On interval valued optimization problems with generalized invex functions

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R E S E A R C H

Open Access

On interval-valued optimization problems

with generalized invex functions

Izhar Ahmad

1,2*

_{, Anurag Jayswal}

3

_{and Jonaki Banerjee}

3

*_{Correspondence:} [email protected]

1_{Department of Mathematics and} Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia

2_{Permanent address: Department of} Mathematics, Aligarh Muslim University, Aligarh, 202 002, India Full list of author information is available at the end of the article

Abstract

This paper is devoted to study interval-valued optimization problems. Suﬃcient optimality conditions are established for LU optimal solution concept under generalized (p,r) –

ρ

– (

η

,

θ

)-invexity. Weak, strong and strict converse duality theorems for Wolfe and Mond-Weir type duals are derived in order to relate the LU optimal solutions of primal and dual problems.

MSC: 90C46; 90C26; 90C30

Keywords: nonlinear programming; interval-valued functions; (p,r) –

ρ

– (

η

,

θ

)-invexity; LU-optimal; suﬃciency; duality

1 Introduction

Many real world decision-making problems need to accomplish many objectives:

mini-mize cost, maximini-mize reliability, minimini-mize deviations from desire levels, minimini-mize risk,etc.

In these cases optimization problems have a large number of applications. The main goal of single objective optimization is to ﬁnd the best solution which corresponds to the min-imum or maxmin-imum value of a single objective function.

In many research fields and real world problems, the methodology for solving optimiza-tion problems has been used. There are three kinds of methodology that are used for solving optimization problems, namely deterministic optimization problem, stochastic optimization problem and interval-valued optimization problem. The optimization prob-lems with interval coefficients are termed an interval-valued optimization problem. In this problem, the coefficient is taken as closed intervals. The solution concept imposed upon the objective function is the main difference between the above said three kinds of problems.

Interval programming methods have been used to tackle speciﬁc issues in multiple ob-jective linear programming: some deal with uncertainty in the obob-jective functions, others handle uncertainty both in the objective functions and in the RHS of the constraints and

others deal with uncertainty in all the coeﬃcients of the model. Charneset al. []

pro-posed an idea for solving the linear programming problems in which the constraints were assumed as closed intervals. Later, Steuer [] developed an algorithm to solve the linear programming problem with interval objective functions.

Urli and Nadeau [] presented a process to solve the multi-objective linear program-ming problems with interval coeﬃcients. Chanas and Kuchta [] generalized the solution concepts of the linear programming problem with interval coeﬃcients in the objective function based on preference relations between intervals. By imposing a partial ordering

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on the set of all closed intervals, Wu [] introduced a solution concept in optimization problems with interval-valued objective functions.

There are several approaches to model uncertainty in optimization problems such as stochastic optimization and fuzzy optimization. Here we consider an optimization prob-lem with interval-valued objective function. Stancu-Minasian and Tigan [, ] investigated this kind of optimization problem.

Wu [] formulated Karush-Kuhn-Tucker optimality conditions for an interval-valued objective function. Later, Wu [, ] formulated a Wolfe-type dual problem related to the interval-valued optimization problem and established duality theorems by using the concept of nondominated solution employed in vector optimization problems. Zhou and Wang [] established a suﬃcient optimality condition and discussed mixed-type dual-ity for a class of nonlinear interval-valued optimization problems. Recently, Bhurjee and Panda [] developed a methodology to study the existence of the solutions of an interval

optimization problem. Very recently, Zhanget al.[] discussed the optimality conditions

and duality results for interval-valued optimization problems under generalized preinvex-ity.

Convexity plays an important role in proving the existence of a solution of a general optimization problem. Hence there is a need to study the convex property of interval op-timization problems. One of the most useful generalizations of convexity was introduced by Hanson []. For details, readers are advised to see []. After the concept ofρ– (η,θ

)-invex function and (p,r)-invex function had been introduced by Zalmai [] and Antczak

[], respectively, Mandal and Nahak[] proposed a new concept of (p,r) –ρ– (η,θ

)-invex function and established some symmetric duality results. Recently, Jayswal et al.

[] derived suﬃcient optimality conditions and duality theorems for interval-valued op-timization problems involving generalized convex functions.

In this paper, we consider an interval-valued optimization problem in which the ob-jective function is an interval-valued function and the constraint functions are real-valued, and derive sufficient optimality conditions and duality theorems under general-ized (p,r) –ρ– (η,θ)-invexity. The organization of the paper is as follows. In Section , we recall some definitions and some basic properties related to interval-valued optimization problems. In Section , some sufficient optimality conditions for a class of interval-valued programming problems are discussed. Wolfe and Mond-Weir type duality theorems are obtained in Sections  and , respectively. Conclusion and future work are proposed in Section .

2 Notation and preliminaries

LetIbe a class of all closed and bounded intervals inR. Throughout this paper, when we

say thatAis a closed interval, we mean thatAis also bounded inR. IfAis a closed interval,

we use the notation A= [aL_,_aU_{], where}_aL _and_aU _{mean the lower and upper bounds}

ofA, respectively. If aL₌_aU ₌_a_{, then}_A_{= [}_a_,_a_{] =}_a_{is a real number. Let}_A_{= [}_aL_,_aU_], B= [bL_,_bU_]_∈_I_{, we deﬁne}

(i) A+B={a+b:a∈Aandb∈B}= [aL₊_bL_,_aU₊_bU_]_,

(ii) –A={–a:a∈A}= [–aU_{, –}_aL_]_.

Therefore we see thatA–B=A+ (–B) = [aL_–_bU_,_aU_–_bL_{]. We also have}

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(ii) kA={ka:a∈A}=

[kaL,kaU] ifk≥,

[kaU_,_kaL_] _if_k_{< ,} wherekis a real number.

LetRn_{denote an}_n_{-dimensional Euclidean space. The function}_F_:_Rn_→_I_{is called an}

interval-valued function,i.e.,F(x) =F(x,x, . . . ,xn) is a closed interval inRfor eachx∈Rn_. The interval-valued functionFcan also be written asF(x) = [FL₍_x_),_FU₍_x_{)], where}_FL₍_x_), FU₍_x_{) are real-valued functions deﬁned on}_Rn_{and satisfy the condition}_FL₍_x₎_≤_FU₍_x_{) for} eachx∈Rn_{. We note that [}_F₍_x_)]L₌_FL₍_x_{) and [}_F₍_x_)]U₌_FU₍_x_).

In interval mathematics, an order relation is often used to rank interval numbers and it implies that an interval number is better than another but not that one is larger than another. ForA= [aL_,_aU_{] and} _B_{= [}_bL_,_bU_{], we write}_A_≤

LUBif and only ifaL≤bLand aU_≤_bU_{. It is easy to see that}_≤

LUis a partial ordering onI. Also, we can writeA<LUBif and only ifA≤LUBandA=B.

Equivalently,A<LUBif and only if

aL<bL, aU≤bU or

aL≤bL, aU<bU or

aL<bL, aU<bU.

Deﬁnition .[] Letf :Rn_→_R_{be a diﬀerentiable function and let}_p_,_r_{be arbitrary} real numbers. If there existη:Rn_×_Rn_→_Rn_,_θ _:_Rn_×_Rn_→_Rn_and_ρ_∈_R_{such that the} relations

 r

er(f(x)–f(y))– (>)≥  p∇f(y)

epη(x,y)–+ρθ(x,y) forp= ,r= , 

r

er(f(x)–f(y))– (>)≥ ∇f(y)η(x,y) +ρθ(x,y) forp= ,r= , f(x) –f(y)(>)≥ 

p∇f(y)

epη(x,y)_–_₊_ρ_θ₍_x_,_y₎

forp= ,r= ,

f(x) –f(y)(>)≥ ∇f(y)η(x,y) +ρθ(x,y) forp= ,r= 

hold, thenf is said to be (strictly) (p,r) –ρ– (η,θ)-invex at the pointyonRn_{with respect} toη,θ.

Deﬁnition . Letf :Rn_→_R_{be a diﬀerentiable function and let}_p_,_r_{be arbitrary real} numbers. If there existη:Rn_×_Rn_→_Rn_,_θ_:_Rn_×_Rn_→_Rn_and_ρ_∈_R_{such that the relations}

 p∇f(y)

epη(x,y)–+ρθ(x,y)≥⇒ r

er(f(x)–f(y))– (>)≥ forp= ,r= ,

∇f(y)η(x,y) +ρθ(x,y)≥⇒ r

er(f(x)–f(y))– (>)≥ forp= ,r= ,

 p∇f(y)

epη(x,y)_–_₊_ρ_θ₍_x_,_y₎

≥⇒f(x) –f(y)(>)≥ forp= ,r= ,

∇f(y)η(x,y) +ρθ(x,y)≥⇒f(x) –f(y)(>)≥ forp= ,r= 

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Deﬁnition . Letf :Rn_→_R_{be a diﬀerentiable function and let}_p_,_r_{be arbitrary real} numbers. If there existη:Rn×Rn→Rn,θ:Rn×Rn→Rnandρ∈Rsuch that the relations

 r

er(f(x)–f(y))– ≤⇒ p∇f(y)

epη(x,y)–≤–ρθ(x,y) forp= ,r= , 

r

er(f(x)–f(y))– ≤⇒ ∇f(y)η(x,y)≤–ρθ(x,y) forp= ,r= , f(x) –f(y)≤⇒ 

p∇f(y)

epη(x,y)_–__≤_–_ρ_θ₍_x_,_y₎

forp= ,r= ,

f(x) –f(y)≤⇒ ∇f(y)η(x,y)≤–ρθ(x,y) forp= ,r= 

hold, thenf is said to be (p,r) –ρ– (η,θ)-quasi-invex at the pointyonRnwith respect to

η,θ.

Remark . It should be noted that the exponentials appearing on the right-hand sides of inequalities above are understood to be taken componentwise and= (, , . . . , )∈Rn.

Remark . All theorems in this paper will be proved only in the case whenp= ,r=  (other cases can be dealt with likewise since the only changes arise in a form of inequality). Moreover, without loss of generality, we shall assume thatr> ,p>  (in the case when r< ,p< , the direction of some of the inequalities in the proof of the theorems should be changed to the opposite one).

In this paper, we consider the following primal optimization problem with interval-valued objective function:

(IVP) minF(x) =FL₍_x_),_FU₍_x₎

subject togj(x)≤, j= , , . . . ,m,

whereF:Rn_→_I_{is an interval-valued function and}_g

j:Rn→Ris a real-valued function. Let={x∈Rn_:_g

j(x)≤,j= , , . . . ,m}be the set of all feasible solutions of (IVP). We also denote byObj(F,) ={F(x) :x∈}the set of all objective values of primal problem (IVP).

Deﬁnition .[] Letx*_{be a feasible solution of the primal problem (IVP). We say that} x*_{is a LU optimal solution of problem (IVP) if there exists no}_x_∈_{such that}_F₍_x_{) <}

LU F(x*_).

Theorem .(Karush-Kuhn-Tucker type conditions []) Assume that x*_{is a LU optimal} solution of primal problem (IVP)and F,gj, j= , , . . . ,m,are differentiable at x*_. Sup-pose that the constraint function gj,j= , , . . . ,m,satisfies the suitable Kuhn-Tucker con-straint qualification[]at x*.Then there exist multipliers <ξL,ξU∈R and≤μj∈R, j= , , . . . ,m,such that

ξL∇FLx*+ξU∇FUx*+ m

j=

μj∇gj

x*= , ()

μjgj

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3 Sufﬁcient optimality conditions

In this section, we shall establish the following suﬃcient optimality conditions for (IVP).

Theorem .(Suﬃciency) Let x*∈ be a feasible solution of (IVP).Suppose that the objective function F and the constraint function gj,j= , , . . . ,m,are diﬀerentiable at x*_. Assume that FL_{and F}U_are₍_p_,_r_{) –}_ρ

– (η,θ)-invex and(p,r) –ρ– (η,θ)-invex, respec-tively,with respect toη,θ and m_j=μjgjis(p,r) –ρ– (η,θ)-invex with respect toη,θ at x* _with₍_ξL_ρ

+ξUρ+ρ)≥.If there exist(Lagrange)multipliers  <ξL,ξU∈R and

μ= (μ,μ, . . . ,μm), ≤μj∈R,j= , , . . . ,m,such that(x*,ξL,ξU,μ)satisﬁes()and(), then x*_{is a LU optimal solution to problem}_(IVP).

Proof Letx*_{be not a LU optimal solution of (IVP). Then there exists a feasible solution}

x∈such that

F(x) <LUF

x*.

That is,

FL₍_x_{) <}_FL₍_x*_), FU₍_x

) <FU(x*), or

FL₍_x₎_≤_FL₍_x*_), FU₍_x

) <FU(x*), or

FL₍_x_{) <}_FL₍_x*_), FU₍_x

)≤FU(x*).

Sincer> , using the property of an exponential function, we obtain

 r[er{F

L_(x

)–FL(x*)}_{– ] < ,}

 r[er{F

U_(x

)–FU(x*)}_{– ] < ,} or

 r[er{F

L_(x

)–FL(x*)}_{– ]}≤_,

 r[er{F

U_(x

)–FU(x*)}_{– ] < ,} or

 r[er{F

L_(x

)–FL(x*)}_{– ] < ,}

 r[er{F

U_(x

)–FU(x*)}_{– ]}≤_.

Using the above inequalities and the (p,r) –ρ– (η,θ)-invexity ofFLand the (p,r) –ρ– (η,θ)-invexity ofFU_at_x*_{, we get}

 p{∇F

L₍_x*₎_}₍_epη(x,x*)_–__{) +}_ρ

 θ(x,x*) < , 

p{∇FU(x*)}(ep

η(x,x*)_–__{) +}_ρ

 θ(x,x*) < , or

 p{∇F

L₍_x*₎_}₍_epη(x,x*)_–__{) +}_ρ

 θ(x,x*) ≤, 

p{∇FU(x*)}(ep

η(x,x*)_–__{) +}_ρ

 θ(x,x*) < , or



p{∇FL(x*)}(ep

η(x,x*)_–__{) +}_ρ

 θ(x,x*) < , 

p{∇FU(x*)}(ep

η(x,x*)_–__{) +}_ρ

 θ(x,x*) ≤.

SinceξL_{>  and}_ξU_{> , from the above inequalities, we get} 

p

ξL∇FLx*+ξU∇FUx*epη(x,x*)_–_

+ξLρθ

x,x* 

+ξUρθ

x,x* 

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On the other hand, from the feasibility ofxto (IVP), we have

gj(x)≤, j= , , . . . ,m.

Sinceμj≥,j= , , . . . ,m, the above inequality together with () yields m

j=

μjgj(x)≤ m

j=

μjgj

x*.

Asr> , using the property of an exponential function, we get

 r

er{ mj=μjgj(x)– jm=μjgj(x*)}_{– }≤_,

which together with the assumption that m_j=μjgjis (p,r) –ρ– (η,θ)-invex atx*gives

 p

m

j=

μj∇gj

x*epη(x,x*)_–_₊_ρ

θ

x,x*≤. ()

On adding () and (), we get

 p

epη(x,x*)_–_

j=

μj∇gj

x*

+ξLρ+ξUρ+ρθ

x,x* 

< .

Therefore, from the hypothesis that (ξL_ρ

+ξUρ+ρ)≥ and the above inequality, we get

 p

epη(x,x*)_–_

j=

μj∇gj

x*

< ,

which contradicts (). Thereforex*_{is a LU optimal solution of (IVP). This completes the}

proof.

Theorem . (Suﬃciency) Let x*_∈_{be a feasible solution of} _(IVP)._{Suppose that the} objective function F and the constraint function gj,j= , , . . . ,m,are diﬀerentiable at x*. Assume that (ξLFL+ξUFU)is(p,r) –ρ– (η,θ)-pseudo-invex with respect toη, θ and

m

j=μjgj is(p,r) –ρ– (η,θ)-quasi-invex with respect toη, θ at x*with (ρ+ρ)≥. If there exist(Lagrange)multipliers <ξL_,_ξU_∈_{R and} _μ_{= (}_μ

,μ, . . . ,μm), ≤μj∈R, j= , , . . . ,m,such that(x*_,_ξL_,_ξU_,_μ₎_satisﬁes_()_and_(),_{then x}*_{is a LU optimal solution} to problem(IVP).

Proof Letx*_{be not a LU optimal solution of (IVP). Then there exists a feasible solution}

x∈such that

F(x) <LUF

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That is,

FL₍_x

) <FL(x*), FU(x) <FU(x*), or

FL₍_x

)≤FL(x*), FU(x) <FU(x*), or

FL₍_x

) <FL(x*), FU(x)≤FU(x*).

SinceξL_{>  and}_ξU_{> , from the above inequalities, we have}

ξLFL(x) +ξUFU(x) <ξLFL

x*+ξUFUx*.

 r

er{(ξLFL_(x

)+ξUFU_(x

))–(ξLFL_(x*₎₊_ξU_FU_(x*_))}

– < ,

which together with the assumption thatξL_FL₊_ξU_FU_{is (}_p_,_r_{) –}_ρ

– (η,θ)-pseudo-invex atx*_gives

 p

ξL∇FLx*+ξU∇FUx*epη(x,x*)_–_₊_ρ

θ

x,x*< . ()

On the other hand, from the feasibility ofxto (IVP), we have

gj(x)≤, j= , , . . . ,m.

Sinceμj≥,j= , , . . . ,m, the above inequality together with () yields m

j=

μjgj(x)≤ m

j=

μjgjx*.

 r

er{ mj=μjgj(x)– jm=μjgj(x*)}_{– }≤_,

which together with the assumption that m_j=μjgjis (p,r) –ρ– (η,θ)-quasi-invex atx* gives

 p

m

j=

μj∇gj

x*epη(x,x*)_–_₊_ρ

θ

x,x* 

≤. ()

On adding () and (), we get

 p

epη(x,x*)_–_

j=

μj∇gj

x*

+ (ρ+ρ)θ

x,x*< .

Since (ρ+ρ)≥, we have 

p

epη(x,x*)_–_

j=

μj∇gj

x*

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which contradicts (). Thereforex*_{is a LU optimal solution of (IVP). This completes the}

proof.

4 Wolfe-type duality

In this section, we consider the following Wolfe-type dual problem:

(WD) maxF(y) +

m

j=

μjgj(y)

subject to

ξL∇FL(y) +ξU∇FU(y) + m

j=

μj∇gj(y) = , ()

ξL> ,ξU> ,μj≥,j= , , . . . ,m, ()

whereF(y) + m_j=μjgj(y) = [FL(y) + _j=m μjgj(y),FU(y) + m_j=μjgj(y)] is an interval-valued function.

Deﬁnition . Let (y*_,_ξ*L_,_ξ*U_,_μ*_{) be a feasible solution of dual problem (WD). We say} that (y*_,_ξ*L_,_ξ*U_,_μ*_{) is a LU optimal solution of dual problem (WD) if there exists no} (y,ξ*L,ξ*U,μ*) such thatF(y*) + m_j=μ*_jgj(y*) <LUF(y) + mj=μ*jgj(y).

Theorem .(Weak duality) Let Xbe an open subset of Rn_._{Let F and gj}_,_j_{= , , . . . ,}_m_,_be diﬀerentiable on X.Suppose that x and(y,ξL_,_ξU_,_μ₎_{are the feasible solutions to}_(IVP)_and (WD),respectively.Further assume thatξL> ,ξU> andμj≥,j= , , . . . ,m,such that

ξLFL+ξUFU+ m_j=μjgjis(p,r) –ρ– (η,θ)-invex with respect toη,θ at y withξL+ξU=  andρ≥.Then

F(x)≥LUF(y) + m

j=

μjgj(y).

Proof Suppose contrary to the result that

F(x) <LUF(y) + m

j=

μjgj(y).

That is,

FL₍_x_{) <}_FL₍_y_{) +} m

j=μjgj(y), FU₍_x_{) <}_FU₍_y_{) +} m

j=μjgj(y), or

FL₍_x₎_≤_FL₍_y_{) +} m

j=μjgj(y), FU₍_x_{) <}_FU₍_y_{) +} m

j=μjgj(y), or

FL(x) <FL(y) + m_j=μjgj(y), FU₍_x₎_≤_FU₍_y_{) +} m

j=μjgj(y).

SinceξL> ,ξU>  andξL+ξU= , the above inequalities together with the feasibility

ofxto (IVP) become

ξLFL(x) +ξUFU(x) + m

j=

μjgj(x) <ξLFL(y) +ξUFU(y) + m

j=

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From the assumption thatξL_FL₊_ξU_FU₊ m

j=μjgjis (p,r) –ρ– (η,θ)-invex aty, we have 

r

er{(ξLFL(x)+ξUFU(x)+ mj=μjgj(x))–(ξLFL(y)+ξUFU(y)+ mj=μjgj(y))}_{– }

≥ 

p

j=

μj∇gj(y)

epη(x,y)– +ρθ(x,y).

The above inequality together with () andρ≥ yields

 r

er{(ξLFL(x)+ξUFU(x)+ mj=μjgj(x))–(ξLFL(y)+ξUFU(y)+ mj=μjgj(y))}_{– }≥_.

Sincer> , using the property of an exponential function, we get

ξLFL(x) +ξUFU(x) + m

j=

μjgj(x)≥ξLFL(y) +ξUFU(y) + m

j=

μjgj(y),

which contradicts the inequality (). This completes the proof.

Theorem .(Strong duality) Let x* _{be a LU optimal solution to}_(IVP)_{at which} Kuhn-Tucker constraints qualiﬁcation are satisﬁed.Then there existξ*L> ,ξ*U> andμ*≥ such that(x*_,_ξ*L_,_ξ*U_,_μ*₎_{is feasible for}_(WD)_{and the two objectives have the same value}_. Further, if the hypothesis of weak duality Theorem. holds for all feasible solutions (y*_,_ξ*L

,ξ*U,μ*),then(x*_,_ξ*L

,ξ*U,μ*)is a LU optimal solution to(WD).

Proof Sincex*_{is a LU optimal solution to (IVP) and the Kuhn-Tucker constraints} qualiﬁ-cation are satisﬁed atx*_{, then by Theorem ., there exist multipliers}_ξ*L

> ,ξ*U>  and

μ*_j≥,j= , , . . . ,m, such that

ξ*L∇FLx*+ξ*U∇FUx*+ m

j=

μ*_j∇gjx*= ,

μ*_jgjx*= ,

which yields that (x*_,_ξ*L_,_ξ*U_,_μ*_{) is a feasible solution for (WD) and corresponding} objec-tive values are the same. Further, if (x*_,_ξ*L_,_ξ*U_,_μ*_{) is not a LU optimal solution of (WD),} then there exists a feasible solution (y*_,_ξ*L

,ξ*U,μ*) for (WD) such that

Fx*<LUF

y*+ m

j=

μ*_jgjy*,

which contradicts weak duality Theorem .. Hence (x*_,_ξ*L_,_ξ*U_,_μ*_{) is a LU optimal}

solu-tion to (WD).

Theorem . (Strict converse duality) Let X be an open subset of Rn_. _{Let F and gj}_, j= , , . . . ,m,be diﬀerentiable on X.Suppose that x* and(y*,ξ*L,ξ*U,μ*)are the feasi-ble solutions to(IVP)and(WD),respectively.Assume thatξ*L_FL₊_ξ*U_FU₊ m

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strictly(p,r) –ρ– (η,θ)-invex with respect toη,θat y*_with_ρ_≥__and

ξ*LFLx*+ξ*UFUx*+ m

j=

μ*_jgjx*≤ξ*LFLy*+ξ*UFUy*+ m

j=

μ*_jgjy*. ()

Then x*₌_y*_.

Proof Now we assume thatx*=y*and exhibit a contradiction. From the assumption that

ξ*LFL+ξ*UFU+ m_j=μ_j*gjis strictly (p,r) –ρ– (η,θ)-invex aty*, we have 

r

er{(ξ* L

FL(x*)+ξ*UFU(x*)+ mj=μ*jgj(x*))–(ξ* L

FL(y*)+ξ*UFU(y*)+ mj=μ*jgj(y*))}_{– }

>  p

ξ*L∇FLy*+ξ*U∇FUy*+ m

j=

μ*_j∇gjy*

epη(x*,y*)– +ρθx*,y*.

From the feasibility of (y*,ξ*L,ξ*U,μ*) to (WD), the above inequality together with the

hypothesisρ≥ yields

 r

er{(ξ* L

FL(x*)+ξ*UFU(x*)+ mj=μ*jgj(x*))–(ξ* L

FL(y*)+ξ*UFU(y*)+ mj=μ*jgj(y*))}_{– }_{> .}

ξ*LFLx*+ξ*UFUx*+ m

j=

μ*_jgjx*>ξ*LFLy*+ξ*UFUy*+ m

j=

μ*_jgjy*,

which contradicts (). This completes the proof.

5 Mond-Weir type duality

In this section, we consider the following Mond-Weir type dual problem for (IVP):

(MWD) maxF(y) =FL(y),FU(y)

subject to

j=

μj∇gj(y) = , ()

μjgj(y)≥,j= , , . . . ,m, ()

ξL> ,ξU> ,μj≥,j= , , . . . ,m. ()

Deﬁnition . Let (y*_,_ξ*L_,_ξ*U_,_μ*_{) be a feasible solution of dual problem (MWD). We say} that (y*_,_ξ*L_,_ξ*U_,_μ*_{) is a LU optimal solution of dual problem (MWD) if there exists no} (y,ξ*L,ξ*U,μ*) such thatF(y*_{) <}

LUF(y).

Theorem .(Weak duality) Let Xbe an open subset of Rn.Let F and gj,j= , , . . . ,m, be diﬀerentiable on X.Suppose that x and(y,ξL,ξU,μ)are the feasible solutions to(IVP) and(MWD),respectively.Further assume that there existξL_{> }_and_ξU_{> }_and_μ

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, , . . . ,m,such that(ξL_FL₊_ξU_FU₎_is₍_p_,_r_)–_ρ

–(η,θ)-pseudo-invex with respect toη,θand m

j=μjgjis(p,r) –ρ– (η,θ)-quasi-invex with respect toη,θat y with(ρ+ρ)≥.Then F(x)≥LUF(y).

Proof Suppose contrary to the result that

F(x) <LUF(y).

That is,

FL₍_x_{) <}_FL₍_y_), FU₍_x_{) <}_FU₍_y_), or

FL₍_x₎_≤_FL₍_y_), FU₍_x_{) <}_FU₍_y_), or

FL₍_x_{) <}_FL₍_y_), FU₍_x₎_≤_FU₍_y_).

SinceξL>  andξU> , from the above inequalities, we have

ξLFL(x) +ξUFU(x) <ξLFL(y) +ξUFU(y). ()

On the other hand, sinceμj≥,j= , , . . . ,m, from the feasibility ofxand (y,ξL,ξU,μ) to (IVP) and (MWD), respectively, we obtain

m

j=

μjgj(x)≤ m

j=

μjgj(y).

 r

er{ mj=μjgj(x)– jm=μjgj(y)}_{– }≤_,

which together with the assumption that m_j=μjgjis (p,r) –ρ– (η,θ)-quasi-invex aty, gives

 p

m

j=

μj∇gj(y)

epη(x,y)_–_₊_ρ

θ(x,y) 

≤.

Therefore, from () and the hypothesis that (ρ+ρ)≥, the above inequality yields 

p

epη(x,y)–ξL∇FL(y) +ξU∇FU(y)+ρθ(x,y) 

≥,

which together with the assumption thatξLFL₊_ξU_FU_{is (}_p_,_r_{) –}_ρ

– (η,θ)-pseudo-invex atygives

 r

er{(ξLFL(x)+ξUFU(x))–(ξLFL(y)+ξUFU(y))}– ≥.

ξLFL(x) +ξUFU(x)≥ξLFL(y) +ξUFU(y),

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Theorem .(Strong duality) Let x*_{be a LU optimal solution to}_(IVP)_{at which} Kuhn-Tucker constraints qualiﬁcation are satisﬁed.Then there existξ*L> ,ξ*U> andμ*≥

such that(x*_,_ξ*L_,_ξ*U_,_μ*₎_{is feasible for}_(MWD)_{and the two objectives have the same} value.Further,if the hypothesis of weak duality Theorem.holds for all feasible solutions (y*_,_ξ*L_,_ξ*U_,_μ*_),_then₍_x*_,_ξ*L_,_ξ*U_,_μ*₎_{is a LU optimal solution to}_(MWD).

Proof Sincex*_{is a LU optimal solution to (IVP) and the Kuhn-Tucker constraints} qual-iﬁcation are satisﬁed atx*_{, then by Theorem ., there exist multipliers}_ξ*L

> ,ξ*U> ,

μ*

j≥,j= , , . . . ,m, such that

ξ*L∇FLx*+ξ*U∇FUx*+ m

j=

μ*_j∇gjx*= ,

μ*_jgjx*= ,

which yields that (x*_,_ξ*L_,_ξ*U_,_μ*_{) is a feasible solution for (MWD) and corresponding} ob-jective values are same. Further, if (x*,ξ*L,ξ*U,μ*) is not a LU optimal solution of (MWD), then there exists a feasible solution (y*_,_ξ*L_,_ξ*U_,_μ*_{) for (MWD) such that}

Fx*<LUF

y*,

which contradicts weak duality Theorem .. Hence (x*_,_ξ*L_,_ξ*U_,_μ*_{) is a LU optimal}

solu-tion to (MWD).

Theorem . (Strict converse duality) Let X be an open subset of Rn_. _{Let F and gj}_, j= , , . . . ,m, be diﬀerentiable on X. Suppose that x* and(y*,ξ*L,ξ*U,μ*) are the fea-sible solutions to(IVP)and(MWD),respectively.Assume thatξ*LFL₊_ξ*U_FU _{is strictly} (p,r) –ρ– (η,θ)-pseudo-invex and mj=μ*jgjis(p,r) –ρ– (η,θ)-quasi-invex with respect toη,θat y*_with₍_ρ

+ρ)≥and

ξ*LFLx*+ξ*UFUx*≤ξ*LFLy*+ξ*UFUy*. ()

Then x*₌_y*_.

Proof Now we assume thatx*₌_y*_{and exhibit a contradiction. From the assumption that} (y*,ξ*L,ξ*U,μ*) is a feasible solution for (MWD), we get

ξ*L∇FLy*+ξ*U∇FUy*+ m

j=

μ*_j∇gjy*= . ()

Since μ*

j≥,j= , , . . . ,m, from the feasibility of x* and (y*,ξ* L

,ξ*U_,_μ*_{) to (IVP) and} (MWD), respectively, we obtain

m

j=

μ*_jgjx*≤ m

j=

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 r

er{ mj=μ*jgj(x*)– m

j=μ*jgj(y*)}_{– }≤_,

which together with the assumption that m_j=μ*

jgjis (p,r) –ρ– (η,θ)-quasi-invex aty* gives

 p

m

j=

μ*_j∇gjy*epη(x*,y*)–+ρθ

x*,y*≤.

Therefore, from () and the hypothesis that (ρ+ρ)≥, the above inequality yields 

p

epη(x*_,y*₎

–ξ*L∇FLy*+ξ*U∇FUy*+ρθ

x*,y*≥,

which together with the assumption thatξ*L_FL₊_ξ*U_FU_{is strictly (}_p_,_r_)–_ρ

–(η,θ )-pseudo-invex aty*_gives

 r

er{(ξ*LFL_(x*₎₊_ξ*U_FU_(x*_))–(_ξ*L_FL_(y*₎₊_ξ*U_FU_(y*_))}

– > .

ξ*LFLx*+ξ*UFUx*>ξ*LFLy*+ξ*UFUy*,

which contradicts (). This completes the proof.

6 Conclusion

In this paper, we have derived suﬃcient optimality conditions for a class of interval-valued optimization problems under generalized invex functions. Weak, strong and strict con-verse duality theorems are discussed for two types of the dual models. It will be interesting to obtain the optimality and duality theorem for a class of interval-valued programming under generalized invexity assumptions in which the involved functions are non-smooth. Moreover, it will also be interesting to see whether the second-order duality results for a class of interval-valued programming problem hold or not. This will orient the future research of the authors.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors carried out the proof. All authors conceived of the study, and participated in its design and coordination. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi} Arabia. 2_{Permanent address: Department of Mathematics, Aligarh Muslim University, Aligarh, 202 002, India.} 3_{Department of Applied Mathematics, Indian School of Mines, Jharkhand, Dhanbad 826 004, India.}

Acknowledgements

Izhar Ahmad thanks the King Fahd University of Petroleum and Minerals, Dhahran-31261, Saudi Arabia for the support under the Internal Project No. IN111037. The authors wish to thank the referees for their several valuable suggestions which have considerably improved the presentation of this article.

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