EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6617 ISSN 1307-5543 – ejpam.com Published by New York Business Global A Fractional Calculus Approach to Interval-Valued Variational Programming Problems Vivekananda Rayanki1, Krishna Kummari2, Izhar Ahmad3,4, Thiti Gaketem5,∗ 1 Department of Mathematics, Vallurupalli Nageswara Rao Vignana Jyothi Institute of Engineering and Technology, Vignana Jyothi Nagar, Pragathi Nagar, Hyderabad - 500090, Telangana, India 2 Department of Mathematics, School of Science, GITAM-Hyderabad Campus, Hyderabad-502329, India. 3 Department of Mathematics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia 4 Center for Intelligent Secure Systems, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia 5 Department of Mathematics, School of Science, University of Phayao, Phayao 56000, Thailand Abstract. This study explores a class of fractional interval-valued variational programming prob- lems involving the Caputo-Fabrizio (C-F) fractional derivative. By employing the concepts of invex and generalized invex functions, we establish sufficient optimality conditions for these problems. Additionally, we develop a Wolfe-type dual formulation and investigate the corresponding duality relationships. In particular, we derive and prove the weak, strong, and converse duality theorems to establish a connection between the primal and dual problems. The theoretical findings are fur- ther illustrated through carefully constructed numerical examples, demonstrating the applicability and effectiveness of the proposed approach. 2020 Mathematics Subject Classifications: 26A51, 49J40, 49K99, 90C46 Key Words and Phrases: Variational programming problem, sufficient optimality conditions, LU-optimality, Caputo-Fabrizio fractional derivative, Wolfe-type duality. 1. Introduction The optimization theory acknowledges interval-valued programming as a crucial compo- nent. In various scientific and mathematical domains, interval-valued optimization has recently gained popularity. Due to the uncertainty of the theory underpinning the param- eters, estimating a physical world system’s parameters is challenging. We can see that ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6617 Email addresses: rayankee@gmail.com (R. Vivekananda), krishna.maths@gmail.com (K. Kummari), drizhar@kfupm.edu.sa (I. Ahmad) thiti.ga@up.ac.th (T. Gaketem) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 2 of 38 there are several applications in many different fields, including decision-making [1], di- agnostic [2], portfolio optimization [3], financial planning, business planning, healthcare, production, hospital planning and management, and many more. In the very recent work, [4] studied a new class of optimization problems governed by interval-valued variational programming and inequalities. Its results work in applications of Control and Optimization problems. And at the same time, [5] discussed some results on solutions associated with interval-valued optimal control problems driven by gener- alized invariant convex (invex) functionals and also investigated necessary and sufficient optimality conditions for the considered optimization problem. Before them,[6] developed a framework for analyzing problems involving interval-valued optimization. Reading the books [7–9] and some recent articles [4–6, 10, 11] will help to learn the basics of interval- valued optimization. Problems with variational programming start with the calculus of variations. Recent ad- vances in variational calculus and optimization theory have given us a cogent framework for analyzing a range of issues in numerous other fields of pure and applied mathematics. The dynamics of rigid bodies, orbit optimization [12], flight design [13, 14], and other prob- lems are among the areas where the calculus of variations is utilized to address problems. Using invexity assumptions, [15] developed some optimality requirements and theorems of duality for interval-valued optimization problems. Later, [11] modified the definitions of pre-invexity and generalized invexity in interval-valued functions and additionally devel- oped Karush-Kuhn-Tucker optimality requirements for the optimization problem under consideration, where the objective function was assumed to be interval-valued. Recently, [16] used extended (p, r)−ρ− (ℵ, θ)-invexity for a problem of interval-valued optimization to study optimality and duality. [17] established duality conclusions for cases involving variational programming. Jimenez et al., [18], developed some duality solutions for the multiobjective variational issue utilizing the pseudo-invexity notion. Treanctua et al. [19] investigated efficiency conditions in interval-valued control models using a modified objec- tive functional and saddle-point criteria. Numerous additional studies have discussed the issues with variational programming (see, for example, [20, 21]). Fractional Calculus (FC) is currently recognized as a fascinating subject by the community of practical engineers. It is a generalization of conventional calculus because derivatives and integrals are employed outside of integer orders. Fractional calculus has numerous uses in science and engineering [22], and it has grown in popularity in recent years as a tool for researching the dynamics of practical problems. [23, 24] looked into a few com- mon variational issues by incorporating fractional derivatives of Riemann-Liouville [25], Caputo, and Riesz types. In other research work [26], variational problems pertaining to a Lagrangian function given fractional derivatives have been analyzed to determine opti- mality conditions. Shalini et al. [27] investigate dynamical control systems using a fuzzy modeling framework, wherein the system dynamics are governed by a fuzzy stochastic process (FSP) driven by fuzzy Brownian motion (FBM). Fuzzy fractional variational problems (FVPs) have been explored in the literature for nec- V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 3 of 38 essary optimality conditions, as seen in the works of [28] and [29]. These studies provide foundational insights into the interplay between fuzzy logic and fractional calculus, par- ticularly in variational settings. Caputo and Fabrizio [30] introduced a novel fractional derivative, characterized by the order θ• ∈ (0, 1), defined using an exponential kernel. This derivative avoids singular kernels and is well-suited for modeling systems with memory ef- fects, which marks a significant departure from classical Riemann–Liouville and Caputo definitions that employ singular kernels. Jayswal and Uniyal [31] presented necessary and sufficient optimality conditions and Mond-Weir duality results for semi-infinite vari- ational programming (SIVP) problems involving Caputo-Fabrizio fractional derivatives. Their analysis leveraged Slater-type constraint qualifications and generalized convexity assumptions. In a follow-up work [32], they extended this framework to investigate opti- mality conditions for broader classes of semi-infinite fractional variational problems with similar structural features. The present study advances the existing literature by consid- ering interval-valued variational programming problems (P) governed by Caputo-Fabrizio fractional derivatives. In contrast to previous works that primarily address crisp-valued problems, the incorporation of interval-valued objective and constraint functions intro- duces a layer of uncertainty and imprecision, which is more reflective of real-world sys- tems. Moreover, unlike earlier criteria that focus on classical convexity, this work applies LU-optimality conditions and generalized-invexity criteria that offer a broader, more flexi- ble framework for establishing optimality in uncertain fractional variational environments. The proposed approach also contributes by establishing Karush-Kuhn-Tucker-type suffi- cient optimality conditions and analyzing Wolfe-type duality in the context of interval- valued programming problems with Caputo-Fabrizio derivatives. These enhancements allow for a more generalized efficiency framework than those discussed in [28] [31], and [29], and facilitate a better understanding of solution robustness under fuzzy uncertainty and memory effects introduced by the exponential kernel. We will now continue to discuss this article’s contents. Several fractional calculus ideas, concepts, and features were reviewed in preliminary section 2. We also go through the LU-Optimal approach to the programming problem of variational calculus. We examine the fractional derivative C-F for the KKT-type sufficient optimality conditions in Section 3. We establish and verify weak, strong, and strict converse duality theorems for the Wolfe-type dual model in section 4 and section 5 contains the article’s conclusion and future direction. 2. Preliminaries This section recollects some notations, symbols, and definitions that will be important in the follow-up to this work. For any number of intervals A = [ aL, aU ] and B = [ bL, bU ] where aL, aU , bL, bU ∈ R, we define the following partial ordering relations on the lines of [33] and [21]: (i) A ⪯LU B ⇐⇒ aL ≤ bLand aU ≤ bU . V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 4 of 38 (ii) A ≺LU B ⇐⇒ aL < bL, aU ≤ bU , or aL ≤ bL, aU < bU , or aL < bL, aU < bU . Throughout the paper, κ : [a1, a2] → R is a function of class C1 and θ• ∈ (0, 1). Definition 1. [8] Left and right Fractional derivatives of Riemann-Liouville of order θ• are defined by a1Dθ• ς κ(ς) = 1 Γ(1− θ•) d dς ∫ ς a1 (ς − ν)−θ•κ(ν)dν, ςDθ• a2κ(θ •) = − 1 Γ(1− θ•) d dς ∫ a2 ς (ν − ς)−θ•κ(ν)dν, θ• ∈ (0, 1). Definition 2. [34] The fractional derivative of Caputo, κ(ς) : [a1, a2] → R of order θ• ∈ (0, 1) is stated to be cDθ• a1+κ(ς) = 1 Γ(1− θ•) d dς ∫ ς a1 1 (ς − ν)θ• [κ(ν)− κ(a)]dν. If κ ∈ C1, then cDθ• a1+κ(ς) = 1 Γ(1− θ•) d dς ∫ ς a1 1 (ς − ν)θ• κ′(ν)dν. As θ• → 1, cDθ• a1+κ(ς) approaches to κ′(ς). Definition 3. [30] The operator of new Caputo-Fabrizio (CF) fractional derivative is described as CFDθ• a1+κ(ς) = κ(θ•) (1− θ•) ∫ ς a1 exp ( − θ•(ς − ν) (1− θ•) ) κ′(ν)dν, θ• ∈ (0, 1), where κ(θ•) signifies the normalization function (1− θ•)+ θ• Γ(θ•) with the property κ(0) = κ(1) = 1. Clearly CFDθ• a1+κ(ς) = 0, if κ(ς) is a constant function, i.e, a constant function’s CF derivative equals zero, but the CF derivative lacks a unique kernel for ς = ν, like the Caputo derivative. Remark 1. As θ• → 1, CFDθ• a1+κ(ς) ⇒ κ′(ς) and as θ• → 0, CFDθ• a1+κ(ς) ⇒ κ(ς)−κ(a1). Definition 4. [1] The right Caputo-Fabrizio fractional derivative is defined as CFDθ• a2−κ(ς) = κ(θ•) (1− θ•) ∫ a2 ς exp ( − θ•(ς − ν) (1− θ•) ) κ′(ν)dν, θ• ∈ (0, 1). Definition 5. The order of Sobolev space 1 ∈ (a1, a2) is defined: H1(a1, a2) = {y ∈ L2(a1, a2) | y ′ ∈ L2(a1, a2)}, y′ is the weak derivative of y. V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 5 of 38 Definition 6. [30] Let κ ∈ H1(a1, a2), a2 > a1, 0 < θ• < 1. The fractional derivative of CF is therefore given as in Definition 3, where κ(θ•) specifies the function of normalization characterized by κ(0) = κ(1) = 1. If the function is κ /∈ H1(a1, a2), then the derivative is written in the following manner: CFDθ• a1+κ(ς) = θ•κ(θ•) (1− θ•) ∫ ς a1 exp ( − θ•(ς − ν) (1− θ•) ) [κ(ς)− κ(ν)] dν, where CF has an exponential kernel. Definition 7. [1, 35] Let κ be a function with the property that κ ∈ H ′ (a1, a2), a1 < a2. In the Caputo-Fabrizio sense, the order of the left Riemann fractional derivative θ• is expressed as CFRDθ• a1+κ(ς) = κ(θ•) (1− θ•) d dς ∫ ς a1 exp ( − θ•(ς − ν) (1− θ•) ) κ(ν)dν, (1) where a1 ≤ ς, θ• (0 < θ• < 1) is a real number and κ(θ•) is a normalization function that depends on θ• such that κ(0) = κ(1) = 1. Similarly, in the Caputo-Fabrizio sense, the order of right Riemann fractional derivative θ• can be stated as follows: CFRDθ• a2−κ(ς) = κ(θ•) (1− θ•) d dς ∫ a2 ς exp ( − θ•(ς − ν) (1− θ•) ) κ(ν)dν, where ς ≤ a2. Remark 2. When θ• → 0, (1) becomes lim θ•→0 CFRDθ• a1+κ(ς) = d dς ∫ ς a1 κ(ν)dν = κ(ς). Proposition 1 (Abdeljawad and Baleanu [1]). Let θ• ∈ (0, 1) and κ, z : [a1, a2] → R be two continuous functions of class C1[a1, a2]. Then the following integration by parts formula holds true:∫ a2 a1 κ(ς)CFDθ• a1+z(ς)dς = [z(ς)I1−θ• a2− κ(ς)] ∣∣∣∣ς=a2 ς=a1 + ∫ a2 a1 z(ς)CFRDθ• a2−κ(ς)dς, where I1−θ• a2− κ(ς) = K(θ•) (1− θ•) ∫ a2 ς exp ( − θ• (1− θ•) (ν − ς) ) κ(ν)dν. Let 𭟋 : ℑ × Rn × Rn → R be a continuously differentiable function where ℑ = [a1, a2] is real valued interval. Now, we are dealing with the function 𭟋(ς,κ(ς),CFDθ• a1+κ(ς)), where κ : ℑ → Rn is an n-dimensional function of class C1[a1, a2] and CFDθ• a1+κ represents the Caputo-Fabrizio derivative of order 0 < θ• < 1 for a function κ. The partial derivatives of 𭟋 are represented by 𭟋ς = ∂𭟋 ∂ς ,𭟋κ = [ ∂𭟋 ∂κ1 , ∂𭟋 ∂κ2 , ∂𭟋 ∂κ3 , ..., ∂𭟋 ∂κn ] , V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 6 of 38 𭟋 CFDθ•a1+ κ = [ ∂𭟋 ∂(CFDθ• a1+ κ1) , ∂𭟋 ∂(CFDθ• a1+ κ2) , ∂𭟋 ∂(CFDθ• a1+ κ3) , ..., ∂𭟋 ∂(CFDθ• a1+ κn) ] . Where κ1,κ2,κ3, ...,κn are components of κ. Consider the space of piecewise smooth functions to be κ : ℑ → Rn along with the norm ∥κ∥ = ∥κ∥∞ + ∥Dκ∥∞, where the differential operator D is described as follows: v = Dκ ⇐⇒ κ(ς) = κ0 + ∫ ς a1 v(s)ds, where κ0 signifies the boundary value. Let 𭟋 : X → R defined by 𭟋(κ) = ∫ a2 a1 𭟋(ς,κ(ς),CFDθ• a1+κ(ς))dς be differentiable. For notational convenience, 𭟋(ς,κ(ς),CFDθ• a1+κ(ς)) will be written as 𭟋(ς,κ,CFDθ• a1+κ). Now, we define the concept of invex and generalized invex functions by using the Caputo- Fabrizio (CF) fractional derivative of order 0 < θ• < 1, in the following way: Definition 8. The functional 𭟋 is stated as invex (strictly invex) with respect to ℵ if a differentiable vector function ℵ(ς,κ, κ̄) ∈ C1[a1, a2] with ℵ(ς,κ,κ) = 0 occurs such that forall κ, κ̄ ∈ X, a2∫ a1 𭟋(ς,κ,CFDθ• a1+κ)dς − a2∫ a1 𭟋(ς, κ̄,CFDθ• a1+κ̄)dς ≥ (>) ∫ a2 a1 [ ℵ(ς,κ, κ̄)𭟋κ̄(ς, κ̄,CFDθ• a1+κ̄) + (CFDθ• a1+ℵ(ς,κ, κ̄))𭟋CFDθ•a1+ κ̄ (ς, κ̄,CFDθ• a1+κ̄) ] dς. The following example demonstrates that, the function 𭟋 is invex under Caputo-Fabrizio fractional derivative. Example 1. : Let 𭟋(κ) : X = [0, 1] → R be defined by 𭟋(κ) = 1∫ 0 {−κ(ς) + 6.5514ς − 16.3785e − ς 3 + 22.9299)}dς and κ(ς) = −ς2 + ς + 1 ∈ X, forall ς ∈ [0, 1] and note that (κ ̸= κ̄) κ̄(ς) = 1 ∈ X in such way that, a2∫ a1 𭟋(ς,κ,CFDθ• a1+κ)dς − a2∫ a1 𭟋(ς, κ̄,CFDθ• a1+κ̄)dς > a2∫ a1 {ℵ(ς,κ, κ̄)𭟋κ̄(ς, κ̄,CFDθ• a1+κ̄) + CFDθ• a1+(ℵ(ς,κ, κ̄)) 𭟋CFDθ• a1+ κ̄(ς, κ̄, CFDθ• a1+κ̄)}dς. In this example, θ• = 1 4 , a1 = 0, a2 = 1, ℵ(ς,κ, κ̄) = κ − κ̄ and κ(ς) = −ς2 + ς + 1 are V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 7 of 38 Figure 1: Graphical view of the function 𭟋 = −κ(ς) + 6.5514ς − 16.3785e − ς 3 + 22.9299 taken relevantly and deduce CFDθ• a1+κ(ς) = −6.5514ς−22.9299e−ς/3+22.9299. But in the given example, the function is not convex for the differentiable function (κ− κ̄)T = κ+ κ̄. Definition 9. The functional 𭟋 is stated as pseudo-invex with regard to ℵ if a differentiable function ℵ(ς,κ, κ̄) ∈ C1[a1, a2] with ℵ(ς,κ,κ) = 0 occurs such that ∀κ, κ̄ ∈ X,∫ a2 a1 [ ℵ(ς,κ, κ̄)𭟋κ̄(ς, κ̄,CFDθ• a1+κ̄) + (CFDθ• a1ℵ(ς,κ, κ̄))𭟋CFDθ•a+κ̄ (ς, κ̄,CFDθ• a1κ̄) ] dς ≥ 0 ⇒ a2∫ a1 𭟋(ς,κ,CFDθ• a1+κ)dς ≥ a2∫ a1 𭟋(ς, κ̄,CFDθ• a1+κ̄)dς, or equivalently, a2∫ a1 𭟋(ς,κ,CFDθ• a1+κ)dς < a2∫ a1 g(ς, κ̄,CFDθ• a1+κ̄)dς ⇒ ∫ a2 a1 [ ℵ(ς,κ, κ̄)𭟋κ̄(ς, κ̄,CFDθ• a1+κ̄) +CFDθ• a1+ℵ(ς,κ, κ̄))𭟋CFDθ•a1+ κ̄ (ς, κ̄,CFDθ• a1+κ̄) ] dς < 0. V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 8 of 38 The following example illustrates that, the function G is pseudo-invex, but not a invex under Caputo-Fabrizio fractional derivative. Example 2. : Let 𭟋(κ) : X = [0, 1] → R be defined by 𭟋(κ) = 1∫ 0 {−κ(ς) + 2.1838ς + 9.8272e − ς 3 − 7.6459)}dς and κ(ς) = 1 3 ς2 − 3.6666667 11 ς +1 ∈ X, forall ς ∈ [0, 1] and note that (κ ̸= κ̄) κ̄(ς) = 1 ∈ X in such way that, a2∫ a1 {ℵ(ς,κ, κ̄)𭟋κ̄ (ς, κ̄,CFDθ• a1+κ̄) + CFDθ• a1+(ℵ(ς, κ, κ̄)) 𭟋CFDθ• a1+ κ̄(ς, κ̄, CF Dθ• a1+ κ̄)} dς > 0 ⇒ a2∫ a1 𭟋(ς,κ,CFDθ• a1+κ)dς > a2∫ a1 𭟋(ς, κ̄,CFDθ• a1+κ̄)dς. But, 𭟋 = 1∫ 0 {−κ(ς) + 2.1838ς + Figure 2: Graphical view of the function 𭟋 = −κ(ς) + 2.1838ς + 9.8272e − ς 3 − 7.6459 9.8272e − ς 3 − 7.6459}dς is not invex at κ̄ = 1 ∈ X, that is a2∫ a1 𭟋(ς,κ,CFDθ• a1+κ)dς − a2∫ a1 𭟋(ς, κ̄,CFDθ• a1+κ̄)dς ≯ a2∫ a1 {ℵ(ς,κ, κ̄)𭟋κ̄(ς, κ̄,CFDθ• a1+κ̄) +CFDθ• a1+(ℵ(ς,κ, κ̄))𭟋CFDθ• a1+ κ̄(ς, κ̄, CFDθ• a1+κ̄)}dς. This example provided, on taking θ• = 1 4 , a1 = 0, a2 = 1, ℵ(ς, κ, κ̄) = κ − κ̄, κ(ς) = V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 9 of 38 1 3 ς2 − 3.6666667 11 ς + 1 and deduce CFDθ• a1+κ(ς) = 2.1838ς + 10.919e−ς/3 − 7.6459. Definition 10. The functional 𭟋 is stated as strictly pseudo-invex with regards to ℵ if a differentiable function ℵ(ς,κ, κ̄) ∈ C1[a1, a2] with ℵ(ς,κ,κ) = 0 occurs such that ∀ κ, κ̄ ∈ X, ∫ a2 a1 [ ℵ(ς,κ, κ̄)𭟋κ̄(ς, κ̄,CFDθ• a1+κ̄) +CFDθ• a1+ℵ(ς,κ, κ̄))𭟋CFDθ•a1+ κ̄ (ς, κ̄,CFDθ• a1+κ̄) ] dς ≥ 0 ⇒ a2∫ a1 𭟋(ς,κ,CFDθ• a1+κ)dς > a2∫ a1 𭟋(ς, κ̄,CFDθ• a1+κ̄)dς, or, alternatively a2∫ a1 𭟋(ς,κ,CFDθ• a1+𭟋)dς ≤ a2∫ a1 𭟋(ς, κ̄,CFDθ• a1+κ̄)dς ⇒ ∫ a2 a1 [ ℵ(ς,κ, κ̄)𭟋κ̄(ς, κ̄,CFDθ• a1+κ̄) +CFDθ• a1+ℵ(ς,κ, κ̄))𭟋CFDθ•a+κ̄ (ς, κ̄,CFDθ• a1+κ̄) ] dς < 0. Definition 11. The functional 𭟋 is stated as quasi-invex in respect to ℵ if a differentiable function ℵ(ς,κ, κ̄) ∈ C1[a1, a2] with ℵ(ς,κ,κ) = 0 occurs in such way that ∀ κ, κ̄ ∈ X,∫ a2 a1 [ ℵ(ς,κ, κ̄)𭟋κ̄(ς, κ̄,CFDθ• a1+κ̄) +CFDθ• a1+ℵ(ς,κ, κ̄))𭟋CFDθ•a1+ κ̄ (ς, κ̄,CFDθ• a1+κ̄) ] dς > 0 ⇒ a2∫ a1 𭟋(ς,κ,CFDθ• a1+κ)dς > a2∫ a1 𭟋(ς, κ̄,CFDθ• a1+κ̄)dς, or equivalently, a2∫ a1 𭟋(ς,κ,CFDθ• a1+κ)dς ≤ a2∫ a1 𭟋(ς, κ̄,CFDθ• a1+κ̄)dς ⇒ ∫ a2 a1 [ ℵ(ς,κ, κ̄)𭟋κ̄(ς, κ̄,CFDθ• a1+κ̄) +CFDθ• a1+ℵ(ς,κ, κ̄))𭟋CFDθ•a1+ κ̄ (ς, κ̄,CFDθ• a1+κ̄) ] dς ≤ 0. V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 10 of 38 The following example shows that G is a quasi-invex function but neither invex nor pseudo- invex. Example 3. : Let 𭟋(κ) : X = [0, 1] → R be defined by 𭟋(κ) = 1∫ 0 {−κ(ς) + 1.871822ς + 6.5514e − ς 3 − 5.6154}dς and κ(ς) = 2 7 ς2 − 3.1428 11 ς + 1 ∈ X, forall ς ∈ [0, 1] and note that (κ ̸= κ̄) κ̄(ς) = 1 ∈ X such that, a2∫ a1 {ℵ(ς,κ, κ̄)𭟋κ̄(ς, κ̄,CFDθ• a1+κ̄) +CFDθ• a1+(ℵ(ς,κ, κ̄))𭟋CFDθ• a1+ κ̄(ς, κ̄, CFDγ a+κ̄)}dς > 0 ⇒ a2∫ a1 𭟋(ς,κ,CFDθ• a1+κ)dς > a2∫ a1 𭟋(ς, κ̄,CFDθ• a1+κ̄)dς. But, 𭟋 is neither invex nor pseudo-invex, that is a2∫ a1 𭟋(ς,κ,CFDθ• a1+κ)dς − a2∫ a1 𭟋(ς, κ̄,CFDθ• a1+κ̄)dς ≯ a2∫ a1 {ℵ(ς,κ, κ̄)𭟋κ̄(ς, κ̄,CFDθ• a1+κ̄) +CFDθ• a1+(ℵ(ς,κ, κ̄))𭟋CFDθ• a1+ κ̄(ς, κ̄, CFDθ• a1+κ̄)}dς. and a2∫ a1 𭟋(ς,κ,CFDθ• a1+κ)dς ≮ a2∫ a1 𭟋(ς, κ̄,CFDθ• a1+κ̄)dς ⇒ a2∫ a1 {ℵ(ς,κ, κ̄)𭟋κ̄(ς, κ̄,CFDθ• a1+κ̄) +CFDθ• a1+(ℵ(ς,κ, κ̄))𭟋CFDθ• a1+ κ̄(ς, κ̄, CFDθ• a1+κ̄)}dς ≮ 0. In construction of this example, relevantly taken θ• = 1 4 , a1 = 0, a2 = 1, ℵ(ς,κ, κ̄) = κ−κ̄, κ(ς) = 2 7 ς2− 3.1428 11 ς+1 and deduce CFDθ• a1+κ(ς) = 1.871822ς+6.5514e − ς 3−5.6154. V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 11 of 38 Figure 3: Graphical view of the function 𭟋 = −κ(ς) + 1.871822ς + 6.5514e − ς 3 − 5.6154 In the above Definitions (8) - (11), CFDθ• a1+ℵ(ς,κ, κ̄) is the vector whose i th component is dθ • dςθ• ℵi(ς,κ, κ̄). Let ϕ(ς, κ,CFDθ• a1+κ(ς)) = [ ϕL(ς,κ, CFDθ• a1+y(ς)), ϕU (ς,κ,CFDθ• a1+κ(ς)) ] be an interval-valued function and h(ς, κ, CFDθ• a1+y(ς)) be a m-dimensional function hav- ing continuous derivatives up to the second order with regard to each of its parameters. Here, κ is a n-dimensional function of ς, and CFDθ• a1+y(ς) is the CF fractional derivative of order θ• with respect to ς, where 0 < θ• < 1. Let us Consider the following problem (P) under Caputo-Fabrizio fractional derivative: (P ) : minφ(κ)κ∈X =  a2∫ a1 ϕL(ς,κ(ς),CFDθ• a1+κ(ς))dς, a2∫ a1 ϕU (ς, y(ς),CFDθ• a1+κ(ς))dς  subject to, κ(a1) = α, κ(a2) = β, (2) h(ς,κ(ς),CFDθ• a1+κ(ς)) ≤ 0, ς ∈ ℑ. (3) The region (feasibility region), where the restrictions are satisfied, is provided by Φ = {κ ∈ X : κ(a1) = α,κ(a2) = β, h(ς,κ(ς),CFDθ• a1+κ(ς)) ≤ 0, ς ∈ ℑ = [a1, a2]}. Definition 12. A feasible point κ̄ is said to be a LU optimal solution of the problem (P), V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 12 of 38 if there is no feasible point κ ∈ F such that, a2∫ a1 ϕL(ς,κ(ς),CFDθ• a1+κ(ς)dς, a2∫ a1 ϕU (ς,κ(ς),CFDθ• a1+κ(ς)dς  ≺LU  a2∫ a1 ϕL(ς, κ̄(ς),CFDθ• a1+κ̄(ς)dς, a2∫ a1 ϕU (ς, κ̄(ς),CFDθ• a1+κ̄(ς)dς  . For convenience we write as, ϕ(ς, κ̄,CFDθ• a1+κ̄) in the place of ϕ(ς, κ̄(ς), CFDθ• a1+ κ̄(ς)). 3. Optimality Conditions The following KKT necessary optimality conditions were established in [36] for the prob- lem(P), which will be used to demonstrate the sufficient conditions and strong duality in the subsequent sections of the paper. Theorem 1 (Karush-Kuhn-Tucker Necessary Optimaliy Conditions). Let κ̄ be the LU optimum solution of (P) with Slater’s constraint qualification satisfied at κ̄. Then, ∃ a function that is piecewise smooth, Θ̄ : ℑ → Rm, such that (κ̄, Θ̄) satisfies, ϕL κ̄(ς, κ̄,CFDθ• a1+κ̄) + ϕU κ̄ (ς, κ̄,CFDθ• a1+κ̄) + Θ̄(ς)hκ̄(ς, κ̄,CFDθ• a1+κ̄) = −CFRDθ• b− [ ϕL CFDθ•a+κ̄(ς) (ς, κ̄,CFDθ• a1+κ̄) + ϕU CFDθ•a+κ̄(ς) (ς, κ̄,CFDθ• a1+κ̄) +Θ̄(ς)hCFDθ• a1+ κ̄(ς)(ς, κ̄, CFDθ• a1+κ̄) ] , (4) Θ̄(ς)h(ς, κ̄,CFDθ• a1+κ̄) = 0, Θ̄(ς) ≥ 0. (5) Theorem 2 (Sufficient Optimality Conditions). Let κ̄ ∈ X be a feasible solution of (P) and there is a piecewise smooth function Θ̄: ℑ → Rm, Θ̄(ς) ≥ 0 such that equations (4) and (5) are satisfied at (κ̄, Θ̄). Also, assume that (i) the functional a2∫ a1 (ϕL + ϕU )(ς,κ(ς),CFDθ• a1+κ(ς))dς is invex at κ̄ on X, (ii) the functional a2∫ a1 Θ̄(ς)h(ς, κ̄,CFDθ• a+κ̄)dς is invex at κ̄ on X, then κ̄ is a LU optimal solution for (P). Proof. If κ̄ is not a LU optimum solution for (P), then by Definition 12 another feasible solution κ for (P) exists, such that a2∫ a1 ϕL(ς,κ,CFDθ• a1+κ)dς, a2∫ a1 ϕU (ς,κ,CFDγ a1+κ)dς  V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 13 of 38 ≺LU  a2∫ a1 ϕL(ς, κ̄,CFDθ• a1+κ̄)dς, a2∫ a1 ϕU (ς, κ̄,CFDθ• a1+κ̄)dς  . Thus, we have  a2∫ a1 ϕL(ς,κ,CFDθ• a1+κ)dς < a2∫ a1 ϕL(ς, κ̄,CFDθ• a1+κ̄)dς a2∫ a1 ϕU (ς,κ,CFDθ• a1+κ)dς ≤ a2∫ a1 ϕU (ς, κ̄,CFDθ• a1+κ̄)dς, or  a2∫ a1 ϕL(ς,κ,CFDθ• a1+κ)dς ≤ a2∫ a1 ϕL(ς, κ̄,CFDθ• a1+κ̄)dς a2∫ a1 ϕU (ς,κ,CFDθ• a1+κ)dς < a2∫ a1 ϕU (ς, κ̄,CFDθ• a1+κ̄)dς, or  a2∫ a1 ϕL(ς,κ,CFDθ• a1+κ)dς < a2∫ a1 ϕL(ς, κ̄,CFDθ• a1+κ̄)dς a2∫ a1 ϕU (ς,κ,CFDθ• a1+κ)dς < a2∫ a1 ϕU (ς, κ̄,CFDθ• a1+κ̄)dς. From the above inequalities, we get a2∫ a1 [ ϕL + ϕU ] (ς,κ,CFDθ• a1+κ)dς < a2∫ a1 [ ϕL + ϕU ] (ς, κ̄,CFDθ• a1+κ̄)dς (6) By hypothesis(i), the functional a2∫ a1 (ϕL + ϕU )(ς,κ,CFDθ• a1+κ)dς is invex at κ̄ ∈ X. Thus, from the Definition 8, we have a2∫ a1 𭟋(ς,κ,CFDθ• a1+κ)dς − a2∫ a1 𭟋(ς, κ̄,CFDθ• a1+κ̄)dς ≥ (>) ∫ a2 a1 [ ℵ(ς,κ, κ̄)𭟋κ̄(ς, κ̄,CFDθ• a1+κ̄) +(CFDθ• a1+ℵ(ς,κ, κ̄))𭟋CFDθ•a1+ κ̄ (ς, κ̄,CFDθ• a1+κ̄) ] dς, which along with (6), gives a2∫ a1 {ℵ(ς,κ, κ̄) [ ϕL κ̄ + ϕU κ̄ ] (ς, κ̄,CFDθ• a1+κ̄) V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 14 of 38 +CFDθ• a1+ℵ(ς,κ, κ̄) [ ϕL CFDθ• a1+ κ̄ + ϕU CFDθ• a1+ κ̄ ] (ς, κ̄,CFDθ• a1+κ̄)}dς < 0 (7) On the other hand, from (4) together with Proposition 1, yields a2∫ a1 ℵ(ς,κ, κ̄)[ϕL κ̄(ς, κ̄,CFDθ• a1+κ̄) + ϕU κ̄ (ς, κ̄,CFDθ• a1+κ̄) + Θ̄(ς)hκ̄(ς, κ̄,CFDθ• a1+κ̄)]dς = a2∫ a1 ℵ(ς,κ, κ̄)(−CFDθ• b−)[ϕ L CFDθ• a1+ κ̄(ς, κ̄, CFDθ• a1+κ̄) + ϕU CFDθ• a1+ κ̄(ς, κ̄, CFDθ• a1+κ̄) + Θ̄(ς)hCFDθ• a1+ κ̄(ς, κ̄, CFDθ• a1+κ̄)]dς = {ℵ(ς,κ, κ̄)I1−θ• b− [ϕL κ̄(ς, κ̄,CFDθ• a1+κ̄) + ϕU κ̄ (ς, κ̄,CFDθ• a1+κ̄) + Θ̄(ς)hκ̄(ς, κ̄,CFDθ• a1+κ̄)]}| b a − a2∫ a1 {ϕL CFDθ• a1+ κ̄(ς, κ̄, CFDθ• a1+κ̄) + ϕU CFDθ• a1+ κ̄(ς, κ̄, CFDθ• a1+κ̄) + Θ̄(ς)hCFDθ• a1+ κ̄(ς, κ̄, CFDθ• a1+κ̄)}. By using (2), we get a2∫ a1 ℵ(ς,κ, κ̄)[ϕL κ̄(ς, κ̄,CFDθ• a1+κ̄) + ϕU κ̄ (ς, κ̄,CFDθ• a1+κ̄) +Θ̄(ς)hκ̄(ς, κ̄,CFDθ• a1+κ̄)]dς = − a2∫ a1 { ϕL CFDθ•a1+ κ̄ (ς, κ̄,CFDθ• a1+κ̄) + ϕU CFDθ•a1+ κ̄ (ς, κ̄,CFDθ• a1+κ̄) +Θ̄(ς)h CFDθ•a1+ κ̄ (ς, κ̄,CFDθ• a1+κ̄) } CFDθ• a1+ℵ(ς,κ, κ̄))dς, that is a2∫ a1 ℵ(ς,κ, κ̄) [ ϕL κ̄(ς, κ̄,CFDθ• a1+κ̄) + ϕU κ̄ (ς, κ̄,CFDθ• a1+κ̄) +Θ̄(ς)hκ̄(ς, κ̄,CFDθ• a1+κ̄) ] dς + a2∫ a1 { ϕL CFDθ•a1+ κ̄ (ς, κ̄,CFDθ• a1+κ̄) + ϕU CFDθ•a1+ κ̄ (ς, κ̄,CFDθ• a1+κ̄) V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 15 of 38 +Θ̄(ς)h CFDθ•a1+ κ̄ (ς, κ̄,CFDθ• a1+κ̄) } CFDθ• a1+ℵ(ς,κ, κ̄)dς = 0. (8) For the feasibility of κ of (P), we have h(ς,κ(ς),CFDθ• a+κ(ς)) ≤ 0, ς ∈ ℑ, wherein, by utilising the fact Θ̄(ς) ∈ Rm, Θ̄(ς) ≥ 0 and (5), we have a2∫ a1 Θ̄(ς)h(ς,κ,CFDθ• a1+)dς − a2∫ a1 Θ̄(ς)h(ς, κ̄,CFDθ• a1+κ̄)dς ≤ 0, which along with the hypothesis(ii), invexity of a2∫ a1 Θ̄(ς)h(ς, κ̄,CFDθ• a+κ̄)dς at κ̄ ∈ X, yeilds 0 ≥ a2∫ a1 Θ̄(ς) [ ℵ(ς,κ, κ̄)hκ̄(ς, κ̄,CFDθ• a1+κ̄)+ CFDθ• a1+ℵ(ς,κ, κ̄)hCFDθ•a1+ κ̄ (ς, κ̄(ς),CFDθ• a1+κ̄(ς)) ] dς. (9) On adding (7) and (9), we get a2∫ a1 { ℵ(ς,κ, κ̄) [ ϕL κ̄(ς, κ̄,CFDθ• a+κ̄) + ϕU κ̄ (ς, κ̄,CFDθ• a1+κ̄) +Θ̄(ς)hκ̄(ς, κ̄,CFDθ• a1+κ̄) ] +(CFDθ• a1+ℵ(ς,κ, κ̄))[ϕ L CFDθ• a1+ κ̄ (ς, κ̄,CFDθ• a1+κ̄) +ϕU CFDθ• a+κ̄ (ς, κ̄,CFDθ• a1+κ̄) + Θ̄(ς)h CFDθ•a1+ κ̄ (ς, κ̄,CFDθ• a1+κ̄)] } dς < 0, which is a contradiction to (8). Hence the theorem. We provide an algorithm for the Theorem 3.2 in the following way: •Algorithm: Input: • Primal objective functional: (P-2): min  a2∫ a1 ϕL(ς,κ(ς),CFDθ• a1+κ(ς))dς, a2∫ a1 ϕU (ς,κ(ς),CFDθ• a1+κ(ς))dς  V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 16 of 38 • Set of constraints: h(ς,κ(ς),CFDθ• a1+κ(ς)) ≤ 0, κ(a1) = α, κ(a2) = β, ς ∈ [a1, a2]. • Set of feasible point: Φ2 = {κ ∈ X : h(ς,κ(ς),CFDθ• a1+κ(ς)) ≤ 0,κ(a1) = α,κ(a2) = β}. • Set of self data: ϕL(ς,κ(ς),CFDθ• a1+κ(ς)), ϕU (ς,κ(ς),CFDθ• a1+κ(ς)), h(ς,κ(ς),CFDθ• a1+κ(ς)). The above functions are continuous differetiable andare invexity for ς ∈ Φ2. output: • Verification of invexity: ϕL(ς,κ(ς),CFDθ• a1+κ(ς)), ϕU (ς,κ(ς),CFDθ• a1+κ(ς)), h(ς,κ(ς),CFDθ• a1+κ(ς)), ς ∈ Φ2. the point κ̄ is satisfying the Definition12. Set of self data. Begin: • Selective stage: select a point that κ̄ ∈ Φ2 If slater’s constraint qualification satisfied atκ̄; then continue to the next, if KKT necessary optimality conditions (4) and (5) holds at κ̄; V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 17 of 38 else stop; end if; • Screening stage: detecting piece-wise smooth function Θ̄ If self data holds at κ̄for (P − 2); else stop; end if; • Conclusive stage: detecting the point κ̄ = 0, is optimal solution for the problem (P-2). else stop; end; The following example illustrates Theorem 3.2. Example 4. Consider the following interval-valued variational programming problem un- der Caputo-Fabrizio fractional derivative: (P-2) : min  a2∫ a1 ϕL(ς,κ(ς),CFDθ• a1+κ(ς))dς, a2∫ a1 ϕU (ς,κ(ς),CFDθ• a1+κ(ς))dς  subject to, ln2− ln(ς + 2) ≤ 0, κ(0) = 0, κ(1) = 1, ς ∈ [0, 1], where, ϕL(ς,κ(ς),CFDθ• a1+κ(ς)) = ς3 + ς, ϕU (ς,κ(ς),CFDθ• a1+κ(ς)) = 3ς3 + 3. Take a note that κ̄ = 0 is a feasible solution of (P-2) and it can be easily observed that there is Θ ∈ R and Θ̄ = 0, such that the relations (4) and (5) hold for the problem (P- 2). Also, it is observed that a2∫ a1 (ϕL + ϕU )(ς,κ(ς),CFDθ• a1+κ(ς))dς is invex at κ̄ on Φ2 and a2∫ a1 Θ̄(ς)h(ς, κ̄,CFDθ• a+κ̄)dς is invex at κ̄ on Φ2. Since all of the observations of Theorem 3.2 are satisfied, then (Θ̄ = 0, κ̄ = 0) is the LU optimum solution to the problem (P-2). V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 18 of 38 Figure 4: Graphical view of the example problem (P-2) The problem (P-2) has a feasible region, Φ2 = {κ ∈ X : ln2 − ln(ς + 2) ≤ 0,κ(0) = 0,κ(1) = 1}. Theorem 3 (Sufficient Optimality Conditions). Let κ̄ ∈ X be a feasible solution of (P) and there is a function that is piecewise smooth, Θ̄: ℑ → Rm, Θ̄(ς) ≥ 0 in such way that (4) and (5) are satisfied at (κ̄, Θ̄). Also, assume that (i) The functional a2∫ a1 (ϕL + ϕU )(ς,κ(ς),CFDθ• a1+κ(ς))dς is pseudo-invex at κ̄ on X, (ii) The functional a2∫ a1 Θ̄(ς)h(ς, κ̄,CFDθ• a+κ̄)dς is quasi-invex at κ̄ on X, then κ̄ is a LU optimal solution for (P). Proof. If κ̄ is not a LU optimal solution for (P), then by Definition 12 there is another feasible solution κ for (P), such that a2∫ a1 ϕL(ς,κ,CFDθ• a1+κ)dς, a2∫ a1 ϕU (ς,κ,CFDθ• a1+κ)dς  ≺LU  a2∫ a1 ϕL(ς, κ̄,CFDθ• a1+κ̄)dς, a2∫ a1 ϕU (ς, κ̄,CFDθ• a1+κ̄)dς  . V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 19 of 38 Thus, we have  a2∫ a1 ϕL(ς,κ,CFDθ• a1+κ)dς < a2∫ a1 ϕL(ς, κ̄,CFDθ• a1+κ̄)dς a2∫ a1 ϕU (ς,κ,CFDθ• a1+κ)dς ≤ a2∫ a1 ϕU (ς, κ̄,CFDθ• a1+κ̄)dς, or  a2∫ a1 ϕL(ς,κ,CFDθ• a1+κ)dς ≤ a2∫ a1 ϕL(ς, κ̄,CFDθ• a1+κ̄)dς a2∫ a1 ϕU (ς,κ,CFDθ• a1+κ)dς < a2∫ a1 ϕU (ς, κ̄,CFDθ• a1+κ̄)dς, or  a2∫ a1 ϕL(ς,κ,CFDθ• a1+κ)dς < a2∫ a1 ϕL(ς, κ̄,CFDθ• a1+κ̄)dς a2∫ a1 ϕU (ς,κ,CFDθ• a1+κ)dς < a2∫ a1 ϕU (ς, κ̄,CFDθ• a1+κ̄)dς. From the above inequalities, we get a2∫ a1 [ ϕL + ϕU ] (ς,κ(ς),CFDθ• a1+κ(ς))dς < a2∫ a1 [ ϕL + ϕU ] (ς, κ̄(ς),CFDθ• a1+κ̄(ς))dς, which, according to the hypothesis (i), there exists ℵ(ς,κ, κ̄) ∈ C1[a, b] such that a2∫ a1 { ℵ(ς,κ, κ̄) [ ϕL κ̄ + ϕU κ̄ ] (ς, κ̄(ς),CFDθ• a1+κ̄(ς)) +CFDθ• a1+ℵ(ς,κ, κ̄)(ϕ L CFDθ• a1+ κ̄ + ϕU CFDθ• a1+ κ̄)(ς, κ̄(ς), CFDθ• a1+κ̄(ς)) } dς < 0. (10) On the other hand, from (4), we have a2∫ a1 ℵ(ς,κ, κ̄) [ ϕL κ̄(ς, κ̄,CFDθ• a1+κ̄) + ϕU κ̄ (ς, κ̄,CFDθ• a1+κ̄) +Θ̄(ς)hκ̄(ς, κ̄,CFDθ• a1+κ̄) ] dς V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 20 of 38 = a2∫ a1 ℵ(ς,κ, κ̄)(−CFDθ• a2−){ϕ L CFDθ• a1+ κ̄(ς)(ς, κ̄, CFDθ• a1+κ̄) +ϕU CFDθ• a1+ κ̄(ς)(ς, κ̄, CFDθ• a1+κ̄) + Θ̄(ς)hCFDθ• a1+ κ̄(ς)(ς, κ̄, CFDθ• a1+κ̄)}dς = {ℵ(ς,κ, κ̄)I1−θ• b− [ ϕL CFDθ•a1+ κ̄ (ς, κ̄,CFDθ• a1+κ̄) + ϕU CFDθ•a1+ κ̄ (ς, κ̄,CFDθ• a1+κ̄) +Θ̄(ς)hκ̄(ς, κ̄,CFDθ• a1+κ̄) ] } ∣∣∣∣a2 a1 − a2∫ a1 { ϕL CFDθ•a1+ κ̄ (ς, κ̄,CFDθ• a1+κ̄) + ϕU CFDθ•a1+ κ̄ (ς, κ̄,CFDθ• a1+κ̄) +Θ̄(ς)h CFDθ•a1+ κ̄ (ς, κ̄,CFDθ• a1+κ̄) } CFDθ• a1+ℵ(ς,κ, κ̄)dς. (by Proposition 1) By using (2), we get a2∫ a1 ℵ(ς,κ, κ̄)[ϕL κ̄(ς, κ̄,CFDθ• a1+κ̄) + ϕU κ̄ (ς, κ̄,CFDθ• a1+κ̄) +Θ̄(ς)hκ̄(ς, κ̄,CFDθ• a1+κ̄)]dς = − a2∫ a1 { ϕL CFDθ•a1+ κ̄ (ς, κ̄,CFDθ• a1+κ̄) + ϕU CFDθ•a1+ κ̄ (ς, κ̄,CFDθ• a1+κ̄) +Θ̄(ς)h CFDθ•a1+ κ̄ (ς, κ̄,CFDθ• a1+κ̄) } CFDθ• a1+ℵ(ς,κ, κ̄))dς, that is a2∫ a1 ℵ(ς,κ, κ̄) [ ϕL κ̄(ς, κ̄,CFDθ• a1+κ̄) + ϕU κ̄ (ς, κ̄,CFDθ• a1+κ̄) +Θ̄(ς)hκ̄(ς, κ̄,CFDθ• a1+κ̄) ] dς + a2∫ a1 { ϕL CFDθ•a1+ κ̄ (ς, κ̄,CFDθ• a1+κ̄) + ϕU CFDςa+κ̄ (ς, κ̄,CFDθ• a1+κ̄) +Θ̄(κ)h CFDθ•a1+ κ̄ (ς, κ̄,CFDθ• a1+κ̄) } CFDθ• a1+ℵ(ς,κ, κ̄)dξ = 0. (11) V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 21 of 38 In order to determine the feasibility of κ in the problem (P), we have h(ς, κ(ς), CFDθ• a+κ(ς)) ≤ 0, ς ∈ ℑ, which by using the fact Θ̄(ς) ∈ Rm, Θ̄(ς) ≥ 0 and (5) we have a2∫ a1 Θ̄(ς)h(ς,κ,CFDθ• a1+)dς ≤ a2∫ a1 Θ̄(ς)h(ς, ς̄ ,CFDθ• a1+κ̄)dς. On the basis of hypothesis (ii), there exists ℵ(ς,κ, κ̄) ∈ C1[a, b] such that a2∫ a1 Θ̄(ς) [ ℵ(ς,κ, κ̄)hκ̄(ς, κ̄,CFDθ• a1+κ̄)+ CFDθ• a1+ℵ(ς,κ, κ̄)hCFDθ•a1+ κ̄ (ς, κ̄(ς),CFDθ• a1+κ̄(ς)) ] dς ≤ 0. (12) On adding (10) and (12), we get a2∫ a1 { ℵ(ς,κ, κ̄) [ ϕL κ̄(ς, κ̄,CFDθ• a1+κ̄) + ϕU κ̄ (ς, κ̄,CFDθ• a1+κ̄) +Θ̄(ς)hκ̄(ς, κ̄,CFDθ• a1+κ̄) ] +(CFDθ• a1+ℵ(ς,κ, κ̄))[ϕ L CFDθ• a1+ κ̄ (ς, κ̄,CFDθ• a1+κ̄) + ϕU CFDθ• a+κ̄ (ς, κ̄,CFDθ• a+κ̄)+ Θ̄(ς)h CFDθ•a1+ κ̄ (ς, κ̄,CFDθ• a1+κ̄)] } dς < 0, which is a contradiction to (11). Hence the theorem. The following example illustrates Theorem 3.3. Example 5. Consider the following problem (P-3): (P-3) = min  a2∫ a1 ϕL(ς, y(ς),CFDθ• a1+κ(ς))dς, a2∫ a1 ϕU (ς,κ(ς),CFDθ• a1+κ(ς))dς  subject to, − y(ς) + 1.871822ς + 6.5514e − ς 3 − 5.6154 ≤ 0, κ(0) = 1, κ(1) = 1, ς ∈ [0, 1], where, ϕL(ς,κ(ς),CFDθ• a1+κ(ς)) = −κ(ς) + 6.5514ς − 16.37859e − ς 3 + 23.9299, V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 22 of 38 ϕU (ς,κ(ς),CFDθ• a1+κ(ς)) = −κ2(ς) + 6.5514ς − 16.3785e − ς 3 + 22.9299), and κ(ς) = −ς2 + ς + 1 ∈ X. The feasible region of (P-3) is Φ3 = {κ ∈ X : −κ(ς)+ 1.871822ς +6.5514e − ς 3− 5.6154) ≤ 0,κ(0) = 1,κ(1) = 1}. Note that κ̄ = 1 is a feasible solution of (P-3) and it can be easily observe that there is Θ ∈ R and Θ̄ = 0, such that the relations (4) and (5) holds for (P-3). Also it is observed that a2∫ a1 (ϕL + ϕU )(ς,κ(ς),CFDθ• a1+κ(ς))dς is pseudo-convex at κ̄ on Φ3 and a2∫ a1 Θ̄(ς)h(ς, κ̄,CFDθ• a+κ̄)dς is quasi-convex at κ̄ on Φ3. Since all the observations of Theorem 3 are satisfied, then (Θ̄ = 0, κ̄ = 1) is a LU optimal solution (P-3). 4. Wolfe-Type Dual Model We are concerned in this part with the Wolfe-type dual problem (WD) with the CF derivative operator in relation to the primary problem (P), it is as follows: (WD) maxG(ε, Θ̄) = a2∫ a1 [ [ ϕL(ς, ε,CFDθ• a1+ε), ϕ U (ς, ε,CFDθ• a1+ε) ] + (Θ̄)Th(ς, ε,CFDθ• a1+ε(ς)) ] dς, subject to, ε(a1) = α, ε(a2) = β, (13) ϕL ε (ς, ε, CFDθ• a1+ε) + ϕU ε (ς, ε, CFDθ• a1+ε) + (Θ̄)T (ς)hε(ς, ε, CFDθ• a1+ε(ς)) = −CFDθ• a2− { ϕL CFDθ• a1+ ε(ς) (ς, ε,CFDθ• a1+ε) + ϕU CFDθ• a1+ ε(ς) (ς, ε,CFDθ• a1+ε) +(Θ̄)T (ς)hCFDθ• a1+ ε(ς)(ς, ε, CFDθ• a1+ε(ς)) } , (14) a2∫ a1 (Θ̄)T (ς)h(ς, ε(ς),CFDθ• a1+ε(ς))dς ≤ 0, (15) Θ̄(ξ) ≥ 0, ς ∈ ℑ. (16) Here, ε(ς) signifies an n-dimensional function, Θ̄(ς) denotes an m-dimensional function. Let Θ(ε̄) = {(Θ̄, ε̄) : Θ̄ ∈ Rm, ε̄ ∈ X: satisfying the constraints of (WD), forall ς ∈ ℑ} be the collection of all feasible points to (WD). V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 23 of 38 Definition 13. A feasible point (ε̄, Θ̄) is stated to be an LU optimal point of a maximum type for (WD), if there is no feasible point (ε,Θ), such that a2∫ a1 ϕL(ς, ε̄,CFDθ• a+ε̄)dς, a2∫ a1 ϕU (ς, ε̄,CFDθ• a1+ε̄)dθ • + a2∫ a1 (Θ̄)Th(ς, ε̄,CFDθ• a1+ε̄)dς ≺LU  a2∫ a1 ϕL(ς, ε,CFDθ• a1+ε)dς, a2∫ a1 ϕU (ς, ν,CFDθ• a1+ε)dς  + a2∫ a1 (Θ̄)Th(ς, ε,CFDθ• a1+ε)dς. The weak, strong, and strict converse duality theorems are studied for (WD) from the standpoint of the CF fractional derivative operator: Theorem 4 (Weak Duality). Let (Θ̄, κ̄) and (Θ̄, ε̄) be the feasible points for (P) and (WD), respectively. Suppose that, (Θ̄)T > 0 and the functional (ϕL + ϕU + (Θ̄)Th)(ς, ε̄,CFDθ• a1+ε̄) is invex at ε̄ on X, then the following can’t hold a2∫ a1 ϕL(ς, κ̄,CFDθ• a1+κ̄)dς, a2∫ a1 ϕU (ς, κ̄,CFDθ• a1+κ̄)dς  ≺LU  a2∫ a1 (ϕL + (Θ̄)Th)(ς, ε̄,CFDθ• a1+ε̄)dς, a2∫ a1 (ϕU + (Θ̄)Th)(ς, ε̄,CFDθ• a1+ε̄)dς  . Proof. Assume, contrary to the outcome, a2∫ a1 ϕL(ς, κ̄,CFDθ• a1+κ̄)dς, a2∫ a1 ϕU (ς, κ̄,CFDθ• a1+κ̄)dς  ≺LU  a2∫ a1 (ϕL + (Θ̄)Th)(ς, ε̄,CFDθ• a1+ε̄)dς, a2∫ a1 (ϕU + (Θ̄)Th)(ς, ε̄,CFDθ• a1+ε̄)dς  . From (P), h(ς,κ(ς),CFDθ• a1+κ(ς)) ≤ 0 and h is continuously differential function. Moreover (Θ̄)T ≥ 0. Then, it follows that a2∫ a1 (ϕL + (Θ̄)Th)(ς, κ̄,CFDθ• a1+κ̄)dς < a2∫ a1 (ϕL + (Θ̄)Th)(ς, ε̄,CFDθ• a1+ε̄)dς a2∫ a1 (ϕU + (Θ̄)Th)(ς, κ̄,CFDθ• a1+κ̄)dς ≤ a2∫ a1 (ϕU + (Θ̄)Th)(ς, ε̄,CFDθ• a1+ε̄)dς, V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 24 of 38 or  a2∫ a1 (ϕL + (Θ̄)Th)(ς, κ̄,CFDθ• a1+κ̄)dς ≤ a2∫ a1 (ϕL + (Θ̄)Th)(ς, ε̄,CFDθ• a1+ν̄)dς a2∫ a1 (ϕU + (Θ̄)Th)(ς, κ̄,CFDθ• a1+κ̄)dς < a2∫ a1 (ϕU + (Θ̄)Th)(ς, ε̄,CFDθ• a1+ε̄)dς, or  a2∫ a1 (ϕL + (Θ̄)Th)(ς, κ̄,CFDθ• a1+κ̄)dς < a2∫ a1 (ϕL + (Θ̄)Th)(ς, ε̄,CFDθ• a1+ε̄)dς a2∫ a1 (ϕU + (Θ̄)Th)(ς, κ̄,CFDθ• a1+κ̄)dς < a2∫ a1 (ϕU + (Θ̄)Th)(ς, ε̄,CFDθ• a1+ε̄)dς. From the above inequalities, we get a2∫ a1 (ϕL + ϕU + (Θ̄)Th))(ς, κ̄,CFDθ• a1+κ̄)dς − a2∫ a1 (ϕL + ϕU + (Θ̄)Th)(ς, ε̄,CFDθ• a1+ε̄)dς < 0, Hence, taking the invexity assumption on a2∫ a1 (ϕL + ϕU + (Θ̄)Th))(ς, ., .)dς at ε̄ on X, there exists ℵ(ς, κ̄, ε̄) ∈ C1[a1, a2] such that a2∫ a1 {ℵ(ς, κ̄, ε̄)T [ ϕL ε̄ + ϕU ε̄ + (Θ̄)Th ] (ς, ε̄,CFDθ• a1+ε̄) +CFDγ a1+ℵ(ς, κ̄, ε̄) T [ ϕL CFDθ• a1+ ε̄ +ϕU CFDθ• a1+ ε̄ +ℵ(ς, κ̄, ε̄)ThCFDθ• a1+ ε̄ ] (ς, ε̄,CFDθ• a1+ε̄)}dς < 0. (17) Further, from the dual constraint (14) and the Proposition 2.1, we get a2∫ a1 { ℵ(ς, κ̄, ε̄)T [ ϕL ε̄ (ς, ε̄, CFDθ• a1+ε̄) + ϕU ε̄ (ς, ε̄, CFDθ• a1+ε̄) (18) +(Θ̄)T (ς)hε̄(ς, ε̄, CFDθ• a1+ε̄) ]} dς = a2∫ a1 ℵ(ς, κ̄, ε̄)T [ − CFDθ• a1− { ϕL CFDθ• a1+ ε̄ (ς, ε̄,CFDς a1+ε̄) V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 25 of 38 + ϕU CFDθ• a1+ ε̄ (ς, ε̄,CFDθ• a1+ε̄) } + (Θ̄)T (ς)hCFDγ a+ε̄(ς, ε̄, CFDθ• a1+ε̄) ] dς = [ ℵ(ς, κ̄, ε̄)T I1−θ• b− { ϕL CFDθ• a1+ ε̄ + ϕU CFDθ• a1+ ε̄ } (ς, ε̄,CFDθ• a1+ε̄) + (Θ̄)T (ς)hCFDθ• a1+ ε̄(ς, ε̄, CFDθ• a1+ε̄) ]b a − a2∫ a1 ℵ(ς, κ̄, ε̄)T [ CFDθ• a1+ { ϕL CFDθ• a1+ ε̄ ++ϕU CFDθ• a1+ ε̄ } (ς, ε̄,CFDθ• a1+ε̄) + (Θ̄)T (ς)hCFDθ• a1+ ε̄(ς, ε̄, CFDθ• a1+ε̄) ] dς. By using (13), it gives a2∫ a1 [ ℵ(ς, κ̄, ε̄)T { ϕL ε̄ (ς, ε̄, CFDθ• a1+ε̄) + ϕU ε̄ (ς, ε̄, CFDθ• a1+ε̄) +(Θ̄)T (ς)hε̄(ς, ε̄, CFDθ• a1+ε̄) }] dς = − a2∫ a1 [ CFDθ• a1+(ε− ε̄) { ϕL CFDθ• a1+ ε̄ + ϕU CFDθ• a1+ ε̄ } (ς, ε̄,CFDθ• a1+ε̄) +(Θ̄)T (ς)hCFDθ• a1+ ε̄(ξ, ε̄, CFDθ• a1+ε̄) ] dς. That is a2∫ a1 ℵ(ς, κ̄, ε̄)T { ϕL ε̄ (ς, ε̄, CFDθ• a1+ε̄) + ϕU ε̄ (ς, ε̄, CFDθ• a1+ε̄) +(Θ̄)T (ς)hε̄(ς, ε̄, CFDθ• a1+ε̄) } dς + a2∫ a1 CFDθ• a1+(ε− ε̄)T { ϕL CFDθ• a1+ ε̄ (ς, ε̄,CFDθ• a1+ε̄) + ϕU CFDθ• a1+ ε̄ (ς, ε̄,CFDθ• a1+ε̄) +(Θ̄)T (ς)hCFDθ• a1+ ε̄(ς, ε̄, CFDθ• a1+ε̄) } dς = 0, (19) which contradicts (17). Hence the theorem. We provide an algorithm for the weak duality theorem in the following way: V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 26 of 38 Algorithm of weak duality Input: • Primal objective functional: (P-4): min  a2∫ a1 ϕL(ς,κ(ς),CFDθ• a1+κ(ς))dς, a2∫ a1 ϕU (ς,κ(ς),CFDθ• a1+κ(ς))dς  • Set of constraints: h(ς,κ(ς),CFDθ• a1+κ(ς)) ≤ 0, κ(a1) = α, κ(a2) = β, ς ∈ [a1, a2]. • Set of feasible point for (P-4): Φ3 = {h(ς,κ(ς),CFDθ• a1+κ(ς)) ≤ 0,κ(a1) = α,κ(a2) = β}. • Wolfe dual objective functional corresponding to the primal (P-4): (WD-1) = max a2∫ a1 [ [ ϕL(ς, ε,CFDθ• a1+ε), ϕU (ς, ε,CFDθ• a1+ε) ] + (Θ̄)Th(ς, ε,CFDθ• a1+ε(ς)) ] dς, • Set of constraints for (WD-1): ϕL ε (ς, ε, CFDθ• a1+ε) + ϕU ε (ς, ε, CFDθ• a1+ε) + (Θ̄)T (ς)hε(ς, ε, CFDθ• a1+ε(ς)) = −CFDθ• a2− { ϕL CFDθ• a1+ ε(ς) (ς, ε,CFDθ• a1+ε) + ϕU CFDθ• a1+ ε(ς) (ς, ε,CFDθ• a1+ε) +(Θ̄)T (ς)hCFDθ• a1+ ε(ς)(ς, ε, CFDθ• a1+ε(ς)) } , a2∫ a1 (Θ̄)T (ς)h(ς, ε(ς),CFDθ• a1+ε(ς))dς ≤ 0, Θ̄(ξ) ≥ 0, ς ∈ ℑ,where a1 = 0, a2 = 1. V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 27 of 38 • Set of feasible point for (WD-1): W3 = {(Θ̄, ε̄) : Θ̄ ∈ Rm, ε ∈ X : satisfying constraints of (WD-1), forall ς ∈ ℑ} • Set of self data: where, ϕLς,κ(ς),CFDθ• a1+κ(ς)), ϕU (ς,κ(ς),CFDθ• a1+κ(ς)), h(ς,κ(ς),CFDθ• a1+κ(ς)) and κ(ς)are continuously differentiable functions and are invexity for. ς ∈ Φ3. output: • Verification of invexity: ϕL(ς,κ(ς),CFDθ• a1+κ(ς)) ϕU (ς,κ(ς),CFDθ• a1+κ(ς) ϕL(ς, ε(ς),CFDθ• a1+ε(ς)) + (Θ̄)T (ς)h(ς, ε(ς), ϕU (ς, ε(ς),CFDθ• a1+ε(ς)) + (Θ̄)T (ς)h(ς, ε(ς), h(ς,κ(ς),CFDθ• a1+κ(ς)), ς ∈ Φ3,W3. the point κ̄, the point ε̄ is satisfying the Definition12. Set of self data. Begin: Main Optimization Loop: Primary problem solution : • while not converged: • Compute Caputo-Fabrizio fractional derivative: CFDθ• a1+κ(ς) • Evaluate objective functions V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 28 of 38 JL = a2∫ a1 ϕL(ς,κ(ς),CFDθ• a1+κ(ς))dς, JU = a2∫ a1 ϕU (ς,κ(ς),CFDθ• a1+κ(ς))dς • Check constraints h(ς,κ(ς),CFDθ• a1+κ(ς)) ≤ 0, ∀ ς ∈ [a1, a2] and • Boundary conditions are satisfied: • Solution is feasible • Store current solution • else: • Apply constraint projection • Update solution using gradient descent or other optimization κ̄k+1 = Update solution(κ̄k, J L, JU , Constraints) • Check convergence criteria Wolfe-Type Dual problem (WD-1) solution: • while not converged: • Compute adjoint system Solve the Euler-Lagrange type equation: ϕL ε (ς, ε, CFDθ• a1+ε) + ϕU ε (ς, ε, CFDθ• a1+ε) + (Θ̄)T (ς)hε(ς, ε, CFDθ• a1+ε(ς)) + CFDθ• a2− { ϕL CFDθ• a1+ ε(ς) (ς, ε,CFDθ• a1+ε) + ϕU CFDθ• a1+ ε(ς) (ς, ε,CFDθ• a1+ε) +(Θ̄)T (ς)hCFDθ• a1+ ε(ς)(ς, ε, CFDθ• a1+ε(ς)) } = 0. • Check dual constraints a2∫ a1 (Θ̄)T (ς)h(ς, ε(ς),CFDθ• a1+ε(ς))dς ≤ 0, • and boundary conditions Θ̄(ξ) ≥ 0, ς ∈ ℑ = [a1, a2]. V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 29 of 38 Solution is feasible Store current dual solution else; • Apply constraint projection • Update solution using gradient descent or other optimization (Θ̄k+1, ε̄k+1) = Update Dual variables(Θ̄k, ε̄k) • Check convergence criteria Verification and Duality Check: • Duality Verification: For feasible solutions (Θ̄k+1, ς̄k+1), for the primal problem (P-4) and(Θ̄k+1, ε̄k+1)for dual problem (WD-1) Verify that the interval objective values are not comparable under the ⪯LU else stop; end; We provide an example for the weak duality theorem in the following way: Example 6. Let us consider the following interval-valued variational programming prob- lem under Caputo-Fabrizio fractional derivative: (P-4) : min  a2∫ a1 ϕL(ς,κ(ς),CFDθ• a1+κ(ς))dς, a2∫ a1 ϕU (ς,κ(ς),CFDθ• a1+κ(ς))dς  subject to, − 2κ2(ς)− 4κ(ς) + 3 √ (π) + 1√ (π) (1− e−ς) ≤ 0, κ(0) = 1, κ(1) = 1, ς ∈ [0, 1], where, ϕL(ς,κ(ς),CFDθ• a1+κ(ς)) = κ(ς) + 1 2 √ (π) + 1√ (π) (1− e−ς), ϕU (ς,κ(ς),CFDθ• a1+κ(ς)) = 1 3 κ2(ς) + 3 √ (π) + 1√ (π) (1− e−ς), and κ(ς) = 7 2 ς2 − 5.5636 1.5896 ς + 1. V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 30 of 38 The feasible region of (P-4) is Φ3 = {κ ∈ X : −2κ2(ς) − 4κ(ς) + 3 √ (π) + 1√ (π) (1 − e−ς) ≤ 0,κ(0) = 1,κ(1) = 1}. For κ ∈ Φ3, the Wolfe-type dual problem for the primary problem (P-4) is given by (WD-1) : max a2∫ a1 [ {ϕL(ς, ε(ς),CFDθ• a1+ε(ς))dς, ϕU (ς, ε(ς),CFDθ• a1+ε(ς))} + (Θ̄)Th(ς, ε,CFDθ• a1+ε(ς)) ] dς, subject to, ε(0) = 1, ε(1) = 1, (1 + 0.7825e−ς) + ( 2 3 ε+ 4.7688e−ς) + (Θ̄)T (−4ε− 4 + 4.7688e−ς) + 0.9292 d dς { 1∫ ς {(1 + 2 3 ε+ 5.5513e−ς) + (Θ̄)T (−2ε− 4 + 1.5641e−ς)}dς} = 0 1∫ ς {(Θ̄)T (−4ε− 4 + 4.7688e−ς)}dς ≤ 0. Let W3 be the collection of all feasible solutions of the problem (WD-1), that is W3 = {(Θ̄, ε̄) : Θ̄ ∈ Rm, ε ∈ X: satisfying constraints of (WD-1), forall ς ∈ ℑ}. For the feasible solutions (Θ̄ = 0, κ̄ = 1) of (P − 4) and (Θ̄ = 0, ε̄ = 1) of (WD-1), one can easily verify that a2∫ a1 [ {ϕL(ς, ε(ς),CFDθ• a1+ε(ς)), ϕU (ς, ε(ς),CFDθ• a1+ε(ς))} +(Θ̄)Th(ς, ε,CFDθ• a1+ε(ς)) ] dς, is invex at ε̄ on Φ3 ∪W3, we observe that a2∫ a1 ϕL(ς, κ̄,CFDθ• a1+κ̄)dς, a2∫ a1 ϕU (ς, κ̄,CFDθ• a1+κ̄)dς  ⊀LU  a2∫ a1 (ϕL + (Θ̄)Th)(ς, ε̄,CFDθ• a1+ε̄)dς, a2∫ a1 (ϕU + (Θ̄)Th)(ς, ε̄,CFDθ• a1+ε̄)dς  . V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 31 of 38 Theorem 5 (Strong Duality). Let κ̄ be an LU optimal point for (P), and the Slater’s constraint qualification is satisfied at κ̄. Then there are piecewise smooth functions Θ̄ : ℑ → Rm, Θ̄ ≥ 0, such that (κ̄, Θ̄) is a feasible point for (WD) and the two objective functions are equivalent at κ̄ and (κ̄, Θ̄) for (P) and (WD), respectively. Furthermore, if weak duality Theorem 4 holds between (P) and (WD), then (κ̄, Θ̄) is LU-optimality for (WD). Proof. According to the hypothesis of the theorem, κ̄ is a LU optimum point for (P), hence according to Theorem 1, there exist piecewise smooth functions Θ̄ : ℑ → Rm such that ϕL κ̄(ς, κ̄,CFDθ• a1+κ̄) + ϕU κ̄ (ς, κ̄,CFDθ• a1+κ̄) + (Θ̄)T (ς)hκ̄(ς, κ̄,CFDθ• a1+κ̄) = −CFRDθ• a2− [ ϕL CFDθ•a+κ̄(ς) (ς, κ̄,CFDθ• a1+κ̄) + ϕU CFDθ•a1+ κ̄(ς) (ς, κ̄,CFDθ• a1+κ̄) +(Θ̄)T (ς)hCFDθ• a1+ κ̄(ς)(ς, κ̄, CFDθ• a+κ̄) ] , (Θ̄)T (ς)h(ς, κ̄,CFDθ• a1+κ̄) = 0, (Θ̄)(ς) ≥ 0. It follows that (κ̄, Θ̄) is a feasible point for (WD) and the objective values of (P) and (WD) are equal. According to Theorem 4, (κ̄, Θ̄) is the optimal point for (WD). Theorem 6 (Strict Converse Duality). Let κ̄ and (Θ̄, ε̄) be the feasibile points for (P) and (WD) respectively, such that a2∫ a1 ϕL(ς, κ̄,CFDθ• a1+κ̄)dς, a2∫ a1 ϕU (ς, κ̄,CFDθ• a1+κ̄)dς  + a2∫ a1 (Θ̄)T (ς)h(ξ, κ̄,CFDθ• a1+κ̄)dς =  a2∫ a1 ϕL(ς, ε̄,CFDθ• a1+ε̄)dς, a2∫ a1 ϕU (ς, ε̄,CFDθ• a1+ε̄)dς  + a2∫ a1 (Θ̄)T (ς)h(ς, ε̄,CFDθ• a1+ε̄)dς. (20) Further, suppose that ε(ς) ∈ X, ε(ς) ≥ 0 and the functional a2∫ a1 [ϕL + ϕU + (Θ̄)Th](ς, ε̄(ς), CFDθ• a1+ ε̄(ς)) dς is strictly-invex at ε̄ on X. Then, κ̄ = ε̄ and ε̄ is an LU optimal point for (P). V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 32 of 38 Proof. Assume, contrary to the outcome, κ̄ ̸= ε̄. By (20), we have[ a2∫ a1 ϕL(ς, κ̄,CFDθ• a+κ̄)dς, a2∫ a1 ϕU (ς, κ̄,CFDθ• a1+κ̄)dς ] + a2∫ a1 (Θ̄)T (ς)h(ς, κ̄,CFDθ• a1+κ̄)dς = [ a2∫ a1 ϕL(ς, ε̄,CFDθ• a1+ε̄)dς, a2∫ a1 ϕU (ς, ε̄,CFDθ• a1+ε̄)dς ] + a2∫ a1 (Θ̄)T (ς)h(ς, ε̄,CFDθ• a1+ε̄)dς. That is, a2∫ a1 (ϕL + (Θ̄)Th)(ς, κ̄,CFDθ• a1+κ̄)dς = a2∫ a1 (ϕL + (Θ̄)Th)(ς, ε̄,CFDθ• a1+ε̄)dς, (21) a2∫ a1 (ϕU + (Θ̄)Th)(ς, κ̄,CFDθ• a1+κ̄)dς = a2∫ a1 (ϕU + (Θ̄)Th)(ς, ε̄,CFDθ• a1+ε̄)dς. (22) On adding (21) and (22), we get a2∫ a1 [ ϕL + ϕU + (Θ̄)Th ] (ς, κ̄,CFDθ• a1+κ̄)dς = a2∫ a1 [ ϕL + ϕU + (Θ̄)Th ] (ς, ε̄,CFDθ• a1+ε̄)dς. (23) On the other hand, by using strictly-invex of (ϕL + ϕU + (Θ̄)Th)(ς, κ̄,CFDθ• a1+κ̄) at ε̄ on κ, we have a2∫ a1 [ ϕL + ϕU + (Θ̄)Th ] (ς, κ̄,CFDθ• a1+κ̄)dς − a2∫ a1 [ ϕL + ϕU + (Θ̄)Th ] (ς, ε̄,CFDθ• a1+ε̄)dς > a2∫ a1 ℵ(ς, κ̄, ε̄)T [ ϕL ε̄ + ϕU ε̄ + (Θ̄)Thε̄ ] (ς, ε̄,CFDθ• a1+ε̄) +CFDθ• a1+ℵ(ς, κ̄, ε̄) T [ ϕL CFDθ• a1+ ε̄ +ϕU CFDθ• a1+ ε̄ V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 33 of 38 +(Θ̄)ThCFDθ• a1+ ε̄ ] (ς, ε̄,CFDθ• a1+ε̄)dς. (24) By using dual constraints (13), (14) with Proposition 2.1, in view of the Weak-duality Theorem 4, we get a2∫ a1 ℵ(ς, κ̄, ε̄)T [ ϕL ε̄ + ϕU ε̄ + (Θ̄)Thε̄ ] (ς, ε̄,CFDθ• a1+ε̄) + CFDθ• a1+ℵ(ς, κ̄, ε̄) T [ ϕL CFDθ• a1+ ε̄ +ϕU CFDθ• a1+ ε̄ + (Θ̄)ThCFDθ• a1+ ε̄ ] (ς, ε̄,CFDθ• a1+ε̄)dς = 0. (25) (24) and (25), give us a2∫ a1 [ ϕL + ϕU + (Θ̄)Th ] (ς, κ̄,CFDθ• a1+κ̄)dς − a2∫ a1 [ ϕL + ϕU + (Θ̄)Th ] (ς, ε̄,CFDθ• a1+ε̄)dς > 0 which contradicts (23). Hence κ̄ = ε̄. Further, if κ̄ is not an LU optimal point for (P), then there is another feasible point κ for (P) such that  a2∫ a1 ϕL(ς,κ,CFDγ a+κ)dς, a2∫ a1 ϕU (ς,κ,CFDθ• a1+κ)dℵ  ≺LU  a2∫ a1 ϕL(ς, ε̄,CFDθ• a1+ε̄)dς, a2∫ a1 ϕU (ς, ε̄,CFDθ• a1+ν̄)dς  . Since κ and (Θ̄, ν̄) represent the respective feasibility points for (P) and (WD), then from the Theorem 4, we have a2∫ a1 ϕL(ς,κ,CFDθ• a1+κ)dς, a2∫ a1 ϕU (ς,κ,CFDθ• a1+κ)dς  ⊀LU  a2∫ a1 ϕL(ς, ε̄,CFDθ• a1+ε̄)dς, a2∫ a1 ϕU (ς, ε̄,CFDθ• a1+ε̄)dς  + a2∫ a1 (Θ̄)T (ς)h(ς, ε̄,CFDθ• a1+ε̄)dς. V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 34 of 38 Considering the feasibility of κ for (P), it follows that a2∫ a1 ϕL(ς,κ,CFDγ a1+κ)dς, a2∫ a1 ϕU (ς,κ,CFDθ• a1+κ)dς  + a2∫ a1 (Θ̄)T (ς)h(ς,κ,CFDθ• a1+κ)dς ⊀LU  a2∫ a1 ϕL(ς, ε̄,CFDθ• a1+ε̄)dς, a2∫ a1 ϕU (ς, ε̄,CFDθ• a1+ε̄)dς  + a2∫ a1 (Θ̄)T (ς)h(ς, ε̄,CFDθ• a1+ε̄)dς, it is in contradiction with (20). This implies that ε̄ is a LU optimum point for (P) and hence the proof. 5. Conclusion LU optimality and generalized-invexity are employed to establish optimality conditions for a broader class of interval-valued variational programming problems involving objective functions with Caputo-Fabrizio fractional derivatives. For the associated Wolfe-type dual problems, we have derived results pertaining to weak, strong, and strict converse duality. These theoretical developments are further substantiated through illustrative examples. The utilization of the Caputo-Fabrizio fractional derivative introduces memory effects without involving singular kernels, thereby enabling more accurate modeling of physical and engineering systems. Meanwhile, incorporating interval-valued objective functions strengthens the robustness of the model in handling uncertainties that frequently arise in real-world scenarios. The introduction of generalized-invexity significantly extends the framework of optimality and duality, surpassing traditional convexity assumptions and offering greater flexibility and generality. The proposed framework relies on the assumption that the interval-valued objective and constraint functions are both well-defined and bounded. However, in real-world scenarios characterized by high volatility or deep uncertainty, these assumptions may not always hold, potentially affecting the model’s accuracy and reliability. Additionally, the use of Caputo-Fabrizio fractional derivatives, though beneficial for capturing memory effects without singularities, may not be appropriate for systems where long-range memory be- havior is best modeled using singular kernels (e.g., power-law decay), thereby limiting its applicability in certain contexts. V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 35 of 38 This model is well-suited for addressing resource allocation problems in uncertain market environments, particularly when costs or returns exhibit memory-dependent behavior and are best represented using interval values. It also has relevance in control system design and structural optimization, where degradation of material properties over time and underlying uncertainty play a critical role. The framework’s flexibility in handling both interval uncertainty and memory effects makes it applicable across various engineering, economic, and decision-making problems involving complex dynamic systems. Future research may extend this work by exploring fuzzy interval level sets, closed and bounded intervals of real numbers that act as a bridge between fuzzy set theory and clas- sical mathematical analysis. Based on existing literature, such as [34] and [17], which describe applications of fuzzy set theory in system analysis, the concept of fuzzy interval level sets can be used to extend results from interval spaces to fuzzy intervals. Conse- quently, this line of inquiry could pave the way for developing optimality conditions for fuzzy interval-valued variational programming problems with Caputo-Fabrizio fractional derivatives, marking a novel direction in this field of study. Conflicts of interest or competing interests The authors declare that they have no conflicts of interest. Data and code Availability No data were used to support this study Supplementary information Not Applicable Ethical Approval This article does not contain any studies with human participants or animals performed by any of the authors Informed Consent The authors are fully aware and satisfied with the contents of the article. Acknowledgements We sincerely thank the anonymous reviewers for their insightful comments and construc- tive suggestions, which have greatly enhanced the quality and clarity of our paper. V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 36 of 38 References [1] T Abdeljawad and D Baleanu. On fractional derivatives with exponential kernel and their discrete versions. Reports on Mathematical Physics, 80(1):11–27, 2017. [2] B Chetia and P K Das. An application of interval-valued fuzzy soft. International Journal of Contemporary Mathematical Sciences, 5(38):1887–1894, 2010. [3] D Yin et al. Application of interval valued fuzzy linear programming for stock portfolio optimization. Applied mathematics, 9(02):101, 2018. [4] T Saeed and S Treanctua. New classes of interval-valued variational problems and inequalities. Results in Control and Optimization, 13:100324, 2023. [5] S Treanta and Marilena Ciontescu. On optimal control problems with generalized invariant convex interval-valued functionals. Journal of Industrial and Management Optimization, 20:3317–3336, 2024. [6] A K Bhurjee and G Panda. Sufficient optimality conditions and duality theory for interval optimization problem. Annals of Operations Research, 243(1):335–348, 2016. [7] R E Moore. Interval analysis. Prentice-Hall, 1966. [8] A Neumaier. Interval methods for systems of equations. Number 37. Cambridge University Press, 1990. [9] I M Stancu-Minasian. Stochastic programming with multiple objective functions, vol- ume 13. Springer, 1984. [10] R Osuna-Gómez, B Hernández-Jiménez, Y Chalco-Cano, and G Ruiz-Garzón. New efficiency conditions for multiobjective interval-valued programming problems. Infor- mation Sciences, 420:235–248, 2017. [11] Y Sun and L Wang. Optimality conditions and duality in nondifferentiable interval- valued programming. Journal of Industrial & Management Optimization, 9(1), 2013. [12] R Esmaelzadeh. Low-thrust orbit transfer optimization using a combined method. International Journal of Computer Applications, 89(4), 2014. [13] L Blasi, S Barbato, and MMattei. A particle swarm approach for flight path optimiza- tion in a constrained environment. Aerospace Science and Technology, 26(1):128–137, 2013. [14] S Khardi. Aircraft flight path optimization. the hamilton-jacobi-bellman considera- tions. Applied Mathematical Sciences, 6(25):pp–1221, 2012. [15] I Ahmad, A Jayswal, S Al-Homidan, and J Banerjee. Sufficiency and duality in interval-valued variational programming. Neural Computing and Applications, 31(8):4423–4433, 2019. [16] I Ahmad, A Jayswal, and J Banerjee. On interval-valued optimization problems with generalized invex functions. Journal of Inequalities and Applications, 2013(1):313, 2013. [17] I Husain and M Masoodi. Second-order duality for continuous programming contain- ing support functions. Applied Mathematics, 1(6):534–541, 2010. [18] M Arana-Jiménez, G Ruiz-Garzón, A Rufián-Lizana, and R Osuna-Gómez. A neces- sary and sufficient condition for duality in multiobjective variational problems. Eu- ropean journal of operational research, 201(3):672–681, 2010. V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 37 of 38 [19] S Treanta, C F Pırje, J C Yao, and B B Upadhyay. Efficiency conditions in new interval-valued control models via modified t-objective functional approach and saddle-point criteria. Mathematical Modelling and Control, 2025. [20] K Khazafi, N Rueda, and P Enflo. Sufficiency and duality for multiobjective control problems under generalized (b, ρ)-type i functions. Journal of Global Optimization, 46(1):111–132, 2010. [21] S Mititelu and M Postolache. Mond-weir dualities with lagrangians for multiobjective fractional and non-fractional variational problems. Journal of Advanced Mathematical Studies, 3(1), 2010. [22] N Pokharna and I P Tripathi. A generalized division approach for interval fractional programming problems. Applied Mathematical Modelling, 144:116048, 2025. [23] Om P Agrawal. Formulation of euler–lagrange equations for fractional variational problems. Journal of Mathematical Analysis and Applications, 272(1):368–379, 2002. [24] OP Agrawal. Fractional variational calculus and the transversality conditions. Journal of Physics A: Mathematical and General, 39(33):10375, 2006. [25] S G Samko. Fractional integrals and derivatives. Theory and applications, 1993. [26] R Almeida. Variational problems involving a caputo-type fractional derivative. Jour- nal of Optimization Theory and Applications, 174(1):276–294, 2017. [27] M M Shalini, B Kandasamy, M Rangasamy, P B Dhandapani, A Zeb, I Khan, and Abdoalrahman SA Omer. Feasibility of variable delay fuzzy fractional stochastic differential system with non-instantaneous impulses. Applied Mathematics in Science and Engineering, 33(1):2458612, 2025. [28] O S Fard and M Salehi. A survey on fuzzy fractional variational problems. Journal of Computational and Applied Mathematics, 271:71–82, 2014. [29] J Soolaki, O S Fard, and A H Borzabadi. Generalized euler–lagrange equations for fuzzy variational problems. SeMA Journal, 73(2):131–148, 2016. [30] M Caputo and M Fabrizio. A new definition of fractional derivative without singular kernel. Progress in fractional differentiation & applications, 1(2):73–85, 2015. [31] A Jayswal and G Uniyal. Optimal conditions and duality results for a semi- infinite variational programming problem and its mond–weir dual involving caputo– fabrizio fractional derivatives. Journal of Computational and Applied Mathematics, 468:116628, 2025. [32] A Jayswal and G Uniyal. Lagrange duality and saddle point criteria for semi-infinite variational programming problem with caputo-fabrizio fractional derivative. Journal of Applied Mathematics and Computing, pages 1–21, 2025. [33] H Ishibuchi and H Tanaka. Multiobjective programming in optimization of the in- terval objective function. European journal of operational research, 48(2):219–225, 1990. [34] A A Kilbas, H M Srivastava, and J J Trujillo. Theory and applications of fractional differential equations, volume 204. elsevier, 2006. [35] E F D Goufo and A Atangana. Analytical and numerical schemes for a derivative with filtering property and no singular kernel with applications to diffusion. The European Physical Journal Plus, 131(8):269, 2016. V. Rayanki et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6617 38 of 38 [36] V Rayanki, I Ahmad, and K Kummari. Interval-valued variational programming problem with caputo–fabrizio fractional derivative. Mathematical Methods in the Applied Sciences, 46(16):17485–17510, 2023.