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Duality for nondifferentiable multiobjective higher-order symmetric programs over cones involving generalized (F,α,ρ,d)-convexity

Abstract

In this paper, a pair of Wolfe type higher-order symmetric nondifferentiable multiobjective programs over arbitrary cones is formulated and appropriate duality relations are then established under higher-order-K-(F,α,ρ,d)-convexity assumptions. A numerical example which is higher-order K-(F,α,ρ,d)-convex but not higher-order K-F-convex has also been illustrated. Special cases are also discussed to show that this paper extends some of the known works that have appeared in the literature.

MSC: 90C29, 90C30, 49N15.

1 Introduction

Mangasarian [1] introduced the concept of second- and higher-order duality for nonlinear problems. He has also indicated that the study is significant due to the computational advantage over the first-order duality as it provides tighter bounds for the value of the objective function when approximations are used. Motivated by the concept in [1], several researchers [219] have worked in this field.

Multiobjective optimization has a large number of applications. As an example, it is generally used in goal programming, risk programming etc. Optimality conditions for multiobjective programming problems can be found in Miettinen [20] and Pardalos et al. [21]. Recently, Chinchuluun and Pardalos [22] discussed recent developments in multiobjective optimization. These include optimality conditions, applications, global optimization techniques, the new concept of epsilon pareto optimal solutions and heuristics.

Chen [7] considered a pair of symmetric higher-order Mond-Weir type nondifferentiable multiobjective programming problems and established usual duality results under higher-order F-convexity assumptions. Gulati and Gupta [9] proved duality theorems for a pair of Wolfe type higher-order nondifferentiable symmetric dual programs. Ahmad et al. [6] formulated a general Mond-Weir type higher-order dual for a nondifferentiable multiobjective programming problem and established usual duality results.

Gulati and Geeta [23] pointed out certain omissions in some papers on symmetric duality in multiobjective programming and discussed their corrective measures. Later on, Ahmad and Husain [5] and Gulati et al. [11] formulated second-order multiobjective symmetric dual programs with cone constraints and established duality results under second-order invexity assumptions. An omission in the strong duality theorem in Yang et al. [17] has been rectified in Gupta and Kailey [12].

The concept of (F,ρ)-convexity was introduced by Preda [24] as an extension of F-convexity [25] and ρ-convexity [26]. Yang et al. [18] formulated several second-order duals for scalar programming problem and proved duality results involving generalized F-convex functions. Zhang and Mond [19] extended the class of (F,ρ)-convex functions to second-order and obtained duality relations for multiobjective dual problems.

Motivated by various concepts of generalized convexity, Liang et al. [27] introduced a unified formulation of generalized convexity, called (F,α,ρ,d)-convexity and obtained corresponding optimality conditions and duality relations for a single objective fractional problem. This was later on extended to multiobjective fractional programming problem in Liang et al. [28]. Inspired by the concept given in [19, 27, 28], Ahmad and Husain [29] introduced second-order (F,α,ρ,d)-convex functions and proved duality relations for Mond-Weir type second-order multiobjective problems. In the recent work of Ahmad and Husain [30], an attempt is made to remove certain omissions and inconsistencies in the work of Mishra and Lai [31].

Agarwal et al. [2] achieved duality results for a pair of Mond-Weir type multiobjective higher-order symmetric dual programs over arbitrary cones under higher-order K-F-convexity assumptions. Recently, Agarwal et al. [32] have filled some gap in the work of Chen [7] and proved a strong duality theorem for Mond-Weir type multiobjective higher-order nondifferentiable symmetric dual programs.

In this paper, we formulate a pair of symmetric higher-order Wolfe type nondifferentiable multiobjective programs over arbitrary cones and prove weak, strong and converse duality theorems under higher-order-K-(F,α,ρ,d)-convexity assumptions. We also give a nontrivial example of a function lying in the class of higher-order K-(F,α,ρ,d)-convex but not in the class of higher-order K-F-convex. Our study extends some of the known results that appeared in [4, 8, 16, 17].

2 Notations and preliminaries

Consider the following multiobjective programming problem:

K -minimize ϕ ( x ) subject to x X 0 = { x S : g ( x ) C } ,
(P)

where S R n is open, ϕ:S R k , g:S R m , K is a closed convex pointed cone in R k with intKϕ and C is a closed convex cone in R m with nonempty interior.

Definition 1 [33]

The positive polar cone C of C is defined as

C = { z : ξ T z 0 ,  for all  ξ C } .

Definition 2 [33]

A point x ¯ X 0 is a weak efficient solution of (P) if there exists no other x X 0 such that

ϕ( x ¯ )ϕ(x)intK.

Definition 3 [34]

A point x ¯ X 0 is an efficient solution of (P) if there exists no other x X 0 such that

ϕ( x ¯ )ϕ(x)K{0}.

Definition 4 [2, 9, 16]

Let D be a compact convex set in R n . The support function of D is defined by

S(xD)=max { x T y : y D } .

A support function, being convex and everywhere finite, has a subdifferential, that is, there exists z R n such that

S(yD)S(xD)+ z T (yx)for all yD.

The subdifferential of S(xD) is given by

S(xD)= { z D : z T x = S ( x D ) } .

For any set S R n , the normal cone to S at a point xS is defined by

N S (x)= { y R n : y T ( z x ) 0  for all  z S } .

It can be easily seen that for a compact convex set D, y is in N D (x) if and only if S(yD)= x T y, or equivalently, x is in S(yD).

Definition 5 [24, 28]

A functional F:X×X× R n R (where X R n ) is sublinear with respect to the third variable if, for all (x,u)X×X,

(i) F(x,u; a 1 + a 2 )F(x,u; a 1 )+F(x,u; a 2 ) for all a 1 , a 2 R n , and

(ii) F(x,u;αa)=αF(x,u;a), for all α R + and a R n .

For notational convenience, we write

F(x,u;a)= F x , u (a).

Let F:S×S× R n R be a sublinear functional with respect to the third variable. Now, we recall the concept of higher-order K-F-convexity introduced in [2].

Definition 6 A differentiable function ϕ:S R k is said to be higher-order K-F-convex at u on S with respect to ζ:S× R n R k if, for all xS and q R n , we have

Remark 1

(i) If K= R + k , then the above definition reduces to higher-order F-convexity given in [7, 9, 32].

(ii) If K= R + k and ζ i (u,q)= 1 2 q T x x ϕ i (u)q, i=1,2,,k, then Definition 6 becomes second-order F-convexity as considered in [12, 17].

(iii) Taking K= R + k , ζ i (u,q)= 1 2 q T x x ϕ i (u)q and F x , u (a)=η ( x , u ) T a, then the definition reduces to second-order invexity (or η bonvexity) given in [5, 11, 15].

Definition 7 A differentiable function ϕ:S R k is said to be higher-order K-(F,α,ρ,d)-convex at u on S with respect to ζ:S× R n R k if for all xS and q R n , there exist vector ρ R k , a real valued function α:S×S R + {0} and d:S×S R k such that

Remark 2

(i) If K= R + , α(x,u)=1 and ζ i (u,q)= 1 2 q T x x ϕ i (u)q, i=1,2,,k, the definition of higher-order K-(F,α,ρ,d)-convexity reduces to second-order (F,ρ)-convexity given by Srivastava and Bhatia [14].

(ii) If K= R + , α(x,u)=1 and ρ=0, then Definition 7 reduces to higher-order F-convexity (see [7, 9]).

(iii) If we take K= R + 1 , α(x,u)=1, ρ=0, ζ 1 (u,q)= 1 2 q T x x ϕ 1 (u)q and F x , u (a)=η ( x , u ) T a, where η is a function from S×S to R n , then the above definition becomes second-order η-convexity given in [15].

Next, we illustrate a nontrivial example of higher-order K-(F,α,ρ,d)-convex functions which are not higher-order K-F-convex.

Example 1 Let X=[2.5,0.5]R, n=m=1, k=2, K={(x,y):x0,y0}. Consider the function ψ:X R 2 to be defined by ψ(x)=( ψ 1 (x), ψ 2 (x)), where

ψ 1 (x)= x 3 sin 2 x , ψ 2 (x)= x 3 ,

and α:X×X R + {0} to be identified by α(x,u)=( u 2 +1). Let the functional F:X×X×RR be defined by

F x , u (a)=12(1u)a.

Suppose d:X×X R 2 is given by d(x,u)=( d 1 (x,u), d 2 (x,u)), where

d 1 (x,u)= ( x 4 + u 2 ) 3 2 , d 2 (x,u)= ( x 2 + u 2 ) 5 2

and ζ:X×R R 2 is defined by ζ(u,q)=( ζ 1 (u,q), ζ 2 (u,q)), where

ζ 1 (u,q)=15u+12q, ζ 2 (u,q)= sin 2 u+ q 2 .

To show ψ is higher-order (F,α,ρ,d)-convex, we need to prove

L = { ψ 1 ( x ) ψ 1 ( u ) ζ 1 ( u , q ) + q T q ζ 1 ( u , q ) F x , u [ α ( x , u ) ( x ψ 1 ( u ) + q ζ 1 ( u , q ) ) ] ρ 1 d 1 2 ( x , u ) , ψ 2 ( x ) ψ 2 ( u ) ζ 2 ( u , q ) + q T q ζ 2 ( u , q ) F x , u [ α ( x , u ) ( x ψ 2 ( u ) + q ζ 2 ( u , q ) ) ] ρ 2 d 2 2 ( x , u ) } K ,

which for ρ 1 =4 and ρ 2 =28 gives

L = { x 3 sin 2 x u 3 sin 2 u 15 u 12 ( 1 u ) [ ( u 2 + 1 ) ( 2 u cos 2 u + 3 u 2 sin 2 u + 12 ) ] + 4 ( x 4 + u 2 ) 3 , x 3 u 3 sin 2 u + q 2 12 ( 1 u ) [ ( u 2 + 1 ) ( 3 u 2 + 2 q ) ] + 28 ( x 2 + u 2 ) 5 } K = { L 1 , L 2 } K ,

where

L 1 = x 3 sin 2 x u 3 sin 2 u 15 u 12 ( 1 u ) [ ( u 2 + 1 ) ( 2 u cos 2 u + 3 u 2 sin 2 u + 12 ) ] + 4 ( x 4 + u 2 ) 3 0 x , u X  as can be seen from Figure 1
(E1)

and

L 2 = x 3 u 3 sin 2 u + q 2 12 ( 1 u ) [ ( u 2 + 1 ) ( 3 u 2 + 2 q ) ] + 28 ( x 2 + u 2 ) 5 = L 21 + L 22 .

But

L 21 = x 3 u 3 sin 2 u 12 ( 1 u ) [ 3 u 2 ( u 2 + 1 ) ] + 28 ( x 2 + u 2 ) 5 0 x , u X  as can be seen from Figure 2
(E2)

and

L 22 = q 2 12 ( 1 u ) [ 2 q ( u 2 + 1 ) ] 0 u X  and  q ( 10 18 , 10 18 )  as can be seen from Figure 3.
(E3)

Hence, L 2 0. Therefore, ψ is higher-order K-(F,α,ρ,d)-convex with respect to ζ.

Figure 1
figure 1

Graph of L 1 (Expression ( E1 )) against x and u .

Figure 2
figure 2

Graph of L 21 (Expression ( E2 )) against x and u .

Figure 3
figure 3

Graph of L 22 (Expression ( E3 )) against q and u .

Next, we need to show that ψ is not higher-order K-F-convex with respect to ζ. To prove it, we will show that

M = { ψ 1 ( x ) ψ 1 ( u ) ζ 1 ( u , q ) + q T q ζ 1 ( u , q ) F x , u [ x ψ 1 ( u ) + q ζ 1 ( u , q ) ] , ψ 2 ( x ) ψ 2 ( u ) ζ 2 ( u , q ) + q T q ζ 2 ( u , q ) F x , u [ x ψ 2 ( u ) + q ζ 2 ( u , q ) ] } K

i.e., either

ψ 1 (x) ψ 1 (u) ζ 1 (u,q)+ q T q ζ 1 (u,q) F x , u [ x ψ 1 ( u ) + q ζ 1 ( u , q ) ] ≧̸0

or

ψ 2 (x) ψ 2 (u) ζ 2 (u,q)+ q T q ζ 2 (u,q) F x , u [ x ψ 2 ( u ) + q ζ 2 ( u , q ) ] ≧̸0.

Since

M = ψ 1 ( x ) ψ 1 ( u ) ζ 1 ( u , q ) + q T q ζ 1 ( u , q ) F x , u [ x ψ 1 ( u ) + q ζ 1 ( u , q ) ] = x 3 sin 2 x u 3 sin 2 u 15 u 12 ( 1 u ) [ 2 u cos 2 u + 3 u 2 sin 2 u + 12 ] = 181.7330 < 0 ( for  x = 2  and  u = 1 ) .
(E4)

In fact, M<0 for all x,uX as can be seen from Figure 4. Therefore, ψ is not higher-order K-F-convex with respect to ζ.

Figure 4
figure 4

Graph of M (Expression ( E4 )) against x and u .

3 Wolfe type higher-order symmetric duality

In this section, we consider the following Wolfe type nondifferentiable multiobjective higher-order symmetric dual programs.

Primal problem (HNWP) K-minimize

G ( x , y , λ , p ) = f ( x , y ) + S ( x D ) e k + ( λ T h ) ( x , y , p ) e k p T p ( λ T h ) ( x , y , p ) e k y T y ( λ T f ) ( x , y ) e k y T p ( λ T h ) ( x , y , p ) e k

subject to

(1)
(2)
(3)
(4)

Dual problem (HNWD) K-maximize

H ( u , v , λ , r ) = f ( u , v ) S ( v E ) e k + ( λ T g ) ( u , v , r ) e k r T r ( λ T g ) ( u , v , r ) e k u T x ( λ T f ) ( u , v ) e k u T r ( λ T g ) ( u , v , r ) e k

subject to

(5)
(6)
(7)
(8)

where

(i) C 1 and C 2 are closed convex cones with nonempty interiors in R n and R m , respectively,

(ii) S 1 R n and S 2 R m are open sets such that C 1 × C 2 S 1 × S 2 ,

(iii) f: S 1 × S 2 R k , h: S 1 × S 2 × R m R k and g: S 1 × S 2 × R n R k are differentiable functions, e k = ( 1 , , 1 ) T R k , λ R k ,

(iv) r and p are vectors in R n and R m , respectively,

(v) D and E are compact convex sets in R n and R m , respectively, and

(vi) S(xD) and S(vE) are the support functions of D and E, respectively.

Remark 3 The problems (HNWP) and (HNWD) stated above are nondifferentiable because their objective function contains the support function S(xD) and S(vE).

We now prove the following duality results for the pair of problems (HNWP) and (HNWD).

Theorem 1 (Weak duality)

Let (x,y,λ,z,p) be feasible for the primal problem (HNWP) and (u,v,λ,w,r) be feasible for the dual problem (HNWD). Let the sublinear functionals F: R n × R n × R n R and G: R m × R m × R m R satisfy the following conditions:

(A)
(B)

Suppose that (i) either i = 1 k λ i [ ρ i ( 1 ) ( d i ( 1 ) ( x , u ) ) 2 + ρ i ( 2 ) ( d i ( 2 ) ( v , y ) ) 2 ]0 or ρ ( 1 ) 0 and ρ ( 2 ) 0, (ii) f(,v)+ ( ) T w e k is higher-order K-(F, α 1 , ρ ( 1 ) , d ( 1 ) )-convex at u with respect to g(u,v,r), and (iii) {f(x,) ( ) T z e k } is higher-order K-(G, α 2 , ρ ( 2 ) , d ( 2 ) )-convex at y with respect to h(x,y,p). Then

G(x,y,λ,p)H(u,v,λ,r)K{0}.
(9)

Proof Since f(,v)+ ( ) T w e k is higher-order K-(F, α 1 , ρ ( 1 ) , d ( 1 ) )-convex with respect to g(u,v,r), we have

It follows from λint K , λ T e k =1, and sublinearity of F that

(10)

As {f(x,) ( ) T z e k } is higher-order K-(G, α 2 , ρ ( 2 ) , d ( 2 ) )-convex with respect to h(x,y,p), therefore we get

Again, using λint K , λ T e k =1, and sublinearity of G, we obtain

(11)

Further, adding the inequalities (10) and (11), we have

(12)

Now, since (x,y,λ,z,p) is feasible for the primal problem (HNWP) and (u,v,λ,w,r) is feasible for the dual problem (HNWD), α 1 (x,u)>0, by the dual constraint (5), the vector a= α 1 (x,u)[ x ( λ T f)(u,v)+w+ r ( λ T g)(u,v,r)] C 1 , and so from the hypothesis (A), we obtain

F x , u (a)+ α 1 1 a T u0.
(13)

Similarly,

G v , y (b)+ α 2 1 b T y0,
(14)

for the vector b= α 2 (v,y)[ y ( λ T f)(x,y)z+ p ( λ T h)(x,y,p)] C 2 .

Using (13), (14) and the hypothesis (i) in (12), we have

Further, substituting the values of a and b, we have

In view of the fact that x T wS(xD), v T zS(vE) and λ T e k =1, the last inequality yields

(15)

Now, suppose contrary to the result that (9) does not hold, that is,

G(x,y,λ,p)H(u,v,λ,r)K{0},

or

It follows from λint K and λ T e k =1 that

which contradicts (15). Hence the result. □

Theorem 2 (Strong duality)

Let f: S 1 × S 2 R k be a twice differentiable function and let ( x ¯ , y ¯ , λ ¯ , z ¯ , p ¯ ) be a weak efficient solution of (HNWP). Suppose that

(i) the matrix p p ( λ ¯ T h)( x ¯ , y ¯ , p ¯ ) is nonsingular,

(ii) the vectors { y f 1 ( x ¯ , y ¯ ),, y f k ( x ¯ , y ¯ )} are linearly independent,

(iii) the vector { y ( λ ¯ T h)( x ¯ , y ¯ , p ¯ ) p ( λ ¯ T h)( x ¯ , y ¯ , p ¯ )+ y y ( λ ¯ T f)( x ¯ , y ¯ ) p ¯ }span{ y f 1 ( x ¯ , y ¯ ),, y f k ( x ¯ , y ¯ )}{0},

(iv) y ( λ ¯ T h)( x ¯ , y ¯ , p ¯ ) p ( λ ¯ T h)( x ¯ , y ¯ , p ¯ )+ y y ( λ ¯ T f)( x ¯ , y ¯ ) p ¯ =0 implies p ¯ =0,

(v) ( λ ¯ T h)( x ¯ , y ¯ ,0)=( λ ¯ T g)( x ¯ , y ¯ ,0), y ( λ ¯ T h)( x ¯ , y ¯ ,0)=0, p ( λ ¯ T h)( x ¯ , y ¯ ,0)=0, x ( λ ¯ T h)( x ¯ , y ¯ ,0)= r ( λ ¯ T g)( x ¯ , y ¯ ,0) and

(vi) K is a closed convex pointed cone with R + k K.

Then

(I) there exists w ¯ D such that ( x ¯ , y ¯ , λ ¯ , w ¯ , r ¯ =0) is feasible for (HNWD), and

(II) G( x ¯ , y ¯ , λ ¯ , p ¯ )=H( x ¯ , y ¯ , λ ¯ , r ¯ ).

Also, if the hypotheses of Theorem 1 are satisfied for all feasible solutions of (HNWP) and (HNWD), then ( x ¯ , y ¯ , λ ¯ , w ¯ , r ¯ =0) is an efficient solution for (HNWD).

Proof Since ( x ¯ , y ¯ , λ ¯ , z ¯ , p ¯ ) is a weak efficient solution of (HNWP), by the Fritz John necessary optimality conditions [33], there exist α ¯ K , β ¯ C 2 , η ¯ R, such that the following conditions are satisfied at ( x ¯ , y ¯ , λ ¯ , z ¯ , p ¯ ):

(16)
(17)
(18)
(19)
(20)
(21)
(22)
(23)
(24)

From (18) and nonsingularity of p p ( λ ¯ T h)( x ¯ , y ¯ , p ¯ ), we have

β ¯ = ( α ¯ T e k ) ( y ¯ + p ¯ ).
(25)

Also, (19) is equivalent to

y f( x ¯ , y ¯ ) [ β ¯ ( α ¯ T e k ) y ¯ ] +h( x ¯ , y ¯ , p ¯ ) ( α ¯ T e k ) + η ¯ e k + p h( x ¯ , y ¯ , p ¯ ) [ β ¯ ( α ¯ T e k ) ( y ¯ + p ¯ ) ] =0.

If α ¯ =0, then (25) yields β ¯ =0. Further, the above equality gives η ¯ e k =0 or η ¯ =0. Consequently, ( α ¯ , β ¯ , η ¯ )=0, contradicting (24). Hence, α ¯ 0.

Since R + k K K R + k , therefore α ¯ K α ¯ 0.

But α ¯ 0 α ¯ 0 or

α ¯ T e k >0.
(26)

Now, using (25) and (26) in (17), we get

(27)

which, by the hypotheses (iii) and (iv), implies

p ¯ =0.
(28)

Using the hypothesis (iii) in (27), we obtain

y f( x ¯ , y ¯ ) [ α ¯ ( α ¯ T e k ) λ ¯ ] =0.

Since the vectors { y f 1 ( x ¯ , y ¯ ),, y f k ( x ¯ , y ¯ )} are linearly independent, therefore the above equation yields

α ¯ = ( α ¯ T e k ) λ ¯ .
(29)

From (28) in (25), we get

β ¯ = ( α ¯ T e k ) y ¯ .
(30)

Using (26) and (28)-(30) in (16), we have

{ x ( λ ¯ T f ) ( x ¯ , y ¯ ) + γ ¯ + x ( λ ¯ T h ) ( x ¯ , y ¯ , p ¯ ) } (x x ¯ )0,for all x C 1 .

From the hypothesis (v), for r ¯ =0, the above inequality yields

{ x ( λ ¯ T f ) ( x ¯ , y ¯ ) + γ ¯ + r ( λ ¯ T g ) ( x ¯ , y ¯ , r ¯ ) } (x x ¯ )0.
(31)

Let x C 1 . Then x+ x ¯ C 1 , and so (31) implies

{ x ( λ ¯ T f ) ( x ¯ , y ¯ ) + γ ¯ + r ( λ ¯ T g ) ( x ¯ , y ¯ , r ¯ ) } T x0,for all x C 1 .

Therefore,

{ x ( λ ¯ T f ) ( x ¯ , y ¯ ) + γ ¯ + r ( λ ¯ T g ) ( x ¯ , y ¯ , r ¯ ) } C 1 .
(32)

Also, from (26), (30) and β ¯ C 2 , we obtain y ¯ C 2 . Thus ( x ¯ , y ¯ , λ ¯ , w ¯ = γ ¯ , r ¯ =0), satisfies the dual constraints from (5) to (8) in (HNWD), and so it is a feasible solution for the dual problem (HNWD). Now, letting x=0 and x=2 x ¯ in (31), we get

(33)

From (22) and (30), ( α ¯ T e k ) y ¯ N E ( z ¯ ). Since α ¯ T e k >0, y ¯ N E ( z ¯ ). Also, as E is a compact convex set in R m , y ¯ T z ¯ =S( y ¯ E).

Further, from (20), (26) and (30) and the above relation, we obtain

y ¯ T [ y ( λ ¯ T f ) ( x ¯ , y ¯ ) + p ( λ ¯ T h ) ( x ¯ , y ¯ , p ¯ ) ] = y ¯ T z ¯ =S( y ¯ E).
(34)

Therefore, using (28), (33), (34) and the hypothesis (v), for r ¯ =0, we get

that is, the two objective values are equal.

Now, let ( x ¯ , y ¯ , λ ¯ , w ¯ , r ¯ =0) be not an efficient solution of (HNWD), then there exists ( u ¯ , v ¯ , λ ¯ , w ¯ , r ¯ =0) feasible for (HNWD) such that

As x ¯ T [ x ( λ ¯ T f)( x ¯ , y ¯ )+ r ( λ ¯ T g)( x ¯ , y ¯ , r ¯ )]=S( x ¯ D), y ¯ T [ y ( λ ¯ T f)( x ¯ , y ¯ )+ p ( λ ¯ T h)( x ¯ , y ¯ , p ¯ )]=S( y ¯ E) and from the hypothesis (v), for r ¯ =0, we obtain

which contradicts Theorem 1. Hence, ( x ¯ , y ¯ , λ ¯ , w ¯ , r ¯ =0) is an efficient solution of (HNWD). □

Theorem 3 (Converse duality)

Let f: S 1 × S 2 R k be a twice differentiable function and let ( u ¯ , v ¯ , λ ¯ , w ¯ , r ¯ ) be a weak efficient solution of (HNWD). Suppose that

(i) the matrix r r ( λ ¯ T g)( u ¯ , v ¯ , r ¯ ) is nonsingular,

(ii) the vectors { x f 1 ( u ¯ , v ¯ ),, x f k ( u ¯ , v ¯ )} are linearly independent,

(iii) the vector { x ( λ ¯ T g)( u ¯ , v ¯ , r ¯ ) r ( λ ¯ T g)( u ¯ , v ¯ , r ¯ )+ x x ( λ ¯ T f)( u ¯ , v ¯ ) r ¯ }span{ x f 1 ( u ¯ , v ¯ ),, x f k ( u ¯ , v ¯ )}{0},

(iv) x ( λ ¯ T g)( u ¯ , v ¯ , r ¯ ) r ( λ ¯ T g)( u ¯ , v ¯ , r ¯ )+ x x ( λ ¯ T f)( u ¯ , v ¯ ) r ¯ =0 implies r ¯ =0,

(v) ( λ ¯ T g)( u ¯ , v ¯ ,0)=( λ ¯ T h)( u ¯ , v ¯ ,0), x ( λ ¯ T g)( u ¯ , v ¯ ,0)=0, r ( λ ¯ T g)( u ¯ , v ¯ ,0)=0, y ( λ ¯ T g)( u ¯ , v ¯ ,0)= p ( λ ¯ T h)( u ¯ , v ¯ ,0) and

(vi) K is a closed convex pointed cone with R + k K.

Then

(I) there exists z ¯ E such that ( u ¯ , v ¯ , λ ¯ , z ¯ , p ¯ =0) is feasible for (HNWP), and

(II) G( u ¯ , v ¯ , λ ¯ , p ¯ )=H( u ¯ , v ¯ , λ ¯ , r ¯ ).

Also, if the hypotheses of Theorem 1 are satisfied for all feasible solutions of (HNWP) and (HNWD), then ( u ¯ , v ¯ , λ ¯ , z ¯ , p ¯ =0) is an efficient solution for (HNWP).

Proof It follows on the lines of Theorem 2. □

4 Special cases

In this section, we consider some of the special cases of the problems studied in Section 3. In all these cases, K= R + k , C 1 = R + n , C 2 = R + m , ( λ T h)(x,y,p)= 1 2 p T y y ( λ T f)(x,y)p and ( λ T g)(u,v,r)= 1 2 r T x x ( λ T f)(u,v)r.

(i) For k=1, D={Ay: y T Ay1}, E={Bx: x T Bx1}, where A and B are positive semidefinite matrices, ( x T A x ) 1 / 2 =S(xD) and ( y T B y ) 1 / 2 =S(yE). In this case, (HNWP) and (HNWD) reduce to the problems considered in Ahmad and Husain [4].

(ii) Let k=1, D={0} and E={0}, then our problems (HNWP) and (HNWD) become the problems studied in Gulati et al. [8].

(iii) If k=1, then our problems (HNWP) and (HNWD) reduce to the programs studied in Yang et al. [16].

(iv) Let D={0} and E={0}, our problems reduce to (MP) and (MD) considered in Yang et al. [17] along with nonnegativity restrictions x0 and v0. However, taking F x , u (a)= ( x u ) T a and G v , y (b)= ( v y ) T b along with the hypotheses (A) and (B) of Theorem 1 in [17] gives x0 and v0.

5 Conclusions

A pair of Wolfe-type multiobjective higher-order symmetric dual programs involving nondifferentiable functions over arbitrary cones has been formulated. Further, an example of higher-order K-(F,α,ρ,d)-convex which is not higher-order K-F-convex has been illustrated. Weak, strong and converse duality theorems under higher-order K-(F,α,ρ,d)-convexity assumptions have also been established. It is to be noted that some of the known results, including those of Ahmad and Husain [4], Gulati et al. [8] and Yang et al. [16, 17], are special cases of our study. This work can be further extended to study nondifferentiable higher-order multiobjective symmetric dual programs over arbitrary cones with different p i ’s and different support functions.

References

  1. Mangasarian OL: Second and higher-order duality in nonlinear programming. J. Math. Anal. Appl. 1975, 51: 607–620. 10.1016/0022-247X(75)90111-0

    MathSciNet  Article  Google Scholar 

  2. Agarwal RP, Ahmad I, Jayswal A: Higher-order symmetric duality in nondifferentiable multiobjective programming problems involving generalized cone convex functions. Math. Comput. Model. 2010, 52: 1644–1650. 10.1016/j.mcm.2010.06.030

    MathSciNet  Article  Google Scholar 

  3. Ahmad I: Second-order symmetric duality in nondifferentiable multiobjective programming. Inf. Sci. 2005, 173: 23–34. 10.1016/j.ins.2004.06.002

    Article  Google Scholar 

  4. Ahmad I, Husain Z: On nondifferentiable second-order symmetric duality in mathematical programming. Indian J. Pure Appl. Math. 2004, 35: 665–676.

    MathSciNet  Google Scholar 

  5. Ahmad I, Husain Z: On multiobjective second-order symmetric duality with cone constraints. Eur. J. Oper. Res. 2010, 204: 402–409. 10.1016/j.ejor.2009.08.019

    MathSciNet  Article  Google Scholar 

  6. Ahmad I, Husain Z, Sharma S: Higher-order duality in nondifferentiable multiobjective programming. Numer. Funct. Anal. Optim. 2007, 28: 989–1002. 10.1080/01630560701563800

    MathSciNet  Article  Google Scholar 

  7. Chen X: Higher-order symmetric duality in nondifferentiable multiobjective programming problems. J. Math. Anal. Appl. 2004, 290: 423–435. 10.1016/j.jmaa.2003.10.004

    MathSciNet  Article  Google Scholar 

  8. Gulati TR, Ahmad I, Husain I: Second-order symmetric duality with generalized convexity. Opsearch 2001, 38: 210–222.

    MathSciNet  Google Scholar 

  9. Gulati TR, Gupta SK: Higher-order nondifferentiable symmetric duality with generalized F -convexity. J. Math. Anal. Appl. 2007, 329: 229–237. 10.1016/j.jmaa.2006.06.032

    MathSciNet  Article  Google Scholar 

  10. Gulati TR, Gupta SK: Higher-order symmetric duality with cone constraints. Appl. Math. Lett. 2009, 22: 776–781. 10.1016/j.aml.2008.08.017

    MathSciNet  Article  Google Scholar 

  11. Gulati TR, Saini H, Gupta SK: Second-order multiobjective symmetric duality with cone constraints. Eur. J. Oper. Res. 2010, 205: 247–252. 10.1016/j.ejor.2009.12.024

    MathSciNet  Article  Google Scholar 

  12. Gupta SK, Kailey N: A note on multiobjective second-order symmetric duality. Eur. J. Oper. Res. 2010, 201: 649–651. 10.1016/j.ejor.2009.03.024

    MathSciNet  Article  Google Scholar 

  13. Hanson MA: Second-order invexity and duality in mathematical programming. Opsearch 1993, 30: 313–320.

    Google Scholar 

  14. Srivastava MK, Bhatia M:Symmetric duality for multiobjective programming using second-order (F,ρ)-convexity. Opsearch 2006, 43: 274–295.

    MathSciNet  Google Scholar 

  15. Suneja SK, Lalitha CS, Khurana S: Second-order symmetric duality in multiobjective programming. Eur. J. Oper. Res. 2003, 144: 492–500. 10.1016/S0377-2217(02)00154-6

    MathSciNet  Article  Google Scholar 

  16. Yang XM, Yang XQ, Teo KL: Non-differentiable second-order symmetric duality in mathematical programming with F -convexity. Eur. J. Oper. Res. 2003, 144: 554–559. 10.1016/S0377-2217(02)00156-X

    MathSciNet  Article  Google Scholar 

  17. Yang XM, Yang XQ, Teo KL, Hou SH: Multiobjective second-order symmetric duality with F -convexity. Eur. J. Oper. Res. 2005, 165: 585–591. 10.1016/j.ejor.2004.01.028

    MathSciNet  Article  Google Scholar 

  18. Yang XM, Yang XQ, Teo KL, Hou SH: Second order duality for nonlinear programming. Indian J. Pure Appl. Math. 2004, 35: 699–708.

    MathSciNet  Google Scholar 

  19. Zhang J, Mond B: Second order duality for multiobjective nonlinear programming involving generalized convexity. In Proceedings of the Optimization Miniconference III. Edited by: Glower BM, Craven BD, Ralph D. University of Ballarat, Ballarat; 1997:79–95.

    Google Scholar 

  20. Miettinen KM: Nonlinear Multiobjective Optimization. Kluwer, Boston; 1999.

    Google Scholar 

  21. Pardalos PM, Siskos Y, Zopounidis C: Advances in Multicriteria Analysis. Kluwer, Netherlands; 1995.

    Book  Google Scholar 

  22. Chinchuluun A, Pardalos PM: A survey of recent developments in multiobjective optimization. Ann. Oper. Res. 2007, 154: 29–50. 10.1007/s10479-007-0186-0

    MathSciNet  Article  Google Scholar 

  23. Gulati TR, Geeta M: On some symmetric dual models in multiobjective programming. Appl. Math. Comput. 2009, 215: 380–383. 10.1016/j.amc.2009.05.032

    MathSciNet  Article  Google Scholar 

  24. Preda V: On efficiency and duality for multiobjective programs. J. Math. Anal. Appl. 1992, 166: 356–377.

    MathSciNet  Article  Google Scholar 

  25. Hanson MA, Mond B: Further generalization of convexity in mathematical programming. J. Inf. Optim. Sci. 1986, 3: 25–32.

    MathSciNet  Google Scholar 

  26. Vial JP: Strong and weak convexity of sets and functions. Math. Oper. Res. 1983, 8: 231–259. 10.1287/moor.8.2.231

    MathSciNet  Article  Google Scholar 

  27. Liang ZA, Huang HX, Pardalos PM: Optimality conditions and duality for a class of nonlinear fractional programming problems. J. Optim. Theory Appl. 2001, 110: 611–619. 10.1023/A:1017540412396

    MathSciNet  Article  Google Scholar 

  28. Liang ZA, Huang HX, Pardalos PM: Efficiency conditions and duality for a class of multiobjective fractional programming problems. J. Glob. Optim. 2003, 27: 447–471. 10.1023/A:1026041403408

    MathSciNet  Article  Google Scholar 

  29. Ahmad I, Husain Z:Second order (F,α,ρ,d)-convexity and duality in multiobjective programming. Inf. Sci. 2006, 176: 3094–3103. 10.1016/j.ins.2005.08.003

    MathSciNet  Article  Google Scholar 

  30. Ahmad I, Husain Z: Second order symmetric duality in multiobjective programming involving cones. Optim. Lett. 2012. doi:10.1007/s11590–012–0508–2; article in press

    Google Scholar 

  31. Mishra SK, Lai KK: Second-order symmetric duality in multiobjective programming involving generalized cone-invex functions. Eur. J. Oper. Res. 2007, 178: 20–26. 10.1016/j.ejor.2005.11.024

    MathSciNet  Article  Google Scholar 

  32. Agarwal RP, Ahmad I, Gupta SK: A note on higher-order nondifferentiable symmetric duality in multiobjective programming. Appl. Math. Lett. 2011, 24: 1308–1311. 10.1016/j.aml.2011.02.021

    MathSciNet  Article  Google Scholar 

  33. Suneja SK, Aggarwal S, Davar S: Multiobjective symmetric duality involving cones. Eur. J. Oper. Res. 2002, 141: 471–479. 10.1016/S0377-2217(01)00258-2

    MathSciNet  Article  Google Scholar 

  34. Khurana S: Symmetric duality in multiobjective programming involving generalized cone-invex functions. Eur. J. Oper. Res. 2005, 165: 592–597. 10.1016/j.ejor.2003.03.004

    MathSciNet  Article  Google Scholar 

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Gupta, S., Kailey, N. & Kumar, S. Duality for nondifferentiable multiobjective higher-order symmetric programs over cones involving generalized (F,α,ρ,d)-convexity. J Inequal Appl 2012, 298 (2012). https://doi.org/10.1186/1029-242X-2012-298

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Keywords

  • higher-order symmetric duality
  • support function
  • multiobjective programming
  • cones
  • efficient solutions