferentiability, for any h ∈ R n such thatx + h ∈ X , we have Thus, for any g ∈ ∂F (x) and g ∈ ∂F (x), we have the corresponding G ∈ ∂F(x).
Thus, we get g ∈ ∂F (x) and g ∈ ∂F (x), which are required.
• Case 2. Let min . Proof contains similar steps as in Case 1.
From Case 1 and Case 2, it is clear that for any G ∈ ∂F(x), we can obtain the subgradients of F and F atx. This completes the proof for the converse part.
Theorem 6 Let X be a nonempty convex subset of R n and F : where F , F : X → R are extended real-valued functions. Then, at anyx ∈ int(dom(F)), Proof Note that F (x) and F (x) are proper convex, and therefore ∂F (x) and ∂F (x) are nonempty. Let g = (g 1 , g 2 , . . . , g n ) ∈ ∂F (x) and g = (g 1 , g 2 , . . . , g n ) ∈ ∂F (x) forx ∈ int(dom(F)). By the property of real-valued convex functions (see [10]), we have Due to Theorem 2, we get We now consider the following two possible cases.
Hence, from Case 1 and Case 2, we have Theorem 7 Let F : X → I(R) be a proper convex IVF andx ∈ int(dom(F)). Then, the gH-subdifferential set of F atx is bounded.
Proof Note that by Theorem 2, forx ∈ int(dom(F)), the directional derivative of F atx exists everywhere. Thus, for all h ∈ R n such thatx + h ∈ X, we have Hence, the gH-subdifferential set of F atx ∈ int(dom(F)) is bounded, i.e., for every G ∈ ∂F(x), there exists an M > 0 such that G I(R) n ≤ M .

Lemma 14
Let F be an IVF on a nonempty set X ⊆ R n such that where c ∈ R. Then, Proof We have F(x) ⊖ gH F(y) c x − y for all x, y ∈ X, which implies that Interchanging x and y in (23), we obtain With the help of (23) and (24), we get which implies F(x) ⊖ gH F(y) I(R) ≤ c x − y for all x, y ∈ X.
Theorem 8 Let X be a nonempty convex subset of R n and F be a convex IVF on X such that F has gH-subgradient at every x ∈ X. Then, F is gH-Lipschitz continuous on X.
Proof Since F has gH-subdradient at every x ∈ X, then there exists a G ∈ I(R) n such that Thus, F is gH-Lipschitz continuous on X.

Weak sharp minima and its characterizations
In this section, we present the main results-primal and dual characterizations of WSM for a gH-lsc and convex IVF.
, where a > 0. Thus,S ⊆ S ⊆ R n . Clearly, the functions F and F are 5 − x 1 x 2 − x 1 and 10 − x 2 1 x 2 − x 2 2 x 1 , respectively. Note that for any α > 0, Thus,S = {0} × [−a, 0] is a set of WSM of both F and F over S with modulus α, for any α > 0. Therefore, by Remark 8,S is a set of WSM of F over S with modulus α > 0.
Theorem 9 (Primal characterization). Let F, S, andS be as in Definition 24. Further, define an IVF F o : R n → I(R) by Then, the setS is a set of WSM of F over the set S with modulus α > 0 if and only if Proof SupposeS is a set of WSM of F over S with modulus α > 0. Then, by Definition 24, for any x ∈S, d ∈ R n , and t > 0, we have For the converse part, let y ∈ S and x ∈S. Therefore, from Lemma 5, we get Since x ∈S is arbitrary, we have for all x ∈S and α > 0 by part (i) of Lemma 7.
Hence,S is the set of WSM of F over S with modulus α > 0, and the proof is complete.
Theorem 10 (Dual characterizations). Let F, S, andS be as in Definition 24. Define an IVF Then, for any α > 0, the following statements are equivalent. (e) For all x ∈S and d ∈ T S (x) ∩ NS(x), where p ∈ P (y |S).
Proof (a) ⇐⇒ (b). Let x ∈S. By hypothesis,S is a set of WSM of F over S. Therefore, by Theorem 9, we get α dist(d, TS(x)) F oD (x)(d) for all d ∈ R n , which along with Theorem 6 imply Notice that for all x ∈S and d ∈ R n , we have That is, Thus, by (26) and (27), we get Next, with the help of Lemma 9, we get the desired result Conversely, we have Also, by Theorem 6, we have Thus, by (31), we get Therefore, by Theorem 9,S is a set of WSM of F over S with modulus α.
(a) ⇐⇒ (c). Let the statement (a) holds. Let x ∈S. Therefore, by Theorem 9, we have Note that for x ∈S, F o (x) = F(x). Thus, By (32) and Theorem 9, we get Conversely, we are given that Note that for x ∈S, we have In view of (33) and (34), we have Hence, by Theorem 9,S is the set of WSM of F over S with modulus α > 0. (b) ⇐⇒ (d). If the statement (b) holds, then obviously the statement (d) also holds. Conversely, let the statement (d) holds. Let x ∈S and G ∈ αB ∩ NS(x). Therefore, there exists aȳ ∈S such that G ∈ ∂F o (ȳ). Thus, by Definition 23, we get In particular, for any z ∈S, F o (z) = F o (ȳ). Thus, (35) reduces to (z −ȳ) ⊤ ⊙ G 0 for all z ∈S.

Conclusion and future scopes
In this article, the conventional concepts of support function and subdifferentiability have been extended for IVFs (Definitions 22 and Definition 23). Also, some important characteristics of the gH-subdifferential set like nonemptyness (Lemma 13), boundedness (Theorem 7), convexity and closedness (Theorem 4) have been presented. Subsequently, we have provided few necessary results (Lemma 8, Theorem 3 and Lemma 9) based on the support function of a subset of I(R) n . It has been reported that the gH-subdifferential set of a gH-differentiable convex IVF is a singleton set containing the gH-gradient (Theorem 5). The relationship between gH-directional derivative and the support function of gH-subdifferential set of convex IVF has been also established (Theorem 6). Further, we have introduced the notion of WSM for convex IVFs (Definition 24). With the help of the proposed concepts of gH-subdifferentiability and support function, a primal characterization (Theorem 9) and a few dual characterizations (Theorem 10) of WSM have been presented.
In future, we shall apply proposed theory on WSM to derive necessary and sufficient conditions under which a global error bound may exist for a convex inequality system as follows: where Λ is an index set, and for each λ ∈ Λ, H λ : X ⊆ R n → I(R) is gH-lsc, convex, proper, and the set C is closed convex subset of X. By a global error bound for the inequality system (41), we mean the existence of a constant β > 0 such that β dist (x, Ω) dist(x, C) ⊕ H λ+ (x) for each λ ∈ Λ and x ∈ X, where Ω = {x : x ∈ C and H λ (x) 0} and H λ+ (x) = max{0, H λ (x)}. For the sake of convenience, we define H(x) = sup {H λ (x) : λ ∈ Λ} for each x ∈ X. Note that if a constant β > 0 exists such thatβ dist (x, Ω) dist(x, C) ⊕ H + (x) for all x ∈ X, Hence, C ⊕ [r, r] B.