Duality for Nondifferentiable Multiobjective Higher-Order Symmetric Programs over Cones Involving Generalized (F, Α, Ρ, D) - Convexity

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.


Introduction
Mangasarian [] 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 [], several researchers [-] 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 [] and Pardalos et al. []. Recently, Chinchuluun and Pardalos [] 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 [] 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 [] proved duality theorems for a pair of Wolfe type higher-order nondifferentiable symmetric dual programs. Ahmad et al. [] formulated a general Mond-Weir type higher-order dual for a nondifferentiable multiobjective programming problem and established usual duality results.
Gulati and Geeta [] pointed out certain omissions in some papers on symmetric duality in multiobjective programming and discussed their corrective measures. Later on, Ahmad and Husain [] and Gulati et al. [] formulated second-order multiobjective symmetric dual programs with cone constraints and established duality results under second-http://www.journalofinequalitiesandapplications.com/content/2012/1/298 Definition  [] A pointx ∈ X  is an efficient solution of (P) if there exists no other x ∈ X  such that Definition  [, , ] 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 ∈ R n such that The subdifferential of S(x | D) is given by For any set S ⊂ R n , the normal cone to S at a point x ∈ 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 | D) = x T y, or equivalently, x is in ∂S(y | D).
for all a  , a  ∈ 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 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 [].
Definition  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 x ∈ S and q ∈ R n , we have
Definition  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 x ∈ S and q ∈ R n , there exist vector ρ ∈ R k , a real valued function α : S × S → R + \{} and d : . . , k, the definition of higher-order K -(F, α, ρ, d)-convexity reduces to second-order (F, ρ)-convexity given by Srivastava and Bhatia []. (ii) If K = R + , α(x, u) =  and ρ = , then Definition  reduces to higher-order where η is a function from S × S to R n , then the above definition becomes second-order η-convexity given in [].
Next, we illustrate a nontrivial example of higher-order K -(F, α, ρ, d)-convex functions which are not higher-order K -F-convex.
To show ψ is higher-order (F, α, ρ, d)-convex, we need to prove which for ρ  = - and ρ  = - gives and But Hence, L  . Therefore, ψ is higher-order K -(F, α, ρ, 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 Since In fact, M <  for all x, u ∈ X as can be seen from Figure . Therefore, ψ is not higher-order K -F-convex with respect to ζ .

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

Dual problem (HNWD) K -maximize
where (i) C  and C  are closed convex cones with nonempty interiors in R n and R m , respectively, functions, e k = (, . . . , ) 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(x | D) and S(v | E) are the support functions of D and E, respectively.
Remark  The problems (HNWP) and (HNWD) stated above are nondifferentiable because their objective function contains the support function S(x | D) and S(v | E).
We now prove the following duality results for the pair of problems (HNWP) and (HNWD). http://www.journalofinequalitiesandapplications.com/content/2012/1/298 Theorem  (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: () )-convex at u with respect to g(u, v, r), and (iii) -{f (x, ·) -(·) T ze k } is higher-order K -(G, α  , ρ () , d () )-convex at y with respect to -h(x, y, p). Then It follows from λ ∈ int K * , λ T e k = , and sublinearity of F that As -{f (x, ·) -(·) T ze k } is higher-order K -(G, α  , ρ () , d () )-convex with respect to -h(x, y, p), therefore we get Again, using λ ∈ int K * , λ T e k = , and sublinearity of G, we obtain Further, adding the inequalities () and (), we have 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  (x, u) > , by the dual constraint (), the vector ∈ C *  , and so from the hypothesis (A), we obtain Similarly, y, p)] ∈ C *  . Using (), () and the hypothesis (i) in (), we have Further, substituting the values of a and b, we have In view of the fact that x T w S(x | D), v T z S(v | E) and λ T e k = , the last inequality yields Now, suppose contrary to the result that () does not hold, that is, y, λ, p) -H(u, v, λ, r) ∈ -K\{}, http://www.journalofinequalitiesandapplications.com/content/2012/1/298 It follows from λ ∈ int K * and λ T e k =  that which contradicts (). Hence the result.

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 [], Gulati et al. [] and Yang et al. [, ], 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.