ON OPTIMALITY AND DUALITY IN INTERVAL-VALUED VARIATIONAL PROBLEM WITH B-(p, r)-INVEXITY

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https://doi.org/10.1051/ro/2021088 www.rairo-ro.org

ON OPTIMALITY AND DUALITY IN INTERVAL-VALUED VARIATIONAL PROBLEM WITH B-(p, r)-INVEXITY

Indira P. Debnath

^∗

and Nisha Pokharna

Abstract.In this paper, we consider a class of interval-valued variational optimization problem. We extend the definition ofB-(p, r)-invexity which was originally defined for scalar optimization problem to the interval-valued variational problem. The necessary and sufficient optimality conditions for the problem have been established under B-(p, r)-invexity assumptions. An application, showing utility of the sufficiency theorem in real-world problem, has also been provided. In addition to this, for the interval-optimization problem Mond–Weir and Wolfe type duals are presented and related duality theorems have been proved. Non-trivial examples verifying the results have also been presented throughout the paper.

Mathematics Subject Classification. 90C30, 90C46.

Received October 21, 2020. Accepted May 30, 2021.

1. Introduction

Interval-valued optimization has a huge number of real-world applications such as in fuzzy logic, artificial intelligence, robotics, genetic algorithm, optimal control, neural computing, image restoration problem, optimal shape design problems, robust optimization, power unit problems, molecular distance geometry problems, engineering designs etc. Also, the variational optimization problems are of much importance in various fields such as structural optimization, epidemic, production and inventory, shape optimization in fluid mechanics, optimal control of processes and material inversion in geophysics. Motivated and inspired by the above observations, in this paper we extend the notion ofB-(p, r)-invexity to the class of interval-valued variational optimization problem.

In variational calculus, the classical solutions to minimization problem are prescribed by boundary value problems involving Euler–Lagrange equations. Euler was the first to develop the general formula for finding the curve along which a given integral expression has its greatest value. Later, Lagrange proposed a method of mul- tipliers that allows for solutions to the problem without having to solve the conditions or constraints explicitly.

These two techniques changed the way of solving the optimization problem and have the last influence on how partial differential equations are viewed. The technique of handling such optimization problem is fundamental in many wide areas of mathematics, physics and engineering. [for more details, refer [7,14]]

Keywords.Interval-valued variational problem, optimality, duality, LU optimal,B-(p, r)-invexity.

Sardar Vallabhbhai National Institute of Technology, Surat, India.

∗Corresponding author:[email protected]

c

The authors. Published by EDP Sciences, ROADEF, SMAI 2021

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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In field of operation research, three kinds of optimization techniques, namely deterministic optimization, stochastic optimization and interval-valued optimization are mostly used to solve any optimization problem.

Among these techniques, interval-valued optimization is an emerging branch of mathematics which deals with uncertainty in optimization problem. In an interval optimization problem, uncertainty may appear in the form of closed intervals either in the objective function or in at least one of the constraints or in both. Various solution concepts have been introduced for interval-valued optimization problems. Ishibuchi and Tanaka [17] used UC-nondominated solution concept for an interval valued optimization problem. In this paper, LU-optimality solution concept is used which was originally defined by Wu [23] and it follows from the concept of a nondominated solution employed in vector optimization problems.

Recently, many researchers have specifically driven attention towards the study of optimality conditions and duality results for interval-valued optimization problems under various generalized convexity assumptions. Wu [23] studied Wolfe-type dual and derived duality theorems for an interval valued optimization problem using nondominated solution concept [24–26]. Bhurjee and Panda [11] studied whether the solution of an interval optimization problem exists. Using the concept of generalized preinvexity, Zhang et al. [28] established the optimality conditions and duality theorems for an interval-valued optimization problem. For an interval-valued optimization problem, Ahmadet al. [2] obtained optimality conditions and duality results by using generalized (p, r)-ρ-(η, θ) invexity. Debnath and Gupta [13] derived the Fritz John and KKT optimality conditions for a nondifferentiable fractional interval-valued programming problem utilizing the concept of LU optimal solution.

For an interval optimization problem with generalized convex function (pseudo-invex function), the sufficient optimality conditions and duality theorems were investigated by Jayswalet al. [18].

In recent times, many authors have worked in establishing the relation between mathematical problems and calculus of variation which was first proposed by Hanson [15]. After Hanson’s work, this problem became of much interest for many researchers and several contributions have been made in this direction. Till the times when Hanson [16] introduces the concept of invexity, optimality conditions and duality results were proved for the optimization problems where the functions involved were convex. Hanson was the first to prove optimality conditions and duality results for a wider class of invex functions. Mond et al. [21] extended this invexity concept to continuous case and proved duality results for a class of variational problem. Bector and Husain [8] extended the concept of duality used in vector optimization problem to variational problem. Thereafter, many researchers derived duality relations for multiobjective problem under generalized invexity assumptions [1,3–5,8–10,12,19,28].

Optimality and duality have various applications in optimization. They are of interest as they play an impor- tant role in analysing the behaviour of original problem and also helps in obtaining an optimal solution. Antczak and Jim´enez [4,5] established optimality conditions and duality results for a multiobjective variational problem utilizingB-(p, r)-invexity and (φ, ρ)-invexity. Bhatia and Mehra [10] derived optimality and duality results under generalized B-invexity assumptions. Kim [19] obtained duality results for a multiobjective variational problem using concept of generalized type I invexity. Mishra and Mukherjee [20] usedV-invexity assumptions to prove various results for a multiobjective variational problem. Ahmad and Sharma [1] discussed the sufficiency and duality for a multiobjective variational control problem involving (F, α, ρ, θ)-V-convexity. Aran-Jim´enez [6]

provided new pseudoinvexity conditions for a multiobjective variational problem using generalized convexity.

Zhanget al. [27] derived various duality results for a multiobjective variational control problem withG-invexity.

Very recently, Ahmadet al. [3] studied sufficiency and duality for an interval-valued variational problem under invexity assumption.

In this paper, we consider a class of variational problem where the objective function is interval-valued. Our main focus is to investigate the optimality conditions and duality results for the considered problem under B-(p, r)-invexity assumptions. The paper is divided section-wise as follows. In Section2, some basic definitions and concepts regarding the interval-valued optimization problem are given. In Section 3, sufficient optimality conditions for a class of interval-valued variational problem are studied. In Section 4, some duality results for defined Mond–Weir dual are obtained. Duality results for defined Wolfe dual are established in Section 5 and finally, in Section6, we conclude our paper.

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2. Definitions and preliminaries

Let J be the class of all closed and bounded interval [a, b] with a ≤b in R. For a closed interval B in R, using standard notationB = [b^L, bÛ] withb^L andbÛ being the lower and upper endpoints ofB, respectively. If b^L =bÛ =b, thenB= [b^L, bÛ] =b is a real number. LetB= [b^L, bÛ]∈J andC= [c^L, cÛ]∈J. Then, we have the following operations of interval analysis by Moore [22]:

(1) B=C =⇒ b^L=c^L andb^U =c^U. (2) −B={−b:b∈B}=

−b^U,−b^L . (3) B+C={b+c:b∈B, c∈C}=

b^L+c^L, b^U+c^U . (4) λ+B={λ+b:b∈B}=

λ+b^L, λ+b^U . (5) λB=λ[b^L, b^U] =

(

λb^L, λb^U

, ifλ≥0 λb^U, λb^L

, ifλ <0, whereλis a real number.

ForB= b^L, b^U

andC= c^L, c^U

, following ordering relation is used to rank interval numbers:

B LUC if and only if, b^L≤c^L, b^U ≤c^U.

It can be easily seen thatLU is a partial ordering onJ. Further,B≺LUCif and only ifB≤CandB6=C or equivalently,B ≺LUC if and only if:

b^L< c^L, bÛ ≤cÛ, or b^L≤c^L, bÛ < cÛ, or b^L< c^L, bÛ < cÛ.

LetR^q be theq-dimensional Euclidean space and A= [a, b] be a real interval. We assume thatC(A,R^q) be the space of piecewise smooth functions u: A → R^q with norm kuk =kuk_∞+kDuk_∞, where k · k_∞ is the uniform norm and the differentiation operatorD is given byz=Du ⇐⇒ u(t) =u(a) +Rb

az(s) ds, withu(a) being a prescribed boundary value. Thus, _dt^d ≡D excluding discontinuities.

A function Φ :A×R^q×R^q →J is said to be interval-valued if range of the function is an interval inRsuch that Φ^L(t, u(t),u(t))˙ ≤Φ^U(t, u(t),u(t)). Thus, an interval-valued function can be written as:˙

Φ (t, u(t),u(t)) =˙

Φ^L(t, u(t),u(t))˙ ,Φ^U(t, u(t),u(t))˙ ,

where u : A → R^q is piecewise smooth function of t and ˙u(t) is derivative of u with respect to t and Φ^L(t, u(t),u(t)) and Φ˙ ^U(t, u(t),u(t)) are real valued functions.˙

We denote the first partial derivative of Φ with respect to uand ˙uby Φu and Φu˙, respectively, such that

Φ_u=







∂Φ

∂u1

∂Φ

∂u2

...

∂Φ

∂u_q







and Φ_u_˙ =







∂Φ

∂u˙1

∂Φ

∂u˙2

...

∂Φ

∂u˙_q





 .

Now, we extend the definition ofB-(p, r)-invexity for the interval-valued variational optimization case which was originally defined for a scalar optimization problem by Antczak and Jim´enez [5].

Definition 2.1. Assume that Φ : A×R^q ×R^q → J be an interval-valued function defined as Φ (t, u,u) =˙ [Φ^L(t, u,u)˙ ,Φ^U(t, u,u)].˙ Also, letφ^L, φ^U :A×R^q×R^q →Rare continuously differentiable functions. If their exists real numbersp, r, a functionη:A×R^q×R^q →R^q withη(t, u(t),u(t)) = 0 for all¯ u∈C(A,R^q) andt∈A withu(t) = ¯u(t) and a functionb:C(A,R^q)×C(A,R^q)→R+\{0} such that foru∈C(A,R^q):

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Case (i): ifp6= 0, r6= 0.

1

rb(u,u)¯

e^r(^R_a^b^Φ^L^(t,u,^{u) dt−}^˙ ^R_a^b^Φ^L(^t,¯^u,^u^¯^˙)^dt)−1

≥¹_pRb

a e^pη(t,u,¯^u)−1^T

Φ^L_u_¯(t,u,¯ u)˙¯ −_dt^d Φ^L_u_¯_˙ (t,u,¯ u)˙¯

dt, and

1

rb(u,u)¯

e^r(^R_a^b^ΦÛ^(t,u,^{u) dt−}^˙ ^R_a^b^ΦÛ(^t,¯û,û^¯^˙)^dt)−1

≥¹_pRb

Φ^U_¯_u(t,u,¯ u)˙¯ −_dt^d Φ^U_¯_u_˙ (t,u,¯ u)˙¯

dt.











(2.1)

Case (ii): ifp= 0, r6= 0.

1

rb(u,u)¯

e^r(^R_a^b^Φ^L^(t,u,˙u) dt−R_b

aΦ^L(^t,¯^u,^u^¯^˙)^dt)−1

≥Rb

a (η(t, u,u))¯ ^T Φ^L_u_¯(t,u,¯ u)˙¯ −_dt^d Φ^L_u_¯_˙ (t,u,¯ u)˙¯

dt, and

1

rb(u,u)¯

e^r(^R_a^b^Φ^U^(t,u,u) dt−^˙ R_b

aΦÛ(^t,¯û,û^¯^˙)^dt)−1

≥Rb

a (η(t, u,u))¯ ^T Φ^U_u_¯(t,u,¯ u)˙¯ −_dt^d Φ^U_u_¯_˙ (t,u,¯ u)˙¯

dt.











(2.2)

Case (iii): ifp6= 0, r= 0.

b(u,u)¯ Rb

aΦ^L(t, u,u) dt˙ −Rb

aΦ^L(t,u,¯ u) dt˙¯

≥¹_pRb

Φ^L_¯_u(t,u,¯ u)˙¯ −_dt^d Φ^L_¯_u_˙ (t,u,¯ u)˙¯

dt, and

b(u,u)¯ Rb

aΦ^U(t, u,u) dt˙ −Rb

a Φ^U(t,u,¯ u) dt˙¯

≥¹_pRb

Φ^U_¯_u(t,u,¯ u)˙¯ −_dt^d Φ^U_¯_u_˙ (t,u,¯ u)˙¯

dt.











(2.3)

Case (iv): ifp= 0, r= 0.

b(u,¯u) Rb

≥Rb

a (η(t, u,u))¯ ^T Φ^L_u_¯(t,u,¯ u)˙¯ −_dt^d Φ^L_u_¯_˙ (t,u,¯ u)˙¯

dt, and

b(u,¯u)Rb

aΦ^U(t,¯u,u) dt˙¯

≥Rb

a (η(t, u,u))¯ ^T Φ^U_u_¯(t,u,¯ u)˙¯ −_dt^d Φ^U_u_¯_˙ (t,u,¯ u)˙¯

dt.











(2.4)

Then, Φ is said to be B-(p, r)-invex funciton with respect toη at ¯uinC(A,R^q).

Definition 2.2. The function Φ is said to be strict B-(p, r)-invex function with respect toηat ¯uin C(A,R^q), if all the inequalities (2.1)–(2.4) in the Definition2.1are strict inequalities.

Example 2.3. LetA= [0,1] andC(A,[0,1]) be the space of all piecewise smooth functionsu:A→[0,1]. Con- sider the interval-valued function Φ :A×[0,1]×[0,1]→J,φ(t, u(t),u(t)) =˙

φ^L(t, u(t),u(t))˙ , φ^U(t, u(t),u(t))˙ , whereφ^L andφ^U are continuously differentiable functions, as follows:

Φ (t, u,u) =˙

u²+ 1, u²+ 2u+ 5 .

Letη:A×[0,1]×[0,1]→Randb:C(A,[0,1])×C(A,[0,1])→R+\{0} be given byη(t, u,u) =¯ ¹₄(¯u²−u²) andb(u,¯u) =u¯u+ 9, respectively. Then, the function φdefined above isB-(4,3) invex at ¯u= 0.

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Consider the following interval-valued variational problem, in which the state vector u(t) travels from a defined initial stateu(a) =γ to some defined final stateu(b) =δ in such a manner so that a given functional can be minimized:

(IVP) min

Z b a

Φ (t, u(t),u(t)) dt˙ =

"

Z b a

Φ^L(t, u(t),u(t)) dt,˙ Z b

a

Φ^U(t, u(t),u(t)) dt˙

#

subject to

u(a) =γ, u(b) =δ, h(t, u(t),u(t))˙ ≤0, t∈A,

where Φ : A×R^q×R^q → J is an interval-valued function and Φ^L,Φ^U : A×R^q ×R^q →R are continuously differentiable function andh:A×R^q×R^q →R^sis continuously differentiables-dimensional function.

Letω denotes the set of all feasible points of the interval-valued variational problem. Then ω={u∈C(A,R^q) :u(a) =γ, u(b) =δ and h(t, u,u)˙ ≤0, t∈A}.

Definition 2.4. Let ¯u∈ωbe a feasible point of the problem (IVP). Then, ¯uis said to be an LU optimal point of (IVP), if there exists no feasible pointu∈ω such that

Z b a

Φ (t, u,u) dt˙ ≺LU

Z b a

Φ (t,u,¯ u) dt.˙¯

3. Optimality conditions

In this section, we give the necessary and sufficient optimality conditions for the considered interval-valued variational problem (IVP). Throughout the paper, all the proofs are presented for thep6= 0 andr6= 0 case of B-(p, r)-invexity. The proofs for the other cases ofB-(p, r)-invexity can be completed in same lines.

Theorem 3.1 (Necessary conditions). Let u¯ be an LU optimal point for the (IVP) and assume that Kuhn–

Tucker constraint qualification is satisfied atu. Then, there exists piecewise smooth functions¯ σ:A→R², σ(t) = σ^L(t), σ^U(t)

≥0, σ6= 0andξ:A→R^s, ξ(t)≥0, such that σ^LΦ^L_u_¯(t,u,¯ u) +˙¯ σ^UΦ^U_u_¯ (t,u,¯ u) +˙¯ ξ^Th_u_¯(t,u,¯ u) =˙¯ d

dt

σ^LΦ^L_u_¯_˙ (t,u,¯ u) +˙¯ σ^UΦ^U_u_¯_˙ (t,u,¯ u) +˙¯ ξ^Thu¯˙ (t,u,¯ u)˙¯ , ξ^Th(t,u,¯ u) = 0, t˙¯ ∈A.

Theorem 3.2 (Sufficiency). Let u¯ be a feasible point for (IVP). Assume that there exists piecewise smooth functions σ : A → R², σ(t) = (σ^L(t), σ^U(t)) ≥ 0, σ 6= 0 and ξ : A → R^s, ξ(t) ≥ 0, such that the following conditions

σ^LΦ^L_u_¯(t,u,¯ u) +˙¯ σ^UΦ^U_u_¯ (t,u,¯ u) +˙¯ ξ^Thu¯(t,u,¯ u) =˙¯ d dt

σ^LΦ^L_u_¯_˙ (t,u,¯ u) +˙¯ σ^UΦ^U_u_¯_˙ (t,u,¯ u) +˙¯ ξ^Thu¯˙(t,u,¯ u)˙¯ , (3.1)

ξ^Th(t,u,¯ u) = 0, t˙¯ ∈A, (3.2)

hold. Further, assume that Φis strictlyB-(p, r)-invex function with respect toη atu¯ onω andξ^ThisB-(p, r)- invex function with respect to sameη atu¯ on ω. Then,u¯ is anLUoptimal point for (IVP).

Proof. Assume that ¯uis not an LU optimal point. Then, there exists another feasible pointusuch that Z b

a

Φ (t, u,u) dt˙ ≺_LU Z b

a

Φ (t,u,¯ u) dt.˙¯

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This means





 Rb

aΦ^L(t, u,u) dt <˙ Rb

Rb

aΦ^U(t, u,u) dt <˙ Rb

aΦ^U(t,u,¯ u) dt˙¯

or 



 Rb

aΦ^L(t, u,u) dt˙ ≤Rb

Rb

aΦ^U(t, u,u) dt <˙ Rb

aΦ^U(t,u,¯ u) dt˙¯

(3.3)

or 



 Rb

aΦ^L(t, u,u) dt <˙ Rb

Rb

aΦ^U(t, u,u) dt˙ ≤Rb

aΦ^U(t,u,¯ u) dt.˙¯

From feasibility ofufor (IVP) with hypothesis (3.2) of Theorem3.2, it implies that Z b

a

ξ^Th(t, u,u) dt˙ ≤ Z b

a

ξ^Th(t,u,¯ u) dt.˙¯ (3.4)

By assumption Φ is strictly B-(p, r)-invex function with respect to η at ¯uon ω. Then, by Definition 2.2, we have

1

rb(u,u)¯

e^r(^R_a^b^Φ^L^(t,u,^{u) dt−}^˙ ^R_a^b^Φ^L(^t,¯^u,^u^¯^˙)^dt)−1

>1 p

Z b a

e^pη(t,u,¯^u)−1T

Φ^L_u_¯(t,u,¯ u)˙¯ − d

dt Φ^L_u_¯_˙ (t,u,¯ u)˙¯

dt, (3.5) and

1

rb(u,u)¯

e^r(^R_a^b^ΦÛ^(t,u,^{u) dt−}^˙ ^R_a^b^ΦÛ(^t,¯û,û^¯^˙)^dt)−1

> 1 p

Z b a

Φ^U_u_¯ (t,u,¯ u)˙¯ − d

dt Φ^U_u_¯_˙ (t,u,¯ u)˙¯

dt.

(3.6) Also,ξ^ThisB-(p, r)-invex function with respect toη at ¯uonω. Thus, following inequality holds

1

rb(u,u)¯

e^r(^R_a^b^ξ^T^h(t,u,^{u) dt−}^˙ ^R_a^b^ξ^T^h(^t,¯^u,^u^¯^˙)^dt)−1

≥ 1 p

Z b a

ξ^Thu¯(t,¯u,u)˙¯ − d

dt ξ^Thu¯˙ (t,u,¯ u)˙¯

dt. (3.7)

Taking into account thatb(u,u)¯ >0 and combining (3.3) and (3.5), (3.3) and (3.6), (3.4) and (3.7), respectively, we obtain

1 p

Z b a

Φ^L_u_¯(t,u,¯ u)˙¯ − d

dt Φ^L_u_¯_˙ (t,u,¯ u)˙¯

dt <0 (3.8)

1 p

Z b a

Φ^U_¯_u(t,u,¯ u)˙¯ − d

dt Φ^U_u_¯_˙ (t,u,¯ u)˙¯

dt <0 (3.9)

1 p

Z b a

(e^pη(t,u,¯^u)−1)^T

ξ^Thu¯(t,u,¯ u)˙¯ − d

dt ξ^Thu¯˙ (t,u,¯ u)˙¯

dt≤0. (3.10)

Multiplying (3.8) byσ^L, (3.9) byσ^U and then adding both sides of resultant inequalities and (3.10), we have 1

p Z b

a

e^pη(t,u,¯^u)−1^T

σ^LΦ^L_u_¯ +σ^UΦ^U_u_¯ +ξ^Thu¯

(t,u,¯ u)˙¯ − d

dt σ^LΦ^L_u_¯_˙ +σ^UΦ^U_u_¯_˙ +ξ^Thu¯˙

(t,u,¯ u)˙¯

dt <0, which is contradiction to (3.1). Hence, ¯uis an LU optimal point of (IVP) and this completes the proof.

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Example 3.3. Let A = [0,1] and C(A,[0,1]) be the space of all piecewise smooth functions u : A → [0,1].

Consider the following interval-valued variational problem:

(P1) min

Z 1 0

Φ (t, u,u) dt˙ = Z 1

0

(u³+e^2u) dt, Z 1

0

u²+e^2u+ 5 dt

subject to

u(0) = 0, u(1) = 1, u−2 ln(u+ 1)≤0, t∈A.

From the formulation of (P1) it is clear that, φ^L(t, u,u) = (u˙ ³+e^2u), φ^U(t, u,u) = (u˙ ²+e^2u + 5) and h(t, u,u) =˙ u−2 ln(u+ 1).

Letω ={u∈C(A,[0,1]) :u(0) = 0, u(1) = 1, u−2 ln(u+ 1)≤0, t∈A}be the feasible set for (P1). Then,

¯

u= 0 is a feasible point for (P1) and the functionφis an interval-valued function.

We check hypothesis (3.1) of Theorem3.2holds at ¯u= 0:

σ^LΦ^L_u_¯(t,u,¯ u) +˙¯ σ^UΦ^U_u_¯(t,u,¯ u) +˙¯ ξ^Th_u_¯(t,u,¯ u)˙¯ − d dt

σ^LΦ^L_¯_u_˙ (t,u,¯ u) +˙¯ σ^UΦ^U_¯_u_˙ (t,u,¯ u) +˙¯ ξ^Thu¯˙(t,u,¯ u)˙¯ = 0.

⇒2σ^L+ 2σ^U =ξ.

From this, letσ^L= 1/4, σ^U = 1/4, ξ= 1.

Next, we check hypothesis (3.2) of Theorem3.2 holds at ¯u= 0:

ξh(t,u,¯ u) =˙¯ ξ(¯u−2 ln(¯u+ 1))

= 0,(¯u= 0).

Letη:A×[0,1]×[0,1]→Randb:C(A,[0,1])×C(A,[0,1])→R+\{0} be defined asη(t, u,u) = 2 sin(u¯ −¯u) andb(u,¯u) =u¯u+ 2, respectively.

Now, we check whether the functionΦ is strictly B-(p, r)-invex and ξh isB-(p, r)-invex atu¯= 0!

Here, computations are presented for functionφ^L: Case (i):p= 1/2, r= 2.

1

rb(u,u)¯

e^r(^R₀¹^Φ^L^(t,u,u) dt−^˙ R₁

0 Φ^L(^t,¯^u,^u^¯^˙)^dt)−1

−1 p

Z 1 0

Φ^L_u_¯(t,u,¯ u)˙¯ − d

dt Φ^L_¯_u_˙ (t,u,¯ u)˙¯

dt,

=1

2(u¯u+ 2)

e²(^R₀¹(û³^+e^2u)^dt−^R₀¹(û^¯³^+e^{2 ¯}û)^dt)−1

− 1 1/2

Z 1 0

e(1/2)(2 sin(u−u))¯ −1

3¯u²+ 2e^¯^u− d dt(0)

dt,

=

e²(^R₀¹(^u³^+e^2u⁻¹)^dt)−1

−4 Z 1

0

e^sin(u)−1 dt >0.

Case (ii):p= 0, r= 2.

1

rb(u,u)¯

e^r(^R₀¹^Φ^L^(t,u,˙u) dt−R₁

0Φ^L(^t,¯^u,˙^u^¯)^dt)−1

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− Z 1

0

(η(t, u,u))¯ ^T

Φ^L_u_¯(t,u,¯ u)˙¯ − d

dt Φ^L_u_¯_˙ (t,u,¯ u)˙¯

dt,

=

e²(^R₀¹(^u³^+e^2u⁻¹)^dt)−1

−4 Z 1

0

sin(u) dt >0.

Case (iii):p= 1/2, r= 0.

b(u,u)¯ Z 1

0

Φ^L(t, u,u) dt˙ − Z 1

0

Φ^L(t,u,¯ u) dt˙¯

−1 p

Z 1 0

e^pη(t,u,¯^u)−1^T

Φ^L_u_¯(t,u,¯ u)˙¯ − d

dt Φ^L_u_¯_˙ (t,u,¯ u)˙¯

dt,

= 2 Z 1

0

u³+e^2u−1 dt−4

Z 1 0

e^sin(u)−1 dt >0.

Case (iv):p= 0, r= 0.

b(u,u)¯ Z 1

0

Φ^L(t, u,u) dt˙ − Z 1

0

Φ^L(t,u,¯ u) dt˙¯

− Z 1

0

(η(t, u,u))¯ ^T

Φ^L_u_¯(t,u,¯ u)˙¯ − d

dt Φ^L_u_¯_˙ (t,u,¯ u)˙¯

dt,

= 2 Z 1

0

u³+e^2u−1 dt−4

Z 1 0

sin(u) dt >0.

It is clear from cases (i) to (iv) that functionφ^Lsatisfy condition of strictB-(1/2,2)-invexity with respect toη at ¯u= 0. Now, to show that functionφis strictB-(1/2,2)-invex function, we have to show thatφÛ also satisfies the conditions of Definition 2.2. For this, we have listed the calculations of all four cases for function φÛ in Table1. Then, from calculations ofφ^L andφÛ, it is clear that functionφis strictB-(1/2,2)-invex function with respect toη at ¯u= 0.

Again, all four cases forξh have been listed in Table1 and it can be seen that ξhis B-(1/2,2)-invex with respect toη at ¯u= 0.

Next, we prove that ¯u= 0 is an LU optimal point for (P1). If possible, suppose ¯u= 0 is not an LU optimal point for (P1), then there exists another feasible point v∈ωsuch that

Z 1 0

Φ (t, v,v) dt˙ ≺LU

Z 1 0

Φ (t,u,¯ u) dt.˙¯

Then, at ¯u= 0 this means

(R1

0 Φ^L(t, v,v) dt <˙ 1 R1

0 Φ^U(t, v,v) dt <˙ 6 or

(R1

0 Φ^L(t, v,v) dt˙ ≤1 R1

0 Φ^U(t, v,v) dt <˙ 6 or

(R1

0 Φ^L(t, v,v) dt <˙ 1 R1

0 Φ^U(t, v,v) dt˙ ≤6.

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Table 1. Resultant inequalities for functionsφ^U andξh.

p= 1/2, r= 2

e²

R₁

0(^u²^+e^2u⁻¹)^dt−1

−4 Z ₁

0

(e^sin(u)−1) dt >0

p= 0, r= 2

e²

R₁

0(^u²^+e^2u⁻¹)^dt−1

−4 Z ₁

0

sin(u) dt >0

φ^U

p= 1/2, r= 0 2 Z ₁

0

u²+e^2u−1 dt−4

Z ₁

0

(e^sin(u)−1) dt >0

p= 0, r= 0 2 Z ₁

0

u²+e^2u−1 dt−4

Z ₁

0

sin(u) dt >0 p= 1/2, r= 2

e²

R₁

0(u−2 ln(u+1)) dt

−1

+ 2 Z ₁

0

(e^sin(u)−1) dt≥0

p= 0, r= 2

e²

R₁

0(u−2 ln(u+1)) dt

−1

+ 2 Z ₁

0

sin(u) dt≥0

ξh

p= 1/2, r= 0 2 Z ₁

0

(u−2 ln(u+ 1)) dt+ 2 Z ₁

0

(e^sin(u)−1) dt≥0

p= 0, r= 0 2 Z ₁

0

(u−2 ln(u+ 1)) dt+ 2 Z ₁

0

sin(u) dt≥0

But there does not exists any other feasible pointvsuch that Z 1

0

Φ (t, v,v) dt˙ ≺LU

Z 1 0

Φ (t,u,¯ u) dt.˙¯

Hence, ¯u= 0 is an LU optimal point for (P1) and thus sufficiency theorem is verified.

4. Mond–Weir type duality

We present the Mond–Weir type dual for the considered (IVP) as follows:

(IMD) max

Z b a

Φ (t, v,v) dt˙ =

"

Z b a

Φ^L(t, v,v) dt,˙ Z b

a

Φ^U(t, v,v) dt˙

#

subject to

v(a) =γ, v(b) =δ,

σ^LΦ^L_v(t, v,v) +˙ σ^UΦ^U_v (t, v,v) +˙ ξ^Th_v(t, v,v)˙

= d dt

σ^LΦ^L_v_˙ (t, v,v) +˙ σ^UΦ^U_v_˙ (t, v,v) +˙ ξ^Thv˙(t, v,v)˙ , (4.1) Z b

a

ξ^Th(t, v,v)˙ ≥0, t∈A, (4.2)

σ= (σ^L, σ^U)≥0, σ6= 0, ξ≥0.

LetωMDdenotes the set of all feasible points of (IMD). Then,ωMD={(v, σ, ξ) :v∈C(A,R^q), σ∈R², ξ∈R^s, satisfying the constraints of (IMD),∀t∈A}.

We denote the setW ={v∈C(A,R^q) : (v, σ, ξ)∈ω_MD}and Ω =ω∪W.

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Definition 4.1. A feasible point (¯v,σ,¯ ξ) is said to be an LU optimal point of a maximum type for (IMD), if¯ there exists no feasible point (v, σ, ξ) such that

Z b a

Φ (t,v,¯ v) dt˙¯ ≺LU

Z b a

Φ (t, v,v) dt.˙

Theorem 4.2 (Weak duality). Let u and (v, σ, ξ) be any feasible points for (IVP) and (IMD), respectively.

Assume thatσ= (σ^L, σ^U)>0,Φandξ^ThareB-(p, r)-invex functions with respect to sameη atvonΩ. Then, the following can not hold

Z b a

Φ (t, u,u) dt˙ ≺_LU Z b

a

Φ (t, v,v) dt.˙ Proof. Suppose that the following hold

Z b a

Φ (t, u,u) dt˙ ≺LU

Z b a

Φ (t, v,v) dt.˙ Then, this implies





 Rb

aΦ^L(t, v,v) dt <˙ 0 Rb

aΦ^U(t, v,v) dt <˙ 0

or 



 Rb

aΦ^L(t, v,v) dt˙ ≤0 Rb

aΦ^U(t, v,v) dt <˙ 0

or 



 Rb

aΦ^L(t, v,v) dt <˙ 0 Rb

aΦ^U(t, v,v) dt˙ ≤0.

The above inequalities are further equivalent to







e^r(^R_a^b^Φ^L^(t,u,u) dt−^˙ R_b

aΦ^L(t,v,˙v) dt)−1

<0

e^r(^R_a^b^Φ^U^(t,u,˙^{u) dt−}^R_a^b^Φ^U^(t,v,^{v) dt}^˙ )−1

<0 or







e^r(^R_a^b^Φ^L^(t,u,^{u) dt−}^˙ ^R_a^b^Φ^L^(t,v,˙^{v) dt})−1

≤0

e^r(^R_a^b^Φ^U^(t,u,˙u) dt−R_b

aΦ^U(t,v,v) dt˙ )−1

<0 or







aΦ^L(t,v,v) dt˙ )−1

<0

e^r(^R_a^b^Φ^U^(t,u,˙^{u) dt−}^R_a^b^Φ^U^(t,v,^{v) dt}^˙ )−1

≤0.

Since, (σ^L, σ^U)>0,the above inequalities implies σ^L

e^r(^R_a^b^Φ^L^(t,u,^{u) dt−}^˙ ^R_a^b^Φ^L^(t,v,^{v) dt}^˙ )−1 +σ^U

e^r(^R_a^b^Φ^U^(t,u,^{u) dt−}^˙ ^R_a^b^Φ^U^(t,v,^{v) dt}^˙ )−1

<0. (4.3) From the feasibility ofufor (IVP) and from the feasibility of (v, σ, ξ) for (IMD), it follows that

Z b a

ξ^Th(t, u,u) dt˙ ≤ Z b

a

ξ^Th(t, v,v) dt.˙

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Then, the above inequality implies that

e^r(^R_a^b^ξ^T^h(t,u,^{u) dt−}^˙ ^R_a^b^ξ^T^h(t,v,˙^{v) dt})−1

≤0. (4.4)

On adding both sides of (4.3) and (4.4), we get σ^L

e^r(^R_a^b^Φ^L^(t,u,^{u) dt−}^˙ ^R_a^b^Φ^L^(t,v,^{v) dt}^˙ )−1 +

e^r(^R_a^b^ξ^T^h(t,u,^{u) dt−}^˙ ^R_a^b^ξ^T^h(t,v,^{v) dt}^˙ )−1

<0. (4.5) From assumption that Φ isB-(p, r)-invex function with respect toηat v on Ω. Then, we have the following

1

rb(u, v)

aΦ^L(t,v,v) dt˙ )−1

≥1 p

Z b a

e^pη(t,u,v)−1^T

Φ^L_v (t, v,v)˙ − d

dt Φ^L_v_˙ (t, v,v)˙

dt (4.6) and

1

rb(u, v)

e^r(^R_a^b^Φ^U^(t,u,u) dt−^˙ R_b

aΦ^U(t,v,˙v) dt)−1

≥ 1 p

Z b a

e^pη(t,u,v)−1^T

Φ^U_v (t, v,v)˙ − d

dt Φ^U_v_˙ (t, v,v)˙

dt.

(4.7) Also,ξ^ThisB-(p, r)-invex function with respect to same η atv on Ω, which implies

1

rb(u, v)

e^r(^R_a^b^ξ^T^h(t,u,^{u) dt−}^˙ ^R_a^b^ξ^T^h(t,v,˙^{v) dt})−1

≥ 1 p

Z b a

e^pη(t,u,v)−1^T

ξ^Th_v(t, v,v)˙ − d

dt ξ^Th_v_˙(t, v,v)˙

dt.

(4.8) Multiplying (4.6) and (4.7) byσ^L andσ^U and adding resultant inequalities, we have

1

rb(u, v)h σ^L

e^r(^R_a^b^Φ^L^(t,u,˙^{u) dt−}^R_a^b^Φ^L^(t,v,^{v) dt}^˙ )−1 +σ^U

e^r(^R_a^b^Φ^L^(t,u,˙^{u) dt−}^R_a^b^Φ^L^(t,v,^{v) dt}^˙ )−1i

≥ 1 p

Z b a

e^pη(t,u,v)−1T

σ^LΦ^L_v +σ^UΦ^U_v

(t, v,v)˙ − d

dt σ^LΦ^L_v_˙ +σ^UΦ^U_v_˙

(t, v,v)˙

dt. (4.9) Adding both sides of (4.8) and (4.9) and from the feasibility of (IMD), we have

1

rb(u, v)h σ^L

e^r(^R_a^b^Φ^L^(t,u,^{u) dt−}^˙ ^R_a^b^Φ^L^(t,v,^{v) dt}^˙ )−1 +

e^r(^R_a^b^ξ^T^h(t,u,u) dt−^˙ R_b

aξ^Th(t,v,v) dt˙ )−1i

≥0, (4.10)

which is contradiction to (4.5).

Hence,Rb

aΦ (t, u,u) dt˙ ≺LURb

aΦ (t, v,v) dt˙ can not hold.

Example 4.3. Let A = [0,1] and C(A,[0,1]) be the space of all piecewise smooth functions u : A → [0,1].

Consider the following interval-valued variational problem:

(P2) min

Z 1 0

Φ (t, u,u) dt˙ = Z 1

0

2u²+ ln(u+ 1) dt,

Z 1 0

2u²+e^u dt

subject to

u(0) = 0, u(1) = 1,

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u²−u≤0, t∈A.

Letω={u∈C(A,[0,1]) :u(0) = 0, u(1) = 1 andu²−u≤0, t∈A}be feasible set for (P2). From the formulation of (P2) and definition of interval-valued function, it is clear that,φ^L(t, u,u) = 2u˙ ²+ ln(u+ 1), φ^U(t, u,u) =˙ 2u²+e^u andh(t, u,u) =˙ u²−u.

The corresponding Mond–Weir-dual is

(MD) max

Z 1 0

Φ (t, v,v) dt˙ = Z 1

0

2v²+ ln(v+ 1) dt,

Z 1 0

2v²+e^v dt

subject to

v(0) = 0, v(1) = 1, v²−v≤0, t∈A, σ^L

4v+ 1 v+ 1

+σ^U(4v+e^v) +ξ(2v−1) = 0.

Atv= 0,

σ^L+σ^U−ξ= 0,

=⇒ σ^L= 1/4, σ^U = 3/4, ξ= 1, Z 1

0

ξ(t)(v²−v) dt≥0, t∈A.

From the formation of Mond–Weir-dual it is clear that,φ^L(t, v,v) = 2v˙ ²+ ln(v+ 1), φ^U(t, v,v) = 2v˙ ²+e^v andh(t, v,v) =˙ v²−v. LetωMDbe the feasible set of (MD). ωMD={(v, σ, ξ) :v ∈C(A,[0,1]), σ∈R², ξ∈R, satisfying the constraints of (MD),t ∈ A}. Then, we denote the set W ={v ∈ C(A,[0,1]) : (v, σ, ξ)∈ ωMD} and Ω =ω∪W.

We observe that (v(t), σ(t), ξ(t)) = (0,(1/4,3/4),1) is a feasible point for (MD) and objective value at this feasible point is [0,1]. Also, u(t) = t is a feasible point for (P2) and the corresponding objective value is [1.05,2.38].

Letη:A×[0,1]×[0,1]→Randb:C(A,[0,1])×C(A,[0,1])→R+\{0}be defined asη(t, u, v) = ln(1 +u−v) andb(u, v) =uv+ 1, respectively.

Next, we show thatφandξh areB-(p, r)-invex function at v= 0 with respect to above definedη.

We check for functionφ^L as follows:

Case (i):p= 1, r= 1/3.

1

rb(u, v)

e^r(^R₀¹^Φ^L^(t,u,u) dt−^˙ R₁

0Φ^L(t,v,v) dt˙ )−1

−1 p

Z 1 0

e^pη(t,u,v)−1^T

Φ^L_v(t, v,v)˙ − d

dt,

= 1

1/3(uv+ 1)

e^(1/3)(^R₀¹^(2u²+ln(u+1)) dt−R₁

0(^2v²^+ln(v+1))^dt)−1

− Z 1

0

e^ln(1+u−v)−1T

4v+ 1 v+ 1 − d

dt(0)

dt,

= 3

e^(1/3)(^R₀¹^(2u²+ln(u+1)) dt)−1

− Z 1

0

udt≥0.

Case (ii):p= 0, r= 1/3.

1

rb(u, v)

e^r(^R₀¹^Φ^L^(t,u,u) dt−^˙ R₁

0Φ^L(t,v,v) dt˙ )−1

− Z 1

0

(η(t, u, v))^T

Φ^L_v(t, v,v)˙ − d

dt,

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Table 2.Resultant inequalities for functionsφ^U andξh

p= 1, r= 1/3 3

e¹³

R₁

0(^2u²^+e^u⁻¹)^dt−1

− Z ₁

0

udt≥0

p= 0, r= 1/3 3

e¹³

R₁

0(^2u²^+e^u⁻¹)−1

− Z ₁

0

ln(u+ 1) dt≥0

φ^U

p= 1, r= 0

Z ₁

0

2u²+e^u−1 dt−

Z ₁

0

udt≥0

p= 0, r= 0

Z ₁

0

2u²+e^u−1

− Z₁

0

ln(u+ 1) dt≥0 p= 1, r= 1/3 3

e¹³

R₁

0(^u²^−u)^dt−1

+ Z ₁

0

udt≥0

p= 0, r= 1/3 3

e¹³ Z ₁

0

u²−u dt−1

+

Z₁

0

ln(u+ 1) dt≥0

ξh

p= 1, r= 0

Z ₁

0

u²dt≥0

p= 0, r= 0

Z ₁

0

u²−u dt+

Z ₁

0

(ln(u+ 1)) dt≥0

= 3

e^(1/3)(^R₀¹^(2u²+ln(u+1)) dt)−1

− Z 1

0

ln(u+ 1) dt≥0.

Case (iii):p= 1, r= 0.

b(u, v) Z 1

0

Φ^L(t, u, v) dt− Z 1

0

Φ^L(t, v,v) dt˙

−1 p

Z 1 0

e^pη(t,u,v)−1T

Φ^L_v (t, v,v)˙ − d

dt,

= Z 1

0

2u²+ ln(u+ 1) dt−

Z 1 0

udt≥0.

Case(iv):p= 0, r= 0.

b(u, v) Z 1

0

Φ^L(t, u,u) dt˙ − Z 1

0

Φ^L(t, v,v) dt˙

− Z 1

0

(η(t, u, v))^T

Φ^L_v (t, v,v)˙ − d

dt,

= Z 1

0

2u²dt≥0.

It is clear from cases (i) to (iv) that functionφ^L satisfies the conditions of Definition2.1. Next, to show that φ isB-(1,1/3)-invex funciton,φ^L andφ^U must be B-(1,1/3)-invex function.We have listed the calulations of all four cases for functionφ^U andξhin Table2. It implies from Table2thatφisB-(1,1/3)-invex function with respect toη atv= 0. Again from Table2, it can be seen thatξhisB-(1,1/3)-invex with respect toη atv= 0.

Moreover, u(t) = t and (v(t), σ(t), ξ(t)) = (0,(1/4,3/4),1) are the feasible points for (P2) and (MD), then we observe that

Z 1 0

Φ (t, u,u) dt˙ ⊀ Z 1

0

Φ (t, v,v) dt.˙ Hence, the weak duality theorem is verified.