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# Duality for nondifferentiable multiobjective higher-order symmetric programs over cones involving generalized (F,\alpha ,\rho ,d)-convexity

*Journal of Inequalities and Applications*
**volume 2012**, Article number: 298 (2012)

## 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,\alpha ,\rho ,d)-convexity assumptions. A numerical example which is higher-order *K*-(F,\alpha ,\rho ,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 [2–19] 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,\rho )-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,\rho )-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,\alpha ,\rho ,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,\alpha ,\rho ,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,\alpha ,\rho ,d)-convexity assumptions. We also give a nontrivial example of a function lying in the class of higher-order *K*-(F,\alpha ,\rho ,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:

where S\subseteq {R}^{n} is open, \varphi :S\to {R}^{k}, g:S\to {R}^{m}, *K* is a closed convex pointed cone in {R}^{k} with intK\ne \varphi and *C* is a closed convex cone in {R}^{m} with nonempty interior.

**Definition 1** [33]

The positive polar cone {C}^{\ast} of *C* is defined as

**Definition 2** [33]

A point \overline{x}\in {X}^{0} is a weak efficient solution of (P) if there exists no other x\in {X}^{0} such that

**Definition 3** [34]

A point \overline{x}\in {X}^{0} is an efficient solution of (P) if there exists no other x\in {X}^{0} such that

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

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

The subdifferential of S(x\mid D) is given by

For any set S\subset {R}^{n}, the normal cone to *S* at a point x\in S is defined by

It can be easily seen that for a compact convex set *D*, *y* is in {N}_{D}(x) if and only if S(y\mid D)={x}^{T}y, or equivalently, *x* is in \partial S(y\mid D).

A functional F:X\times X\times {R}^{n}\mapsto R (where X\subseteq {R}^{n}) is sublinear with respect to the third variable if, for all (x,u)\in X\times X,

(i) F(x,u;{a}_{1}+{a}_{2})\leqq F(x,u;{a}_{1})+F(x,u;{a}_{2}) for all {a}_{1},{a}_{2}\in {R}^{n}, and

(ii) F(x,u;\alpha a)=\alpha F(x,u;a), for all \alpha \in {R}_{+} and a\in {R}^{n}.

For notational convenience, we write

Let F:S\times S\times {R}^{n}\mapsto 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 \varphi :S\to {R}^{k} is said to be higher-order *K*-*F*-convex at *u* on *S* with respect to \zeta :S\times {R}^{n}\mapsto {R}^{k} if, for all x\in S and q\in {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 {\zeta}_{i}(u,q)=\frac{1}{2}{q}^{T}{\mathrm{\nabla}}_{xx}{\varphi}_{i}(u)q, i=1,2,\dots ,k, then Definition 6 becomes second-order *F*-convexity as considered in [12, 17].

(iii) Taking K={R}_{+}^{k}, {\zeta}_{i}(u,q)=\frac{1}{2}{q}^{T}{\mathrm{\nabla}}_{xx}{\varphi}_{i}(u)q and {F}_{x,u}(a)=\eta {(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 \varphi :S\to {R}^{k} is said to be higher-order *K*-(F,\alpha ,\rho ,d)-convex at *u* on *S* with respect to \zeta :S\times {R}^{n}\mapsto {R}^{k} if for all x\in S and q\in {R}^{n}, there exist vector \rho \in {R}^{k}, a real valued function \alpha :S\times S\to {R}_{+}\mathrm{\setminus}\{0\} and d:S\times S\to {R}^{k} such that

**Remark 2**

(i) If K={R}_{+}, \alpha (x,u)=1 and {\zeta}_{i}(u,q)=\frac{1}{2}{q}^{T}{\mathrm{\nabla}}_{xx}{\varphi}_{i}(u)q, i=1,2,\dots ,k, the definition of higher-order *K*-(F,\alpha ,\rho ,d)-convexity reduces to second-order (F,\rho )-convexity given by Srivastava and Bhatia [14].

(ii) If K={R}_{+}, \alpha (x,u)=1 and \rho =0, then Definition 7 reduces to higher-order *F*-convexity (see [7, 9]).

(iii) If we take K={R}_{+}^{1}, \alpha (x,u)=1, \rho =0, {\zeta}_{1}(u,q)=\frac{1}{2}{q}^{T}{\mathrm{\nabla}}_{xx}{\varphi}_{1}(u)q and {F}_{x,u}(a)=\eta {(x,u)}^{T}a, where *η* is a function from S\times 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,\alpha ,\rho ,d)-convex functions which are not higher-order *K*-*F*-convex.

**Example 1** Let X=[-2.5,-0.5]\subset R, n=m=1, k=2, K=\{(x,y):x\geqq 0,y\geqq 0\}. Consider the function \psi :X\to {R}^{2} to be defined by \psi (x)=({\psi}_{1}(x),{\psi}_{2}(x)), where

and \alpha :X\times X\to {R}_{+}\mathrm{\setminus}\{0\} to be identified by \alpha (x,u)=({u}^{2}+1). Let the functional F:X\times X\times R\to R be defined by

Suppose d:X\times X\to {R}^{2} is given by d(x,u)=({d}_{1}(x,u),{d}_{2}(x,u)), where

and \zeta :X\times R\to {R}^{2} is defined by \zeta (u,q)=({\zeta}_{1}(u,q),{\zeta}_{2}(u,q)), where

To show *ψ* is higher-order (F,\alpha ,\rho ,d)-convex, we need to prove

which for {\rho}_{1}=-4 and {\rho}_{2}=-28 gives

where

and

But

and

Hence, {L}_{2}\geqq 0. Therefore, *ψ* is higher-order *K*-(F,\alpha ,\rho ,d)-convex with respect to *ζ*.

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

*i.e.*, either

or

Since

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

## 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

subject to

**Dual problem (HNWD)** *K*-maximize

subject to

where

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

(ii) {S}_{1}\subseteq {R}^{n} and {S}_{2}\subseteq {R}^{m} are open sets such that {C}_{1}\times {C}_{2}\subset {S}_{1}\times {S}_{2},

(iii) f:{S}_{1}\times {S}_{2}\to {R}^{k}, h:{S}_{1}\times {S}_{2}\times {R}^{m}\to {R}^{k} and g:{S}_{1}\times {S}_{2}\times {R}^{n}\to {R}^{k} are differentiable functions, {e}_{k}={(1,\dots ,1)}^{T}\in {R}^{k}, \lambda \in {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(x\mid D) and S(v\mid E) 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(x\mid D) and S(v\mid E).

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

**Theorem 1** (Weak duality)

*Let* (x,y,\lambda ,z,p) *be feasible for the primal problem* (HNWP) *and* (u,v,\lambda ,w,r) *be feasible for the dual problem* (HNWD). *Let the sublinear functionals* F:{R}^{n}\times {R}^{n}\times {R}^{n}\mapsto R *and* G:{R}^{m}\times {R}^{m}\times {R}^{m}\mapsto R *satisfy the following conditions*:

*Suppose that* (i) *either* {\sum}_{i=1}^{k}{\lambda}_{i}[{\rho}_{i}^{(1)}{({d}_{i}^{(1)}(x,u))}^{2}+{\rho}_{i}^{(2)}{({d}_{i}^{(2)}(v,y))}^{2}]\geqq 0 *or* {\rho}^{(1)}\geqq 0 *and* {\rho}^{(2)}\geqq 0, (ii) f(\cdot ,v)+{(\cdot )}^{T}w{e}_{k} *is higher*-*order* *K*-(F,{\alpha}_{1},{\rho}^{(1)},{d}^{(1)})-*convex at* *u* *with respect to* g(u,v,r), *and* (iii) -\{f(x,\cdot )-{(\cdot )}^{T}z{e}_{k}\} *is higher*-*order* *K*-(G,{\alpha}_{2},{\rho}^{(2)},{d}^{(2)})-*convex at* *y* *with respect to* -h(x,y,p). *Then*

*Proof* Since f(\cdot ,v)+{(\cdot )}^{T}w{e}_{k} is higher-order *K*-(F,{\alpha}_{1},{\rho}^{(1)},{d}^{(1)})-convex with respect to g(u,v,r), we have

It follows from \lambda \in int{K}^{\ast}, {\lambda}^{T}{e}_{k}=1, and sublinearity of *F* that

As -\{f(x,\cdot )-{(\cdot )}^{T}z{e}_{k}\} is higher-order *K*-(G,{\alpha}_{2},{\rho}^{(2)},{d}^{(2)})-convex with respect to -h(x,y,p), therefore we get

Again, using \lambda \in int{K}^{\ast}, {\lambda}^{T}{e}_{k}=1, and sublinearity of *G*, we obtain

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

Now, since (x,y,\lambda ,z,p) is feasible for the primal problem (HNWP) and (u,v,\lambda ,w,r) is feasible for the dual problem (HNWD), {\alpha}_{1}(x,u)>0, by the dual constraint (5), the vector a={\alpha}_{1}(x,u)[{\mathrm{\nabla}}_{x}({\lambda}^{T}f)(u,v)+w+{\mathrm{\nabla}}_{r}({\lambda}^{T}g)(u,v,r)]\in {C}_{1}^{\ast}, and so from the hypothesis (A), we obtain

Similarly,

for the vector b=-{\alpha}_{2}(v,y)[{\mathrm{\nabla}}_{y}({\lambda}^{T}f)(x,y)-z+{\mathrm{\nabla}}_{p}({\lambda}^{T}h)(x,y,p)]\in {C}_{2}^{\ast}.

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}w\leqq S(x\mid D), {v}^{T}z\leqq S(v\mid E) and {\lambda}^{T}{e}_{k}=1, the last inequality yields

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

or

It follows from \lambda \in int{K}^{\ast} and {\lambda}^{T}{e}_{k}=1 that

which contradicts (15). Hence the result. □

**Theorem 2** (Strong duality)

*Let* f:{S}_{1}\times {S}_{2}\to {R}^{k} *be a twice differentiable function and let* (\overline{x},\overline{y},\overline{\lambda},\overline{z},\overline{p}) *be a weak efficient solution of* (HNWP). *Suppose that*

(i) *the matrix* {\mathrm{\nabla}}_{pp}({\overline{\lambda}}^{T}h)(\overline{x},\overline{y},\overline{p}) *is nonsingular*,

(ii) *the vectors* \{{\mathrm{\nabla}}_{y}{f}_{1}(\overline{x},\overline{y}),\dots ,{\mathrm{\nabla}}_{y}{f}_{k}(\overline{x},\overline{y})\} *are linearly independent*,

(iii) *the vector* \{{\mathrm{\nabla}}_{y}({\overline{\lambda}}^{T}h)(\overline{x},\overline{y},\overline{p})-{\mathrm{\nabla}}_{p}({\overline{\lambda}}^{T}h)(\overline{x},\overline{y},\overline{p})+{\mathrm{\nabla}}_{yy}({\overline{\lambda}}^{T}f)(\overline{x},\overline{y})\overline{p}\}\notin span\{{\mathrm{\nabla}}_{y}{f}_{1}(\overline{x},\overline{y}),\dots ,{\mathrm{\nabla}}_{y}{f}_{k}(\overline{x},\overline{y})\}\mathrm{\setminus}\{0\},

(iv) {\mathrm{\nabla}}_{y}({\overline{\lambda}}^{T}h)(\overline{x},\overline{y},\overline{p})-{\mathrm{\nabla}}_{p}({\overline{\lambda}}^{T}h)(\overline{x},\overline{y},\overline{p})+{\mathrm{\nabla}}_{yy}({\overline{\lambda}}^{T}f)(\overline{x},\overline{y})\overline{p}=0 *implies* \overline{p}=0,

(v) ({\overline{\lambda}}^{T}h)(\overline{x},\overline{y},0)=({\overline{\lambda}}^{T}g)(\overline{x},\overline{y},0), {\mathrm{\nabla}}_{y}({\overline{\lambda}}^{T}h)(\overline{x},\overline{y},0)=0, {\mathrm{\nabla}}_{p}({\overline{\lambda}}^{T}h)(\overline{x},\overline{y},0)=0, {\mathrm{\nabla}}_{x}({\overline{\lambda}}^{T}h)(\overline{x},\overline{y},0)={\mathrm{\nabla}}_{r}({\overline{\lambda}}^{T}g)(\overline{x},\overline{y},0) *and*

(vi) *K* *is a closed convex pointed cone with* {R}_{+}^{k}\subseteq K.

*Then*

(I) *there exists* \overline{w}\in D *such that* (\overline{x},\overline{y},\overline{\lambda},\overline{w},\overline{r}=0) *is feasible for* (HNWD), *and*

(II) G(\overline{x},\overline{y},\overline{\lambda},\overline{p})=H(\overline{x},\overline{y},\overline{\lambda},\overline{r}).

*Also*, *if the hypotheses of Theorem* 1 *are satisfied for all feasible solutions of* (HNWP) *and* (HNWD), *then* (\overline{x},\overline{y},\overline{\lambda},\overline{w},\overline{r}=0) *is an efficient solution for* (HNWD).

*Proof* Since (\overline{x},\overline{y},\overline{\lambda},\overline{z},\overline{p}) is a weak efficient solution of (HNWP), by the Fritz John necessary optimality conditions [33], there exist \overline{\alpha}\in {K}^{\ast}, \overline{\beta}\in {C}_{2}, \overline{\eta}\in R, such that the following conditions are satisfied at (\overline{x},\overline{y},\overline{\lambda},\overline{z},\overline{p}):

From (18) and nonsingularity of {\mathrm{\nabla}}_{pp}({\overline{\lambda}}^{T}h)(\overline{x},\overline{y},\overline{p}), we have

Also, (19) is equivalent to

If \overline{\alpha}=0, then (25) yields \overline{\beta}=0. Further, the above equality gives \overline{\eta}{e}_{k}=0 or \overline{\eta}=0. Consequently, (\overline{\alpha},\overline{\beta},\overline{\eta})=0, contradicting (24). Hence, \overline{\alpha}\ne 0.

Since {R}_{+}^{k}\subseteq K\Rightarrow {K}^{\ast}\subseteq {R}_{+}^{k}, therefore \overline{\alpha}\in {K}^{\ast}\Rightarrow \overline{\alpha}\geqq 0.

But \overline{\alpha}\ne 0\Rightarrow \overline{\alpha}\ge 0 or

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

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

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

Since the vectors \{{\mathrm{\nabla}}_{y}{f}_{1}(\overline{x},\overline{y}),\dots ,{\mathrm{\nabla}}_{y}{f}_{k}(\overline{x},\overline{y})\} are linearly independent, therefore the above equation yields

From (28) in (25), we get

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

From the hypothesis (v), for \overline{r}=0, the above inequality yields

Let x\in {C}_{1}. Then x+\overline{x}\in {C}_{1}, and so (31) implies

Therefore,

Also, from (26), (30) and \overline{\beta}\in {C}_{2}, we obtain \overline{y}\in {C}_{2}. Thus (\overline{x},\overline{y},\overline{\lambda},\overline{w}=\overline{\gamma},\overline{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\overline{x} in (31), we get

From (22) and (30), ({\overline{\alpha}}^{T}{e}_{k})\overline{y}\in {N}_{E}(\overline{z}). Since {\overline{\alpha}}^{T}{e}_{k}>0, \overline{y}\in {N}_{E}(\overline{z}). Also, as *E* is a compact convex set in {R}^{m}, {\overline{y}}^{T}\overline{z}=S(\overline{y}\mid E).

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

Therefore, using (28), (33), (34) and the hypothesis (v), for \overline{r}=0, we get

that is, the two objective values are equal.

Now, let (\overline{x},\overline{y},\overline{\lambda},\overline{w},\overline{r}=0) be not an efficient solution of (HNWD), then there exists (\overline{u},\overline{v},\overline{\lambda},\overline{w},\overline{r}=0) feasible for (HNWD) such that

As {\overline{x}}^{T}[{\mathrm{\nabla}}_{x}({\overline{\lambda}}^{T}f)(\overline{x},\overline{y})+{\mathrm{\nabla}}_{r}({\overline{\lambda}}^{T}g)(\overline{x},\overline{y},\overline{r})]=-S(\overline{x}\mid D), {\overline{y}}^{T}[{\mathrm{\nabla}}_{y}({\overline{\lambda}}^{T}f)(\overline{x},\overline{y})+{\mathrm{\nabla}}_{p}({\overline{\lambda}}^{T}h)(\overline{x},\overline{y},\overline{p})]=S(\overline{y}\mid E) and from the hypothesis (v), for \overline{r}=0, we obtain

which contradicts Theorem 1. Hence, (\overline{x},\overline{y},\overline{\lambda},\overline{w},\overline{r}=0) is an efficient solution of (HNWD). □

**Theorem 3** (Converse duality)

*Let* f:{S}_{1}\times {S}_{2}\to {R}^{k} *be a twice differentiable function and let* (\overline{u},\overline{v},\overline{\lambda},\overline{w},\overline{r}) *be a weak efficient solution of* (HNWD). *Suppose that*

(i) *the matrix* {\mathrm{\nabla}}_{rr}({\overline{\lambda}}^{T}g)(\overline{u},\overline{v},\overline{r}) *is nonsingular*,

(ii) *the vectors* \{{\mathrm{\nabla}}_{x}{f}_{1}(\overline{u},\overline{v}),\dots ,{\mathrm{\nabla}}_{x}{f}_{k}(\overline{u},\overline{v})\} *are linearly independent*,

(iii) *the vector* \{{\mathrm{\nabla}}_{x}({\overline{\lambda}}^{T}g)(\overline{u},\overline{v},\overline{r})-{\mathrm{\nabla}}_{r}({\overline{\lambda}}^{T}g)(\overline{u},\overline{v},\overline{r})+{\mathrm{\nabla}}_{xx}({\overline{\lambda}}^{T}f)(\overline{u},\overline{v})\overline{r}\}\notin span\{{\mathrm{\nabla}}_{x}{f}_{1}(\overline{u},\overline{v}),\dots ,{\mathrm{\nabla}}_{x}{f}_{k}(\overline{u},\overline{v})\}\mathrm{\setminus}\{0\},

(iv) {\mathrm{\nabla}}_{x}({\overline{\lambda}}^{T}g)(\overline{u},\overline{v},\overline{r})-{\mathrm{\nabla}}_{r}({\overline{\lambda}}^{T}g)(\overline{u},\overline{v},\overline{r})+{\mathrm{\nabla}}_{xx}({\overline{\lambda}}^{T}f)(\overline{u},\overline{v})\overline{r}=0 *implies* \overline{r}=0,

(v) ({\overline{\lambda}}^{T}g)(\overline{u},\overline{v},0)=({\overline{\lambda}}^{T}h)(\overline{u},\overline{v},0), {\mathrm{\nabla}}_{x}({\overline{\lambda}}^{T}g)(\overline{u},\overline{v},0)=0, {\mathrm{\nabla}}_{r}({\overline{\lambda}}^{T}g)(\overline{u},\overline{v},0)=0, {\mathrm{\nabla}}_{y}({\overline{\lambda}}^{T}g)(\overline{u},\overline{v},0)={\mathrm{\nabla}}_{p}({\overline{\lambda}}^{T}h)(\overline{u},\overline{v},0) *and*

(vi) *K* *is a closed convex pointed cone with* {R}_{+}^{k}\subseteq K.

*Then*

(I) *there exists* \overline{z}\in E *such that* (\overline{u},\overline{v},\overline{\lambda},\overline{z},\overline{p}=0) *is feasible for* (HNWP), *and*

(II) G(\overline{u},\overline{v},\overline{\lambda},\overline{p})=H(\overline{u},\overline{v},\overline{\lambda},\overline{r}).

*Also*, *if the hypotheses of Theorem* 1 *are satisfied for all feasible solutions of* (HNWP) *and* (HNWD), *then* (\overline{u},\overline{v},\overline{\lambda},\overline{z},\overline{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}, ({\lambda}^{T}h)(x,y,p)=\frac{1}{2}{p}^{T}{\mathrm{\nabla}}_{yy}({\lambda}^{T}f)(x,y)p and ({\lambda}^{T}g)(u,v,r)=\frac{1}{2}{r}^{T}{\mathrm{\nabla}}_{xx}({\lambda}^{T}f)(u,v)r.

(i) For k=1, D=\{Ay:{y}^{T}Ay\leqq 1\}, E=\{Bx:{x}^{T}Bx\leqq 1\}, where *A* and *B* are positive semidefinite matrices, {({x}^{T}Ax)}^{1/2}=S(x\mid D) and {({y}^{T}By)}^{1/2}=S(y\mid E). 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 x\geqq 0 and v\geqq 0. 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 x\geqq 0 and v\geqq 0.

## 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,\alpha ,\rho ,d)-convex which is not higher-order *K*-*F*-convex has been illustrated. Weak, strong and converse duality theorems under higher-order *K*-(F,\alpha ,\rho ,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.

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

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DOI: https://doi.org/10.1186/1029-242X-2012-298

### Keywords

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