Abstract
In this article, we extend the generalized invexity and duality results for multiobjective variational problems with fractional derivative pertaining to an exponential kernel by using the concept of weak minima. Multiobjective variational problems find their applications in economic planning, flight control design, industrial process control, control of space structures, control of production and inventory, advertising investment, impulsive control problems, mechanics, and several other engineering and scientific problems. The proposed work considers the newly derived Caputo–Fabrizio (CF) fractional derivative operator. It is actually a convolution of the exponential function and the first-order derivative. The significant characteristic of this fractional derivative operator is that it provides a non-singular exponential kernel, which describes the dynamics of a system in a better way. Moreover, the proposed work also presents various weak, strong, and converse duality theorems under the diverse generalized invexity conditions in view of the CF fractional derivative operator.
1 Introduction
The present scenario indicates that the fractional differential equations (FDEs) and fractional variational problems (FVPs) are being used to delineate the physical models and engineering processes in a better way. The clear reason is that the standard mathematical models of integer-order derivatives incorporating models of non-linear nature do not perform efficiently in many instances according to desired results. Recently, the field of fractional calculus has portrayed a significant part in various areas of knowledge such as chemistry [1], biology [2,3], mechanics [4,5], and finance [6]. The application area of fractional modelling and fractional operators encompasses anomalous diffusion [7], physics [8], heat conduction [9], geophysics [10], epidemiology [11], fractals and fractional derivative [12], computational fractional derivative equations [13], fractional predator–prey system [14], and porous media [15]. The models related to these fields utilize fractional derivative operators frequently.
There are various types of fractional derivative operators in the literature of fractional calculus founded by so many famous mathematicians. But the most popular definitions of them are Riemann–Liouville (RL) fractional derivative and Riesz fractional derivative described in the studies of Samko et al. [16] and Podlubny [17], Caputo fractional derivative in refs. [18,19], Weyl fractional derivative [20], Hadamard fractional derivative [16,17], Jumarie’s fractional derivative propounded in works of Jumarie [21,22], Atangana–Baleanu derivative proposed in ref. [23], and Liouville–Caputo derivative described in ref. [24]. The interesting fact is that these definitions of fractional derivative operators have their own significance and their uses vary according to the structure and behaviour of particular models along with initial conditions. A wide literature is available on different perceptions of fractional derivatives. But the most celebrated fractional calculi are the Caputo fractional derivative and the RL derivative. The Caputo fractional derivative handles initial value problems efficiently in comparison to the RL derivative. The newly introduced Caputo–Fabrizio (CF) fractional derivative operator propounded by Caputo and Fabrizio in ref. [25] is actually a convolution of an exponential function and the first-order derivative. In this definition, the derivative of a constant is equal to zero like the usual Liouville–Caputo definition but it also provides the non-singular kernel which was not a characteristic of the Liouville–Caputo fractional derivative. The main purpose of the CF definition was to introduce a new fractional derivative with an exponential kernel to describe even better the dynamics of systems with memory effect.
Recently, some authors presented a new analysis on fractional modelling of real-world problems and application of fractional order Lagrangian approach towards study of problems arising in physical sciences and engineering. Some recent works related to these fields are necessary to be cited here. Jajarmi et al. [26] suggested a general fractional formulation for immunogenic tumour dynamics. Baleanu et al. [27] presented a new study on the general fractional model of COVID-19 with isolation and quarantine effects. Erturk et al. [28] utilized a new fractional-order Lagrangian to describe the dynamics of a beam on nanowire. Jajarmi et al. [29] implemented a new fractional Lagrangian approach to study the case of capacitor microphone. Dubey et al. [30] solved the fractional model of Phytoplankton–Toxic Phytoplankton–Zooplankton system with convergence analysis. Moreover, a fractional model of atmospheric dynamics of carbon dioxide gas [31] and a fractional-order hepatitis E virus model [32] were also recently investigated with efficient computational methods.
Multiobjective variational problems proficiently handle the problems of science, engineering, logistics, and economics where optimal decisions have to be decided between two or more clashing objectives. To derive the optimality conditions it is necessary to study the behaviour of functions and their derivatives at that point. In the theory of mathematical optimization, the duality principle indicates two perspectives of optimization problems: the primal problem and the dual problem. If the primal is a minimization problem, then the dual is a maximization problem, and if the primal is a maximization problem, then the dual is a minimization problem. The concept of duality considers a problem with less number of variables and constraints and so it is much advantageous regarding computational procedure. Duality results play a major role in construction of numerical algorithms for solving some specific types of optimization problems. The duality theory is applied mainly in economics, management, physics, etc. On the other hand, calculus of variations significantly deals with the solution of several problems arising in theory of variations, optimization of orbits, dynamics of rigid bodies, etc. It is closely related to optimization of functional and is expressed in terms of definite integrals pertaining to functions and their derivatives. In the past few years, a number of contributions have been made towards the duality results for multiobjective variational problems. For the first time, Hanson [33] established and developed the linkage between classical calculus of variation and mathematical programming. After that, Mond and Hanson [34] derived optimality and duality results for scalar valued variational problems in view of convexity assumptions. Chandra et al. [35] studied optimality and duality for a class of non-differentiable variational problems. In this sequence, Bector and Husain [36] investigated duality for multiobjective variational problems. Nahak and Nanda [37] and Chen [38] constructed duality results for multiobjective variational problems with invexity. Some years later, Bhatia and Mehra [39] extended further the results of Mond et al. [40] and explored the optimality conditions and duality results for multiobjective variational problems with generalized B-invexity.
The concept of invexity is of great significance in variational problems and mathematical programming. Hanson [41] introduced the notion of invexity to mathematical programming. Mishra and Mukherjee [42] presented duality results for multiobjective FVPs. Furthermore, Mond and Husain [43] also investigated sufficient optimality criteria and duality for variational problems with generalized invexity. It is clearly observed that the duality results derived for variational problems presented in refs. [40,42,43] that hold for convex functions are also well-fitted for the wide range of invex functions. Weir and Mond [44] considered the concept of weak minima to derive the duality results for multiobjective programming problems. Different scalar duality results have also been extended for multiobjective programming problems by Weir and Mond [44]. Mukherjee and Mishra [45] have considered the concept of weak minima in the continuous case and have delivered a complete generalization of the results of Weir and Mond [44] to multiobjective variational problems. Moreover, they also relaxed the generalized convexity conditions to generalized invexity conditions.
Recently, Kumar [46] extended the invexity for continuous functions to invexity of order m. They further generalized the invexity of order m to ρ-pseudoinvexity type-I of order m, ρ-pseudoinvexity type-II of order m, as well as ρ-quasi-invexity type-I and type-II of order m. In 2016, Kumar et al. [47] also analysed the multiobjective FVP under F-Kuhn–Tucker (KT) pseudoinvexity conditions. Hachimi and Aghezzaf [48] established the mixed duality results and the sufficient optimality conditions concerning multiobjective variational problems under generalized (F,α,ρ,d)-type I functions which assimilate the several concepts of generalized type-I functions successfully. Later on, Mishra et al. [49] extended the generalized type-I invexity and duality for non-differentiable multiobjective variational problems. In 2014, Wolfe-type and Mond–Weir-type duality results were formulated for multiobjective variational control models under (ϕ,ρ)-invexity conditions by Antczak [50]. More recently, Upadhyay et al. [51] presented optimality conditions and duality for multiobjective semi-infinite programming problems on Hadamard manifolds utilizing generalized geodesic convexity. Moreover, Upadhyay et al. [52] also investigated Minty’s variational principle for non-smooth multiobjective optimization problems on Hadamard manifolds. Guo et al. [53] showed applications of symmetric gH-derivative to dual interval-valued optimization problems in a very efficient way. Furthermore, optimality conditions and duality for a class of generalized convex interval-valued optimization problems are recently investigated in works of Guo et al. [54].
The main purpose of this study is to derive the weak and strong duality results for multiobjective variational problems pertaining to a CF fractional derivative operator with exponential kernel. The CF fractional derivative possesses the non-singular kernel and so is better than the Caputo and RL fractional derivative operators. The proposed work presents the derivation of duality as well as strict converse duality theorems for variational problems with CF fractional derivative by employing some propositions and theorems of fractional calculus. In this article, we propounded first the optimality conditions for the variational problem. Furthermore, we present Theorem 1 which proves that a minimizer of the variational problem is a solution of the fractional Euler–Lagrange equation containing the CF fractional derivative. Now, we derived the formula for integration by parts for the CF fractional derivative in Proposition 1. The extended invexity definitions in view of CF fractional derivative operator along with Proposition 1 and Theorem 1 have been the key motivation behind the study of the variational problems with fractional calculus approach. Theorem 2 proves the fact that if a function is convex, the solution of the fractional Euler–Lagrange equation containing the CF fractional derivative will be a minimizer of the variational problem. Theorems 3 and 4 present the results for primal variational problems having a weak minimum. Furthermore, Theorems 5–10 are concerned with weak and strong duality results depending on the CF fractional derivative. Finally, Theorems 11 and 12 provide the strict converse duality results in view of the CF fractional derivative.
In the present work, the concept of weak minima has been considered and the generalizations of weak, strong, and strict converse duality results of Mukherjee and Mishra [45] have been extended to multiobjective variational problems pertaining to the CF fractional derivative operator. The remaining part of the article is organized as follows: In Section 2, we present the elemental definitions, formulae, and theorems regarding invexity and fractional derivative operators. Section 3 derives weak and strong duality results. Section 4 presents a strict converse duality result. Finally, Section 5 records the epilogue for the proposed work.
2 Basic definitions, theorems, and symbols
We follow these definitions and symbols in the present article.
Definition 1
[55]: The left and the right RL fractional derivatives of order α are defined by
Definition 2
[56]: The Caputo fractional derivative of
If y ∈ C 1, then
As α → 1,
Definition 3
[25]: The new CF fractional derivative operator is described as follows:
where K(α) signifies the normalization function with the property K(0) = K(1) = 1. Clearly,
Remark 1
Here we consider the value of K(α) as
Remark 2
As α → 1,
Definition 4
Abdeljawad and Baleanu [57] have defined the right CF fractional derivative as
Definition 5
The first-order Sobolev space defined in the interval (a, b) is stated as H 1(a, b) = {x ∈ L 2(a, b) |x′ ∈ L 2(a, b)}, where x′ denotes the weak derivative of x.
Definition 6
[25]: Let y ∈ H 1(a, b), b > a, 0 < α < 1, then the CF fractional derivative is stated as in Definition (3), where K(α) specifies the normalization function with characteristic K(0) = K(1) = 1. If the function y ∉ H 1(a, b), then the derivative is formulated as follows:
Here, the CF fractional derivative has an exponential kernel.
Definition 7
[57]: Let x be a function in such a way that x ∈ H 1(a, b) a < b. The left Riemann fractional derivative of order α in the CF sense is given by
where
Similarly, the right Riemann fractional derivative of order α in the CF sense can be written as follows:
where
Remark 3
When
Proposition 1
Let
Proof
We define the left and right auxiliary fractional integrals as
Now in view of Definition (3) and Eq. (1), it is concluded that
In the next step, we evaluate the integral
Using Eq. (3) along with further utilization of Theorem 1 of ref. [57] and integration by parts for classical derivatives, we obtain
Now in view of Definition 7, we obtain
Definition 8
(Optimality conditions for variational problems):
The following variational problem with the CF fractional derivative is considered here for given
with
Given any
Here, we denote
Theorem 1
Let y be a minimizer of the variational V(y) defined on
Proof
Let y be a solution for the functional
Now utilizing the boundary conditions
Remark 4
Eq. (5) is called the Euler–Lagrange equation associated with the variational
Remark 5
It is notable that Eq. (5) provides necessary criterion only. Now to obtain sufficient criterion, the concept of convex function is necessary to recall.
Definition 9
A function
Theorem 2
If the function Q as described in Eq. (
4
) is convex in
Proof
Let y be a solution for the fractional Euler–Lagrange Eq. (5). Assume
Since
Hence in view of Eq. (7), Eq. (6) provides the inequality as follows:
Consequently,
Definition 10
Invexity definitions
Let
Let Y be the space of piecewise smooth functions
where y 0 signifies the boundary value.
Let
holds for every
Let
Now utilizing the linearity property of the CF derivative operator and further the convexity assumption of
Clearly,
Clearly as
which is the definition of invexity in the continuous case extended by Mond et al. [40]. It is notable that if the function g is independent of ξ, the above given definition of invexity transforms to the inequality
Example 1
The proposed inequality which is derived earlier is given as follows:
Let
Then
Now utilizing the formula of CF derivative, we obtain
where
Thus,
and
Now
Now, we evaluate the term
In this sequence, we also evaluate the integral term of the proposed inequality (9) in view of Eqs. (10) and (15) as follows:
Case I: For
Case II: For
Consequently, it is concluded that the proposed inequality (9) with CF fractional derivative holds well for the function
Now, we extend the definitions of invex, pseudoinvex (PIX), strictly pseudoinvex (SPIX), and quasi-invex (QIX) as described in ref. [58] with the CF fractional derivative of order
Definition 11
Invex
The functional G is stated as invex with respect to η if there exists a differentiable vector function
Definition 12
PIX
The functional G is stated as PIX w.r.t.
or equivalently,
Definition 13
SPIX
The functional G is stated as SPIX w.r.t.
or equivalently,
Definition 14
QIX
The functional G is stated as QIX w.r.t.
or equivalently,
In the aforementioned definitions,
Now we deal with the multiobjective variational primal problem, as discussed in the work of Mukherjee and Mishra [45], with the CF fractional derivative operator in the following way:
(P) Minimize
subject to
where
For the primal problem (P), a point y
0 is referred to as a weak minimum if there exists no other feasible point
Now, we frame the continuous versions of Theorems 2.1 and 2.2, as described in the work of Weir and Mond [44], involving fractional derivative operators with exponential kernel in the following way:
Theorem 3
Let
Proof
The proof is easily established through Theorem 1.□
Theorem 4
Let the primal problem (P) have a weak minimum at a point
where
Proof
The proof is easily established through Theorem 1.□
3 Duality
In relation to the primal problem (P), the dual problem (D) as discussed in ref. [45] is considered with fractional derivative operators pertaining to exponential kernel in the following way:
(D) Maximize
subject to
where
In upcoming steps, we discuss the duality theorems, as discussed in the work of Mukherjee and Mishra [45], with the CF fractional derivative operator of order
Theorem 5
(Weak duality):
If, for all feasible
Proof
(a) Let y be feasible for (P) and
From Eq. (27), we have
Suppose contrary to the result given in statement of Theorem 5, i.e.
Now, use of pseudoinvexity of
or
Now using Proposition 1 in the above obtained inequality (32), we have
we obtain
Now in view of Eq. (30) and inequality (33), we have
which is a contradiction.
Hence,
Thus, the supposition (31), which is contrary to the result given in the statement of Theorem 5, is wrong and consequently
(b) Let
Multiplying Eq. (30) by
Suppose contrary to the result given in the statement of Theorem 5, i.e.
Multiplying the aforementioned obtained inequality (35) by
Now, the pseudoinvexity of
or
Now by using Proposition 1 in the above obtained inequality (37), we have
Now in view of Eq. (34) and inequality (39), we have
which is a contradiction. Thus, the supposition (35) is false. Consequently,
Theorem 6
(Strong duality):
Let the primal (P) have a weak minimum at
then
Proof
Since the primal (P) has a weak minimum at
and
If
(a) The pseudoinvexity of
Since
Now by using Proposition 1 in inequality (42), we have
Since
Using Eq. (45) in inequality (44), we obtain
This is clearly a contradiction. Hence, the supposition of the existence of feasibility of
(b) The proof of part (b) is very similar to that of part (a).
We now consider the following dual problem (D1) in relation to the primal problem (P).
(D1) Maximize
The next step is to establish the weak and strong duality theorems, as discussed in the work of Mukherjee and Mishra [45], with the CF fractional derivative operators.□
Theorem 7
(Weak duality):
If for all feasible
Then
Proof
(a) Let y be feasible for primal (P) and
Suppose contrary to the result, i.e.
Thus, in view of the pseudoinvexity of
From the constraint of the primal problem (P), we have
Now from the constraint of the dual problem (D1), we have
Combining both inequalities (53) and (54), we obtain
or
or
Now, inequality (55) together with the quasi-invexity of
Now adding inequalities (52) and (56), we obtain
Now by using Proposition 1 in inequality (57), we have
Using Eq. (47) and
which is a contradiction. Thus,
(b) The proof of part (b) is very similar to the proof of part (a).
(c) Let y be feasible for (P) and
Suppose contrary to the result, i.e.
Then in view of the quasi-invexity of
Thus, the strict pseudoinvexity of
Now adding inequalities (60) and (62), we obtain
Now using Proposition 1 in inequality (63) along with
which is a contradiction. Consequently,
The proof of part (d) is very similar to the proof of part (c).□
Theorem 8
(Strong duality):
Let the primal (P) have a weak minimum at the point
then
Proof
Since the primal (P) have a weak minimum at y
0 for which the KT constraint qualification is satisfied, then by Theorem 4, there exists
and
If
(a) The pseudoinvexity of
Since
From the constraint of the primal problem (P), we have
Since y
0 is feasible for primal problem (P) thus it satisfies the aforementioned constraint and so
thus
Now from the constraint of the dual problem (D1), we have
Since
Combining both inequalities (67) and (68), we obtain
or
or
The quasi-invexity of
Now adding inequalities (66) and (70), we obtain
Now using Proposition 1 in the above obtained inequality (71) and
Since
Using Eq. (73) in inequality (72), we obtain
which is a discrepancy and consequently contradicts the feasibility of
(b) The proof of part (b) is very similar to the proof of part (a).
(c) The quasi-invexity of
Since
Since
(
Thus, in view of strictly pseudoinvexity of
Now combining inequalities (75) and (77), we obtain
Now using Proposition 1 in inequality (78) and utilizing
Now in view of Eq. (73), inequality (79) can be expressed as
which clearly contradicts the feasibility of
(d) The proof of part (d) is very much identical to the proof of part (c).
Now we state the continuous form of a general primal (CP) and dual (CD) for the multiobjective variational optimization problem, as discussed in the work of Mukherjee and Mishra [45], involving fractional derivative operator with exponential kernel. Consider the problem given as follows:
(CP) Minimize
subject to
where
Let
In connection to (CP), we investigate the problem:
(CD) Maximize
subject to
The weak and strong duality theorems, as described in ref. [45], are articulated here with the CF fractional derivative operators without proof because their proof may be delivered in a very identical manner to that of Theorems (5)–(8).□
Theorem 9
(Weak duality):
If, for all feasible
is QIX and
is QIX and
then
Theorem 10
(Strong duality):
Let (CP) have a weak minimum at
Now, we establish the strict converse duality theorem as stated in ref. [45] with CF fractional derivative operator in the forthcoming section.
4 Strict converse duality
Theorem 11
Let
Assume that
then
Proof
It is assumed that
Now, the aforementioned inequalities imply that
(a) Now, inequality (80) in view of quasi-invexity of
Since it is given that
Now combining inequalities (81) and (82), we obtain
Now using Proposition 1 in inequality (83) and
Since
In view of Eq. (85), inequality (84) can be written as:
which contradicts the assumption
(b) Since
Since it is given that
Now combining inequalities (86) and (87), we obtain
Now using Proposition 1 in inequality (88) along with
In view of Eq. (85), inequality (89) can be written as
which contradicts the assumption
In the next step, we state the theorem as stated in ref. [45] with the CF fractional derivative operator in the following way:
Theorem 12
Let
If
is QIX at v
0
and each
Corollary
Suppose y
0
is a weak minimum for (P) and
If
Proof
It is assumed that
Multiplying Eq. (90) by
Since
and also
and since
Now using Proposition 1 in inequality (93) and
Now using Eq. (91) in inequality (94), we have
which is a contradiction. Thus, the assumption
5 Conclusions
The proposed work extends and derives the generalized invexity and duality results for multiobjective variational problems with the framing of a non-singular fractional derivative pertaining to the exponential kernel by utilizing the concept of weak minima. This work considers the CF fractional derivative operator possessing a non-singular exponential kernel. Moreover, several duality results of weak, strong, and converse categories have also been derived for various types of generalized invexity conditions in view of the CF fractional derivative operator. Some basic theorems and formulas for integration by parts for the fractional derivative with an exponential kernel have played a significant role in proving the weak, strong, and converse duality theorems. This article also presents the derivation of strict converse duality theorems for multiobjective variational problems with the CF fractional derivative by employing some propositions and theorems of fractional calculus. The variational problems with CF fractional derivative may be helpful in analysing the optimization problems and physical processes. Problems related to production planning, oil refinery scheduling, portfolio selection, management sciences, and economics can be modelled successfully in the form of multiobjective variational problems. The results derived in this article are important for the growth of generalized invexity and duality results for a class of multiobjective variational problems involving a non-singular fractional derivative. However, it is difficult to illustrate any practical application on the basis of derived results. But there is an ample scope to explore the optimality and duality results for multiobjective variational problems within the scope of fractional calculus. This work can be further extended to study the multiobjective variational problems and non-differentiable multiobjective variational problems involving other kinds of fractional derivatives. As a future scope of the work, FVPs can also be studied with the theorems and propositions applied in this article.
Acknowledgments
The authors are grateful to Prof. S. K. Mishra, Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi-221005, India, for fruitful ideas and valuable discussions at initial stages of their work on this article.
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Funding information: The authors state no funding involved.
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Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
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Conflict of interest: The authors state no conflict of interest.
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Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.
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