International Journal of Analysis and Applications Volume 18, Number 5 (2020), 784-798 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-18-2020-784 SUFFICIENCY AND DUALITY FOR INTERVAL-VALUED OPTIMIZATION PROBLEMS WITH VANISHING CONSTRAINTS USING WEAK CONSTRAINT QUALIFICATIONS IZHAR AHMAD1,∗, KRISHNA KUMMARI2 AND S. AL-HOMIDAN1 1Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia 2Department of Mathematics, GITAM-Hyderabad Campus, Hyderabad-502329, India ∗Corresponding author: drizhar@kfupm.edu.sa Abstract. In this paper, we are concerned with one of the difficult class of optimization problems called the interval-valued optimization problem with vanishing constraints. Sufficient optimality conditions for a LU optimal solution are derived under generalized convexity assumptions. Moreover, appropriate duality results are discussed for a Mond-Weir type dual problem. In addition, numerical examples are given to support the sufficient optimality conditions and weak duality theorem. 1. Introduction Due to the mathematical challenges and important roles in various fields, mathematical programs with vanishing constraints have attracted many mathematicians in the past decade. Mathematical programming problem with vanishing constraints is a constrained optimization problem and it is closely related to the Mathematical programs with equilibrium constraints, see for example [9, 10, 14]. This problem was first studied by Achtziger and Kanzow in [2] and this serves as a model for many problems from topology and structural optimization (see [2, 5]). For Mathematical programming problems with vanishing constraints, Received October 14th, 2019; accepted November 13th, 2019; published July 17th, 2020. 2010 Mathematics Subject Classification. 26A51, 49J35, 90C32. Key words and phrases. interval-valued optimization problem; vanishing constraints; constraint qualifications; generalized convexity; sufficiency; duality. ©2020 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 784 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-784 Int. J. Anal. Appl. 18 (5) (2020) 785 it is well known that the usual nonlinear programming constraint qualifications such as Slater constraint qualification, Mangasarian-Fromovitz constraint qualification, Cottle constraint qualification and linear in- dependence constraint qualification do not hold (see [12]), while Mishra et al. [12] proved that the standard generalized Guignard constraint qualification holds in many situations, and some sufficient conditions are presented in [12]. The Guignard constraint qualification (GCQ) was introduced by Guignard [8] and is one of the weakest among the most prominent constraint qualifications such as the Slater constraint qualification [19], Abadie constraint qualification [1], Mangasarian-Fromovitz constraint qualification [11], Cottle constraint qualifica- tion [7] and linear independence constraint qualification etc. For more information and inter-relation between these constraint qualifications one can see the survey papers [15, 22]. In recent years, a number of approaches have been developed to deal with interval-valued optimization problems. In [24, 25], Wu derived Karush-Kuhn-Tucker type optimality conditions for a optimization prob- lem with an interval-valued objective function. Further, the Karush- Kuhn-Tucker type necessary optimality conditions for a optimization problem in which objective and constraints functions are assumed to be interval valued were investigated by Singh et al. [17]. However, optimality conditions for an interval-valued multiob- jective programming with generalized differentiable functions (viz. gH-differentiable functions) are discussed in [18]. Bhurjee and Panda [6] provided an overview of an interval-valued optimization problem by developing a methodology to study the efficient solution for an interval-valued optimization problem. For more details related to interval-valued optimization problems, we refer to the papers (see, for example [3,13,16,20,23,26]). To the author’s knowledge, there are no results for an interval-valued mathematical programming problem with vanishing constraints in the literature. Therefore, this paper focuses on an interval-valued mathemat- ical programming problem with vanishing constraints to explore the sufficient optimality conditions and Mond-Weir type duality results. The rest of the article is organized as follows: Some background material and preliminary definitions are provided in Section 2. The sufficient optimality conditions for a LU optimal solution for considered problem under generalized convexity assumptions are given in Section 3. In Section 4, weak, strong and strict converse duality theorems are discussed for a Mond-Weir type dual model. Finally, Section 5 is devoted to the conclusion. 2. Preliminaries For a nonempty subset Q of Rn, we use the notations clQ and clcoQ to denote the closure of Q and closure of the convex hull of Q, respectively. Let Θ be the set of all closed and bounded intervals in R. Let Θ1 = [τ L,τU ], Θ2 = [ρ L,ρU ] ∈ Θ, then we have (i ) Θ1 + Θ2 = {τ + ρ | τ ∈ Θ1 and ρ ∈ Θ2} = [τL + ρL,τU + ρU ], Int. J. Anal. Appl. 18 (5) (2020) 786 (ii ) −Θ1 = {−τ | τ ∈ Θ1} = [−τU,−τL], (iii ) Θ1 − Θ2 = Θ1 + (−Θ2) = [τL −ρU,τU −ρL], (iv ) k + Θ1 = {k + τ | τ ∈ Θ1} = [k + τL,k + τU ], (v ) kΘ1 = {kτ | τ ∈ Θ1} =   [kτL,kτU ], if k ≥ 0, [kτU,kτL], if k < 0, where k is a real number. For Θ1 = [τ L,τU ] and Θ2 = [ρ L,ρU ], the order relation ≤LU is defined as follows: (i) Θ1 ≤LU Θ2 if and only if τL ≤ ρL and τU ≤ ρU . (ii) Θ1 0}, Λ0 = {i ∈{1, 2, ...,r}|`i(x∗) = 0}. Furthermore, the index set Λ+ can be divided into the following subsets Λ+0 = {i ∈{1, 2, ...,r}|`i(x∗) > 0, Φi(x∗) = 0}, Λ+− = {i ∈{1, 2, ...,r}|`i(x∗) > 0, Φi(x∗) < 0}. Similarly, the index set Λ0 can be partitioned in the following way Λ0+ = {i ∈{1, 2, ...,r}|`i(x∗) = 0, Φi(x∗) > 0}, Λ00 = {i ∈{1, 2, ...,r}|`i(x∗) = 0, Φi(x∗) = 0}, Λ0− = {i ∈{1, 2, ...,r}|`i(x∗) = 0, Φi(x∗) < 0}. Also, for x∗ ∈ F, we define the sets Qk, Q k , k = L,U and Q as follows: Qk = { x ∈ Rn|Ψi(x) ≤ Ψi(x∗),∀i = L,U, i 6= k, ϕi(x) ≤ 0, ∀i = 1, 2, ...,p, ζi(x) = 0, ∀i = 1, 2, ...,q, `i(x) ≥ 0, ∀i = 1, 2, ...,r, Φi(x)`i(x) ≤ 0, ∀i = 1, 2, ...,r } . Q k = { x ∈ Rn|Ψi(x) ≤ Ψi(x∗),∀i = L,U, i 6= k, ϕi(x) ≤ 0, ∀i = 1, 2, ...,p, ζi(x) = 0, ∀i = 1, 2, ...,q, `i(x) = 0, Φi(x) ≥ 0, ∀i ∈ Λ0+, Φi(x) ≤ 0,`i(x) ≥ 0, ∀i ∈ Λ0− ∪ Λ00 ∪ Λ+0 ∪ Λ+− } . and Q = { x ∈ Rn|ΨL(x) ≤ ΨL(x∗), ΨU (x) ≤ ΨU (x∗), Int. J. Anal. Appl. 18 (5) (2020) 788 ϕi(x) ≤ 0, ∀i = 1, 2, ...,p, ζi(x) = 0, ∀i = 1, 2, ...,q, `i(x) = 0, Φi(x) ≥ 0, ∀i ∈ Λ0+, Φi(x) ≤ 0,`i(x) ≥ 0, ∀i ∈ Λ0− ∪ Λ00 ∪ Λ+0 ∪ Λ+− } . The linearizing cone Q k , k = L,U at x∗ ∈ F is given by L(Q k ; x∗) = { δ ∈ Rn|∇Ψi(x∗) T δ ≤ 0,∀i = L,U, i 6= k, ∇ϕi(x∗)Tδ ≤ 0, ∀i ∈ Λϕ, ∇ζi(x∗)Tδ = 0, ∀i ∈ Λζ, ∇`i(x∗)Tδ = 0, ∀i ∈ Λ0+, ∇`i(x∗)Tδ ≥ 0, ∀i ∈ Λ00 ∪ Λ0−, ∇Φi(x∗)Tδ ≤ 0, ∀i ∈ Λ+0 ∪ Λ00 } . and the symbol T denotes the transpose of a matrix. The linearizing cone to Q at x∗ ∈ Q, given by. L(Q; x∗) = L(Q L ; x∗) ∩L(Q U ; x∗). Definition 2.2. The tangent cone to Q at x∗ ∈ clQ is defined by T(x∗) = { δ ∈ Rn|∃{xn}⊆ F,{tn} ↓ 0 : xn → x∗ and xn −x∗ tn → δ } The modified Guignard constraint qualification was introduced by Mishra et al. ( [12], Definition 6.14) for a mathematical programming problem with vanishing constraints. From this perspective, we define the modified Guignard constraint qualification (IVVC-GCQ) for an interval-valued optimization problem (IVVC) as follows. Definition 2.3. The modified Guignard constraint qualification (IVVC-GCQ) is said to holds at x∗ ∈ F, if L(Q; x∗) ⊆ clcoT(QL; x∗) ∩ clcoT(QU ; x∗). Mishra et al. [12] proved the Karush-Kuhn-Tucker type necessary optimality conditions for a multiob- jective optimization problem with vanishing constraints under modified Guignard constraint qualification. Along the lines of Mishra et al. ( [12] Theorem 6.4), if we set m = 2, we acquire the following Karush-Kuhn- Tucker type necessary optimality conditions for (IVVC) as follow: Int. J. Anal. Appl. 18 (5) (2020) 789 Theorem 2.1. Let x∗ ∈ F be a LU optimal solution of (IVVC) such that (IVVC-GCQ) holds at x∗. Then there exist 0 < λL,λL ∈ R, µi ∈ R+, i = 1, 2, ...,p, γi ∈ R,i = 1, 2, ...,q and η`i,η Φ i ∈ R,i = 1, 2, ...,r such that λL∇ΨL(x∗) + λU∇ΨU (x∗) + p∑ i=1 µi∇ϕi(x∗) + q∑ i=1 γi∇ζi(x∗) − r∑ i=1 η`i∇`i(x ∗) + r∑ i=1 ηΦi ∇Φi(x ∗) = 0, (2.1) ϕi(x ∗) ≤ 0, µiϕi(x∗) = 0, ∀i = 1, 2, ...,p, (2.2) ζi(x ∗) = 0, ∀i = 1, 2, ...,q, (2.3) η`i = 0, i ∈ Λ+, η ` i ≥ 0, i ∈ Λ00 ∪ Λ0−, η ` i free, i ∈ Λ0+, (2.4) ηΦi = 0, i ∈ Λ+− ∪ Λ0− ∪ Λ0+, η Φ i ≥ 0, i ∈ Λ+0 ∪ Λ00, (2.5) η`i`i(x ∗) = 0,ηΦi Φi(x ∗) = 0,∀i = 1, 2, ...,r. (2.6) We define the following index sets which will be useful to prove the sufficient optimality conditions and duality results. Λ+ϕ = {i ∈{1, 2, ...,p}|µi > 0}, Λ + ζ = {i ∈ Λζ|γi > 0}, Λ−ζ = {i ∈ Λζ|γi < 0}, Λ + + = {i ∈ Λ+|η ` i > 0} Λ+0 = {i ∈ Λ0|η ` i > 0}, Λ − 0 = {i ∈ Λ0|η ` i < 0}, Λ−0+ = {i ∈ Λ0+|η Φ i < 0}, Λ − 00 = {i ∈ Λ00|η Φ i < 0}, Λ−+0 = {i ∈ Λ+0|η Φ i < 0}, Λ + 00 = {i ∈ Λ00|η Φ i > 0}, Λ−00 = {i ∈ Λ00|η Φ i < 0}, Λ + +0 = {i ∈ Λ+0|η Φ i > 0}, Λ−+0 = {i ∈ Λ+0|η Φ i < 0}, Λ + 0− = {i ∈ Λ0−|η Φ i > 0}, Λ++− = {i ∈ Λ+−|η Φ i > 0}. We now turn our attention to define some well-known concepts of convexity and generalized convexity for a real valued differentiable function (see, for example, [4]). Int. J. Anal. Appl. 18 (5) (2020) 790 Definition 2.4. Let Ω : X ⊆ Rn → R be a continuously differentiable function. Then, Ω is said to be a (strictly) convex at (x 6= x∗ ∈ X) x∗ ∈ X if for any x ∈ X, we have Ω(x) − Ω(x∗)(>) ≥ (x−x∗)T∇Ω(x∗). Definition 2.5. Let Ω : X ⊆ Rn → R be a continuously differentiable function. Then, Ω is said to be a quasiconvex at x∗ ∈ X if for any x ∈ X, we have Ω(x) ≤ Ω(x∗) ⇒ (x−x∗)T∇Ω(x∗) ≤ 0, equivalently (x−x∗)T∇Ω(x∗) > 0 ⇒ Ω(x) > Ω(x∗). Definition 2.6. Let Ω : X ⊆ Rn → R be a continuously differentiable function. Then, Ω is said to be a (strictly) pseudoconvex at x∗ ∈ X if for any x ∈ X, we have (x−x∗)T∇Ω(x∗) ≥ 0 ⇒ Ω(x)(>) ≥ Ω(x∗), equivalently Ω(x)(≤) < Ω(x∗) ⇒ (x−x∗)T∇Ω(x∗) < 0. 3. Sufficient optimality conditions In this section, we establish sufficient optimality conditions for the problem (IVVC) using the concept of generalized convexity. Theorem 3.1 (Sufficient optimality conditions). Let x̃ ∈ F and there exist 0 < λL,λU ∈ R, µi ∈ R+, i = 1, 2, ...,p, γi ∈ R,i = 1, 2, ...,q and η`i, η Φ i ∈ R,i = 1, 2, ...,r such that λL∇ΨL(x̃) + λU∇ΨU (x̃) + p∑ i=1 µi∇ϕi(x̃) + q∑ i=1 γi∇ζi(x̃) − r∑ i=1 η`i∇`i(x̃) + r∑ i=1 ηΦi ∇Φi(x̃) = 0, (3.1) ϕi(x̃) ≤ 0, µiϕi(x̃) = 0, ∀i = 1, 2, ...,p, (3.2) ζi(x̃) = 0, ∀i = 1, 2, ...,q, (3.3) η`i = 0, i ∈ Λ+, η ` i ≥ 0, i ∈ Λ00 ∪ Λ0−, η ` i free, i ∈ Λ0+, (3.4) ηΦi = 0, i ∈ Λ+− ∪ Λ0− ∪ Λ0+, η Φ i ≥ 0, i ∈ Λ+0 ∪ Λ00, (3.5) Int. J. Anal. Appl. 18 (5) (2020) 791 η`i`i(x̃) = 0,η Φ i Φi(x̃) = 0,∀i = 1, 2, ...,r. (3.6) Further, assume that λLΨL(.) + λU ΨU (.) is pseudoconvex at x̃ on F and that p∑ i=1 µiϕi(.), ζi(.)(i ∈ Λ+ζ ), −ζi(.)(i ∈ Λ−ζ ), −`i(.)(i ∈ Λ + + ∪Λ + 0 ), `i(.)(i ∈ Λ − 0 ), −Φi(.)(i ∈ Λ − 0+ ∪Λ − 00 ∪Λ − +0), Φi(.)(i ∈ Λ + 00 ∪Λ + 0−∪Λ + +0 ∪ Λ++−) are quasiconvex at x̃ on F. Then x̃ is a LU optimal solution of the problem (IVVC). Proof. Suppose contrary to the result that x̃ is not a LU optimal solution to the problem (IVVC), then by Definition 2.1 there exists x0 ∈ F such that Ψ(x0) 0, λU > 0, therefore the above inequalities yield λLΨL(x0) + λ U ΨU (x0) < λ LΨL(x̃) + λU ΨU (x̃), which by pseudoconvexity of λLΨL(.) + λU ΨU (.) at x̃ on F, we obtain (x0 − x̃)T [ λL∇ΨL(x̃) + λU∇ΨU (x̃) ] < 0. (3.7) For x0 ∈ F, µi ∈ R+, i = 1, 2, ...,p, we have µiϕi(x0) ≤ 0, i = 1, 2, ...,p, which in view of (3.2) implies that p∑ i=1 µiϕi(x0) ≤ p∑ i=1 µiϕi(x̃), which by quasiconvexity of p∑ i=1 µiϕi(.) at x̃ on F, we get (x0 − x̃)T p∑ i=1 µi∇ϕi(x̃) ≤ 0. (3.8) By similar arguments, we have (x0 − x̃)T∇ζi(x̃) ≤ 0,∀i ∈ Λ+ζ , −(x0 − x̃)T∇ζi(x̃) ≤ 0,∀i ∈ Λ−ζ , −(x0 − x̃)T∇`i(x̃) ≤ 0,∀i ∈ Λ++ ∪ Λ + 0 , (x0 − x̃)T∇`i(x̃) ≤ 0,∀i ∈ Λ−0 , −(x0 − x̃)T∇Φi(x̃) ≤ 0,∀i ∈ Λ−0+ ∪ Λ − 00 ∪ Λ − +0, (x0 − x̃)T∇Φi(x̃) ≤ 0,∀i ∈ Λ+00 ∪ Λ + 0− ∪ Λ + +0 ∪ Λ + +−, Int. J. Anal. Appl. 18 (5) (2020) 792 which by the definition of index sets one has (x0 − x̃)T [ q∑ i=1 γi∇ζi(x̃) − r∑ i=1 η`i∇`i(x̃) + r∑ i=1 ηΦi ∇Φi(x̃) ] ≤ 0. (3.9) On adding (3.7), (3.8) and (3.9), we get (x0 − x̃)T [ λL∇ΨL(x̃) + λU∇ΨU (x̃) + p∑ i=1 µi∇ϕi(x̃) + q∑ i=1 γi∇ζi(x̃) − r∑ i=1 η`i∇`i(x̃) + r∑ i=1 ηΦi ∇Φi(x̃) ] < 0, which contradicts (3.1). This completes the proof of this theorem. � Now, we verify the sufficient optimality conditions by the following example. Example 3.1. Consider the following interval-valued optimization problem: (IVVC-1) min x∈F1 Ψ(x) = [ΨL(x), ΨU (x)] = [x + x3,x5] subject to `1(x) = 1 + x 3 ≥ 0, Φ1(x)`1(x) = x(1 + x 3) ≤ 0, which is the form of (IVVC) with n = 1, p = q = 0 and r = 1. The feasible region of (IVVC-1) is F1 = {x ∈ R|`1(x) ≥ 0, Φ1(x)`1(x) ≤ 0}. Note that x̃ = 0 is a feasible solution of (IVVC-1) and it can be easily observe that there exist 0 < λL,λU ∈ R, η`1, and η Φ 1 ∈ R such that the relations (3.1)-(3.6) hold for the problem (IVVC-1). Also, it is not difficult to see that λLΨL(.) + λU ΨU (.) is pseudoconvex at x̃ on F1 and `1(x), Φ1(x) are quasiconvex at x̃ on F1. Since all the assumptions of Theorem 3.1 are satisfied, then x̃ = 0 is a LU optimal solution of the problem (IVVC-1). 4. Mond-Weir type duality We present the following Mond-Weir type dual for (IVVC). (IMWDVC) max Ψ(y) = [ ΨL(y), ΨU (y) ] subject to λL∇ΨL(y) + λU∇ΨU (y) + p∑ i=1 µi∇ϕi(y) + q∑ i=1 γi∇ζi(y) − r∑ i=1 η`i∇`i(y) + r∑ i=1 ηΦi ∇Φi(y) = 0, (4.1) Int. J. Anal. Appl. 18 (5) (2020) 793 µi ≥ 0,µiϕi(y) ≥ 0, ∀i = 1, 2, ...,p, (4.2) γi ∈ R,γiζi(y) = 0, ∀i = 1, 2, ...,q, (4.3) η`i ≥ 0, ∀i ∈ Λ+,η ` i ∈ R,∀i ∈ Λ0, (4.4) −η`i`i(y) ≥ 0, ∀i = 1, 2, ...,r, (4.5) 0 < λL,λU ∈ R,ηΦi ≤ 0, ∀i ∈ Λ0+, η Φ i ≥ 0, ∀i ∈ (Λ0− ∪ Λ+−), (4.6) ηΦi ∈ R, ∀i ∈ (Λ00 ∪ Λ+0), η Φ i Φi(y) ≥ 0, ∀i = 1, 2, ...,r. (4.7) We denote by W1 the set of all feasible solutions of the problem (IMWDVC) and let prW1 = {y ∈ Rn|(y,λL,λU,µ,γ,η`,ηΦ) ∈ W1} be the projection of the set W1 on Rn. Now, we prove duality results between problems (IVVC) and (IMWDVC) under certain generalized con- vexity assumptions imposed on the involved functions. Theorem 4.1 (Weak Duality). Let x ∈ F and (y,λL,λU,µ,γ,η`,ηΦ) ∈ W1. Further, assume that λLΨL(.)+ λU ΨU (.) is pseudoconvex at y on F ∪ prW1 and that p∑ i=1 µiϕi(.), ζi(.)(i ∈ Λ+ζ ), −ζi(.)(i ∈ Λ − ζ ), −`i(.)(i ∈ Λ++ ∪Λ + 0 ), `i(.)(i ∈ Λ − 0 ), −Φi(.)(i ∈ Λ − 0+ ∪Λ − 00 ∪Λ − +0), Φi(.)(i ∈ Λ + 00 ∪Λ + 0−∪Λ + +0 ∪Λ + +−) are quasiconvex at y on F∪ prW1, then Ψ(x) ≥LU Ψ(y). Proof. Suppose, contrary to the result, that Ψ(x) 0, λU > 0, therefore the above inequalities yield λLΨL(x) + λU ΨU (x) < λLΨL(y) + λU ΨU (y), which by pseudoconvexity of λLΨL(.) + λU ΨU (.) at y on F∪ prW1, we obtain (x−y)T [ λL∇ΨL(y) + λU∇ΨU (y) ] < 0. (4.8) Int. J. Anal. Appl. 18 (5) (2020) 794 For x ∈ F, µi ≥ 0, i = 1, 2, ...,p, we have µiϕi(x) ≤ 0, i = 1, 2, ...,p, which in view of (4.2) implies that p∑ i=1 µiϕi(x) ≤ p∑ i=1 µiϕi(y), which by quasiconvexity of p∑ i=1 µiϕi(.) at y on F∪ prW1, we get (x−y)T p∑ i=1 µi∇ϕi(y) ≤ 0. (4.9) By similar arguments, we have (x−y)T∇ζi(y) ≤ 0,∀i ∈ Λ+ζ , −(x−y)T∇ζi(y) ≤ 0,∀i ∈ Λ−ζ , −(x−y)T∇`i(y) ≤ 0,∀i ∈ Λ++ ∪ Λ + 0 , (x−y)T∇`i(y) ≤ 0,∀i ∈ Λ−0 , −(x−y)T∇Φi(y) ≤ 0,∀i ∈ Λ−0+ ∪ Λ − 00 ∪ Λ − +0, (x−y)T∇Φi(y) ≤ 0,∀i ∈ Λ+00 ∪ Λ + 0− ∪ Λ + +0 ∪ Λ + +−, which by the definition of index sets one has (x−y)T [ q∑ i=1 γi∇ζi(y) − r∑ i=1 η`i∇`i(y) + r∑ i=1 ηΦi ∇Φi(y) ] ≤ 0. (4.10) On adding (4.8), (4.9) and (4.10), we get (x−y)T [ λL∇ΨL(y) + λU∇ΨU (y) + p∑ i=1 µi∇ϕi(y) + q∑ i=1 γi∇ζi(y) − r∑ i=1 η`i∇`i(y) + r∑ i=1 ηΦi ∇Φi(y) ] < 0, which contradicts (4.1). This completes the proof of this theorem. � Now, we verify the weak duality theorem by the following example. Example 4.1. Consider the following interval-valued optimization problem: (IVVC-2) min x∈F2 Ψ(x) = [ΨL(x), ΨU (x)] = [x + x3,x5] subject to `1(x) = x 3 ≥ 0, Φ1(x)`1(x) = (−2 + x)x3 ≤ 0, which is the form of (IVVC) with n = 1, p = q = 0 and r = 1. The feasible region of (IVVC-2) is F2 = {x ∈ R|`1(x) ≥ 0, Φ1(x)`1(x) ≤ 0}. Int. J. Anal. Appl. 18 (5) (2020) 795 For any feasible x ∈ F2, the corresponding Mond-Weir type dual problem for the primal problem (IVVC-2) is given by (IMWDVC-1) max Ψ(y) = [ ΨL(y), ΨU (y) ] = [y + y3,y5] subject to λL∇ΨL(y) + λU∇ΨU (y) −η`1∇`1(y) + η Φ 1 ∇Φ1(y) = λL + 3λLy2 + 5λUy4 − 3η`1y 2 + ηΦ1 = 0, η`1 ≥ 0, if 1 ∈ Λ+, η ` 1 ∈ R, if i ∈ Λ0, ηΦ1 ≤ 0, if 1 ∈ Λ0+, η Φ 1 ≥ 0, if 1 ∈ (Λ0− ∪ Λ+−), η Φ 1 ∈ R, if i ∈ (Λ00 ∪ Λ+0), 0 < λL,λU ∈ R, −η`1`1(y) ≥ 0, η Φ 1 Φ1(y) ≥ 0. Let W2 be the set of all feasible solutions of the problem (IMWDVC-1) and note that, (y,λL,λU,η`1,η Φ 1 ) = (0, 1 2 , 1 2 , 1,−1 2 ) is a feasible solution for (IMWDVC-1). Furthermore, it is not difficult to see that λLΨL(.) + λU ΨU (.) is pseudoconvex at y on F2 ∪ prW2 and `1(.), Φ1(.) are quasiconvex at y on F1 ∪ prW2. Also, for the feasible solutions x = 1 for (IVVC-2) and (y,λL,λU,η`1,η Φ 1 ) = (0, 1 2 , 1 2 , 1,−1 2 ) for (IMWDVC- 1), we observe that Ψ(x) >LU Ψ(y). Hence the weak duality Theorem 4.1 is verified. Theorem 4.2 (Strong Duality). Let x̃ be a LU optimal solution to the problem (IVVC) and the generalized Guignard constraint qualification (IVVC-GCQ) is satisfied at x̃. Then there exist λ̃U > 0, λ̃L > 0, µ̃ ∈ Rp+, γ ∈ Rq, η` ∈ Rr, and ηΦ ∈ Rr such that (x̃,λL,λU,µ,γ,η`,ηΦ) is a feasible solution for (IMWDVC) and the two objective values are same. Further, if all the assumptions of the Theorem 4.1 are fulfilled, then the point (x̃,λL,λU,µ,γ,η`,ηΦ) is a LU optimal solution of (IMWDVC). Proof. By assumption x̃ is a LU optimal solution for (IVVC) and the generalized Guignard constraint qualification (IVVC-GCQ) is satisfied at this point, then by Theorem 2.1, there exist λ̃U > 0, λ̃L > 0, µ̃ ∈ Rp+, γ ∈ Rq, η` ∈ Rr, and ηΦ ∈ Rr such that the conditions (2.1)-(2.6) are satisfied. Thus, (x̃,λL,λU,µ,γ,η`,ηΦ) is feasible in (IMWDVC), moreover, the corresponding objective values of (IVVC) and (IMWDVC) are equal. Further, if (x̃,λL,λU,µ,γ,η`,ηΦ) is not a LU optimal solution to (IMWDVC), then there exists a feasible solution (ỹ,λL,λU,µ,γ,η`,ηΦ) for (IMWDVC), such that the following inequality is satisfied Ψ(x̃) λ̃LΨL(ỹ) + λ̃U ΨU (ỹ), which contradicts (4.11). This completes the proof of this theorem. � 5. Conclusion In this paper, we have derived sufficient optimality conditions for an interesting class of interval-valued optimization problems with vanishing constraints under generalized convexity assumptions. 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Optim. 14 (3) (2018), 1157-1178. 1. Introduction 2. Preliminaries 3. Sufficient optimality conditions 4. Mond-Weir type duality 5. Conclusion References