Debnath et al. Journal of Inequalities and Applications 2015, 2015:3 http://www.journalofinequalitiesandapplications.com/content/2015/1/3
RESEARCH
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Symmetric duality for a higher-order nondifferentiable multiobjective programming problem Indira P Debnath1 , Shiv K Gupta1* and Sumit Kumar2 * Correspondence:
[email protected] 1 Department of Mathematics, Indian Institute of Technology, Roorkee, 247 667, India Full list of author information is available at the end of the article
Abstract In this paper, a pair of Wolfe type higher-order nondifferentiable symmetric dual programs over arbitrary cones has been studied and then well-suited duality relations have been established considering K-F convexity assumptions. An example which satisfies the weak duality relation has also been depicted. MSC: 90C29; 90C30; 49N15 Keywords: symmetric duality; higher-order K-F convexity; multiobjective programming; support function; efficient solutions
1 Introduction Consider the following multiobjective programming problem: (P) K-minimize f (x) subject to x ∈ X = x ∈ S : –g(x) ∈ C , where S ⊂ Rn be open, f : S → Rk , g : S → Rm , K , and C are closed convex pointed cones with nonempty interiors in Rk and Rm , respectively. Several researchers have studied the duality relations for different dual problems of (P) under various generalized convexity assumptions. Chen [] considered a pair of symmetric higher-order Mond-Weir type nondifferentiable multiobjecive programming problems and established duality relations under higher-order F-convexity assumptions. Later on, Agarwal et al. [] have filled some of the gap in the work of Chen [] and proved a strong duality theorem for a Mond-Weir type multiobjective higher-order nondifferentiable symmetric dual program. Khurana [] considered a pair of Mond-Weir type symmetric dual multiobjective programs over arbitrary cones and established duality results under conepseudoinvex and strongly cone-pseudoinvex assumptions. Later on, Kim and Kim [] extended the results in Khurana [] to the nondifferentiable multiobjective symmetric dual problem. Gupta and Jayswal [] studied the higher-order Mond-Weir type multiobjective symmetric duality over cones using higher-order cone-preinvex and cone-pseudoinvex functions, which further extends some of the results in [, , ]. Agarwal et al. [] formulated a pair of Mond-Weir type nondifferentiable multiobjective higher-order symmetric dual programs over arbitrary cones and established duality ©2015 Debnath et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Debnath et al. Journal of Inequalities and Applications 2015, 2015:3 http://www.journalofinequalitiesandapplications.com/content/2015/1/3
theorems under higher-order K -F convexity assumptions. In the recent work of Suneja and Louhan [], the authors have considered Wolfe and Mond-Weir type differentiable symmetric higher-order dual pairs. The Mond-Weir type model studied in [] is similar to the problem considered in Gupta and Jayswal []. However, the strong duality result in [] is for arbitrary cones in Rk instead of only those cones which contain the nonnegative orthant of Rk as considered in []. In the present paper, a pair of Wolfe type higher-order multiobjective nondifferentiable symmetric dual program have been formulated and we established weak, strong, and converse duality theorems under K -F convexity assumptions. We also illustrate a nontrivial example of a function which satisfies the weak duality relation.
2 Definitions and preliminaries Let C ⊆ Rn and C ⊆ Rm be closed convex cones with nonempty interiors and let S and S be nonempty open sets in Rn and Rm , respectively such that C × C ⊆ S × S . For a real valued twice differentiable function f (x, y) defined on S × S , ∇x f (x, y) denotes the gradient vector of f with respect to x at (x, y), ∇xx f (x, y) denotes the Hessian matrix with respect to x at (x, y). Similarly, ∇y f (x, y), ∇xy f (x, y), and ∇yy f (x, y) are also defined. Definition . [] A point x¯ ∈ X is a weak efficient solution of (P) if there exists no x ∈ X such that f (¯x) – f (x) ∈ int K. Definition . [] A point x¯ ∈ X is an efficient solution of (P) if there exists no x ∈ X such that f (¯x) – f (x) ∈ K\{}. Definition . The positive dual cone K + of K is defined by K + = y : xT y for all x ∈ K . Definition . For all (x, u) ∈ S × S , a functional F : S × S × Rn → R is said to be sublinear with respect to the third variable, if (i) F(x, u; a + a ) F(x, u; a ) + F(x, u; a ) for all a , a ∈ Rn , (ii) F(x, u; βa) = βF(x, u; a), for all β ∈ R+ and for all a ∈ Rn . For convenience, we write F(x, u; a) = Fx,u (a). Definition . [] Let F : S × S × Rn → R be a sublinear functional with respect to the third variable. Also, let hi : S × Rn → R, i = , , . . . , k be a differentiable function. Then the function f : S × S → Rk is said to be higher-order K -F convex in the first variable at u ∈ S for fixed v ∈ S with respect to h, such that for x ∈ S , pi ∈ Rn , i = , , . . . , k,
f (x, v) – f (u, v) – Fx,u ∇x f (u, v) + ∇p h (u, p ) – h (u, p ) + pT ∇p h (u, p ) , . . . , fk (x, v) – fk (u, v) – Fx,u ∇x fk (u, v) + ∇pk hk (u, pk ) – hk (u, pk ) + pTk ∇pk hk (u, pk ) ∈ K.
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Definition . [] Let ϕ be a compact convex set in Rn . The support function of ϕ is defined by S(x|ϕ) = max xT y : y ∈ ϕ . The subdifferentiable of S(x|ϕ) is given by ∂S(x|ϕ) = z ∈ ϕ : zT x = S(x|ϕ) . For any set S ⊂ Rn , the normal cone to S at a point x ∈ S is defined by NS (x) = y ∈ Rn : yT (z – x) for all z ∈ S . For each i = , , . . . , k, let fi : S × S → R, hi : S × S × Rm → R and gi : S × S × Rn → R be differentiable functions. p = (p , p , . . . , pk ) and r = (r , r , . . . , rk ), for pi ∈ Rm and ri ∈ Rn , i = , , . . . , k. C+ and C+ are the positive dual cones of C and C , respectively. D and E are the compact convex sets in Rn and Rm , respectively. Also, we use the following notations: h(x, y, p) = h (x, y, p ), h (x, y, p ), . . . , hk (x, y, pk ) , g(u, v, r) = g (u, v, r ), g (u, v, r ), . . . , gk (u, v, rk ) , ∇p h(x, y, p) = ∇p h (x, y, p ), ∇p h (x, y, p ), . . . , ∇pk hk (x, y, pk ) , ∇r g(u, v, r) = ∇r g (u, v, r ), ∇r g (u, v, r ), . . . , ∇rk gk (u, v, rk ) , pT ∇p h(x, y, p) = pT ∇p h (x, y, p ), pT ∇p h (x, y, p ), . . . , pTk ∇pk hk (x, y, pk ) rT ∇r g(u, v, r) = rT ∇r g (u, v, r ), rT ∇r g (u, v, r ), . . . , rkT ∇rk gk (u, v, rk ) .
and
3 Problem formulation Consider the following pair of Wolfe type higher-order nondifferentiable multiobjective symmetric dual programs: (WHP)
K-minimize f (x, y) + h(x, y, p) + S(x|D)e – pT ∇p h(x, y, p) k
T λi ∇y fi (x, y) + ∇pi hi (x, y, pi ) e –y i=
subject to
k λi ∇y fi (x, y) + ∇pi hi (x, y, pi ) – z ∈ C+ , –
()
i=
λ = (λ , λ , . . . , λk ) ∈ int K + , (WHD)
λT e = ,
x ∈ C ,
K-maximize f (u, v) + g(u, v, r) – S(v|E)e – r T ∇r g(u, v, r) k
T λi ∇u fi (u, v) + ∇ri gi (u, v, ri ) e –u i=
subject to
z ∈ E,
()
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k λi ∇u fi (u, v) + ∇ri gi (u, v, ri ) + w ∈ C+ ,
()
i=
λ = (λ , λ , . . . , λk ) ∈ int K + ,
λT e = ,
v ∈ C ,
w ∈ D,
()
where e = (e , e , . . . , ek ) ∈ int K is fixed. Remark . If D = {} and E = {}, then our problems (WHP) and (WHD) become the problem studied in Suneja and Louhan []. Next, we will prove weak, strong, and converse duality results between (WHP) and (WHD). Theorem . (Weak duality) Let (x, y, λ, z, p) and (u, v, λ, w, r) be feasible solutions for (WHP) and (WHD), respectively. Assume the following conditions hold: (I) f (·, v) + (·)T we is higher-order K -F convex at u with respect to g(u, v, r) for fixed v, (II) –f (x, ·) + (·)T ze is higher-order K -G convex at y with respect to –h(x, y, p) for fixed x, (III) Rk+ ⊆ K , where F : S × S × Rn → R and G : S × S × Rm → R are the sublinear functionals with respect to the third variable and satisfy the following conditions: Fx,u (a) + uT a
for all a ∈ C+ ,
(A)
Gv,y (b) + bT y
for all b ∈ C+ .
(B)
Then
k f (u, v) + g(u, v, r) – S(v|E)e – r ∇r g(u, v, r) – u λi ∇u f (u, v) + ∇ri gi (u, v, ri ) e T
T
i=
– f (x, y) + h(x, y, p) + S(x|D)e – pT ∇p h(x, y, p) k λi ∇y f (x, y) + ∇pi hi (x, y, pi ) e –y T
i=
∈/ K\{}.
()
Proof We shall obtain the proof by contradiction. Let () not hold. Then
k λi ∇u f (u, v) + ∇ri gi (u, v, ri ) e f (u, v) + g(u, v, r) – S(v|E)e – r ∇r g(u, v, r) – u T
T
i=
– f (x, y) + h(x, y, p) + S(x|D)e – pT ∇p h(x, y, p) k λi ∇y f (x, y) + ∇pi hi (x, y, pi ) e –y T
i=
∈ K\{}.
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It follows from λ ∈ int K + and λT e = that
k λi fi (u, v) + gi (u, v, ri ) – riT ∇ri gi (u, v, ri ) – S(v|E) i=
k T λi ∇u fi (u, v) + ∇ri gi (u, v, ri ) –u i=
k – λi fi (x, y) + hi (x, y, pi ) – pTi ∇pi hi (x, y, pi ) i=
k + S(x|D) – y λi ∇y fi (x, y) + ∇pi hi (x, y, pi ) > . T
()
i=
Now, since f (·, v) + (·)T we is higher-order K -F convex at u with respect to g(u, v, r) for fixed v, we get
f (x, v) + xT we – f (u, v) – uT we – Fx,u ∇u f (u, v) + we + ∇r g (u, v, r ) – g (u, v, r ) + rT ∇r g (u, v, r ), . . . , fk (x, v) + xT wek – fk (u, v) – uT wek – Fx,u ∇u fk (u, v) + wek + ∇rk gk (u, v, rk ) – gk (u, v, rk ) + rkT ∇rk gk (u, v, rk ) ∈ K.
Using λ ∈ int K + and λT e = , it follows that k λi fi (x, v) – fi (u, v) – gi (u, v, ri ) + riT ∇ri gi (u, v, ri ) + xT w – uT w i=
k
λi Fx,u ∇u fi (u, v) + ∇ri gi (u, v, ri ) + wei .
i=
Since λ ∈ int K + ⊆ int Rk+ (by hypothesis (III)), hence λ > . Therefore, using () and sublinearity of F in the above expression, we obtain k λi fi (x, v) – fi (u, v) – gi (u, v, ri ) + riT ∇ri gi (u, v, ri ) + xT w – uT w i=
k
Fx,u λi ∇u fi (u, v) + ∇ri gi (u, v, ri ) + w .
i=
It follows from (A) and the dual constraint () that k λi fi (x, v) – fi (u, v) – gi (u, v, ri ) + riT ∇ri gi (u, v, ri ) + xT w – uT w i=
k
λi ∇u fi (u, v) + ∇ri gi (u, v, ri ) + w –uT i=
for a = (
k
i= λi [∇u fi (u, v) + ∇ri gi (u, v, ri )] + w)
∈ C+ .
()
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Similarly, using hypothesis (II), (B), λ > , (), (), and sublinearity of G, we obtain k λi fi (x, y) – fi (x, v) + hi (x, y, pi ) – pTi ∇pi hi (x, y, pi ) + vT z – yT z i=
k
λi ∇y fi (x, y) + ∇pi hi (x, y, pi ) – z y T
()
i=
for b = –( ki= λi [∇y fi (x, y) + ∇pi hi (x, y, pi )] – z) ∈ C+ . Now, adding () and (), we have k λi fi (x, y) – fi (u, v) – gi (u, v, ri ) + riT ∇ri gi (u, v, ri ) i=
+ hi (x, y, pi ) – pTi ∇pi hi (x, y, pi ) + xT w + vT z
–uT
k λi ∇u fi (u, v) + ∇ri gi (u, v, ri ) i=
+ yT
k λi ∇y fi (x, y) + ∇pi hi (x, y, pi ) . i=
Finally, it follows from xT w S(x|D) and vT z S(v|E) that k λi fi (x, y) + hi (x, y, pi ) – pTi ∇pi hi (x, y, pi ) i=
+ S(x|D) – yT
k λi ∇y fi (x, y) + ∇pi hi (x, y, pi ) i=
k λi fi (u, v) + gi (u, v, ri ) – riT ∇ri gi (u, v, ri ) i=
– S(v|E) – uT
k λi ∇u fi (u, v) + ∇ri gi (u, v, ri ) , i=
which contradicts (). Hence the result.
Example . Let k = , n = m = . Let S = S = R+ = {x ∈ R : x }, C = C = R+ , and K = {(x, y) ∈ R : x , y –x}. Then C+ = C+ = R+ and K + = {(x, y) ∈ R : x , x y}. Obviously, R+ ⊆ K . Let f : S × S → R, g : S × S × Rn → R and h : S × S × Rm → R be defined as f (x, y) = x – y , x ,
g(u, v, r) = (–r u, –r u)
and
h(x, y, p) = (p y, p y).
and S(v|E) = . Suppose (e , e ) = (, ) ∈ Let D = [, ] and E = {}. Then S(x|D) = x+|x| int K . Also, suppose the sublinear functionals F and G are defined as Fx,u (a) = xT a and
Gv,y (b) = bT v.
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Now, substituting the above defined expressions in the problems (WHP) and (WHD), we get x + |x| x + |x| + (λ – λ )y , x + + (λ – λ )y (EP) K-minimize x – y + subject to (λ – λ )y , x , K-maximize u – v – (λ + λ )u , u – (λ + λ )u λ > ,
(ED)
λ – λ > ,
λ + λ = ,
subject to (λ + λ )u + w , λ > ,
λ – λ > ,
λ + λ = ,
v ,
w ∈ [, ].
Now, we shall show that for the primal-dual pair (EP) and (ED), the hypotheses of Theorem . hold. (A.) f (·, v) + (·)T we is higher-order K -F convex at u = ∈ S with respect to g(u, v, r) for fixed v and for all x ∈ S , r , r ∈ R, and we have
f (x, v) + xT we – f (u, v) – uT we – Fx,u ∇u f (u, v) + we + ∇r g (u, v, r ) – g (u, v, r ) + rT ∇r g (u, v, r ), f (x, v) + xT we – f (u, v) – uT we – Fx,u ∇u f (u, v) + we + ∇r g (u, v, r ) – g (u, v, r ) + rT ∇r g (u, v, r ) = x , x ∈ K.
(A.) –f (x, ·) + (·)T ze is higher-order K -G convex at y = ∈ S with respect to –h(x, y, p) for fixed x and for all v ∈ S , p , p ∈ R, and we have
–f (x, v) + vT ze + f (x, y) – yT ze – Gv,y –∇y f (x, y) + ze – ∇p h (x, y, p ) + h (x, y, p ) – pT ∇p h (x, y, p ), –f (x, v) + vT ze + f (x, y) – yT ze – Gv,y –∇y f (x, y) + ze – ∇p h (x, y, p ) + h (x, y, p ) – pT ∇p h (x, y, p ) = v , ∈ K.
(A.) Fx,u (a) + uT a = (x + u)T a ,
∀a ∈ C+ and ∀x, u ∈ S ,
Gv,y (b) + bT y = (v + y)T b ,
∀b ∈ C+ and ∀v, y ∈ S .
The points (x, y, λ , λ , z, p , p ) = (, , , , , , ) and (u, v, λ , λ , w, r , r ) = (, , , , , , ) are feasible for the problems (EP) and (ED), respectively. These feasible points do
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satisfy the result of the weak duality theorem since k λi ∇u f (u, v) + ∇ri gi (u, v, ri ) e f (u, v) + g(u, v, r) – S(v|E)e – r ∇r g(u, v, r) – u
T
T
i=
– f (x, y) + h(x, y, p) + S(x|D)e – pT ∇p h(x, y, p) k λi ∇y f (x, y) + ∇pi hi (x, y, pi ) e –y T
i=
= (–, –) ∈/ K\{}. Theorem . (Strong duality) Let (¯x, y¯ , λ¯ , z¯ , p¯ ) be a weak efficient solution of (WHD). Let (I) the Hessian matrix ∇pi pi hi (¯x, y¯ , p¯ i ) for all i = , , . . . , k be positive or negative definite;
(II) p¯ i = , for some i ∈ {, , . . . , k} imply that ki= ξi ∇yy fi (¯x, y¯ )¯pi = for all ξ ∈ K + ;
k x, y¯ )¯pi ∈/ span{∇y fi (¯x, y¯ ) + ∇y hi (¯x, y¯ , p¯ i ), ∇y fi (¯x, y¯ ) + ∇pi hi (¯x, y¯ , p¯ i ), (III) i= ξi ∇yy fi (¯ i = , , . . . , k}\{}, for all ξ ∈ K + ; (IV) the set of vectors {∇y fi (¯x, y¯ ) : i = , , . . . , k} be linearly independent; (V) ∇y hi (¯x, y¯ , ) = = ∇pi hi (¯x, y¯ , ), hi (¯x, y¯ , ) = gi (¯x, y¯ , ), ∇x hi (¯x, y¯ , ) = ∇ri gi (¯x, y¯ , ), for all i = {, , . . . , k}. Then ¯ w, ¯ ∈ D such that (¯x, y¯ , λ, ¯ r¯ = ) is feasible for (WHD) and (I) there exists w (II) the objective values of (WHP) and (WHD) are equal. Also, if the hypotheses of Theorem . are satisfied for all feasible solutions of (WHP) and ¯ w, ¯ r¯ = ) is an efficient solution for (WHD). (WHD), then (¯x, y¯ , λ, ¯ z¯ , p¯ ) is a weak efficient solution for (WHP), by the Fritz John necessary Proof Since (¯x, y¯ , λ, optimality conditions [], there exist α = (α , α , . . . , αk ) ∈ K + , β ∈ C , and η ∈ R such that
k k λ¯ i β – α T e y¯ ∇xy fi (¯x, y¯ ) αi ∇x fi (¯x, y¯ ) + ∇x hi (¯x, y¯ , p¯ i ) + α T e γ + i=
+
i=
k
∇pi x hi (¯x, y¯ , p¯ i ) λ¯ i β – λ¯ i α T e y¯ – αi p¯ i
T (x – x¯ )
for all x ∈ C ,
()
i= k k T αi ∇y fi (¯x, y¯ ) + ∇y hi (¯x, y¯ , p¯ i ) – α e λ¯ i ∇y fi (¯x, y¯ ) + ∇pi hi (¯x, y¯ , p¯ i ) i=
i=
+
k
λ¯ i ∇yy fi (¯x, y¯ ) β – α T e y¯
i=
+
k
∇pi y hi (¯x, y¯ , p¯ i ) λ¯ i β – λ¯ i α T e y¯ – αi p¯ i = ,
T β – α T e y¯ ∇y fi (¯x, y¯ ) + ∇pi hi (¯x, y¯ , p¯ i ) + ηei = , i = , , . . . , k, ∇pi pi hi (¯x, y¯ , p¯ i ) λ¯ i β – λ¯ i α T e y¯ – αi p¯ i = , i = , , . . . , k,
()
i=
() ()
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β
T
k ¯λi ∇y fi (¯x, y¯ ) + ∇pi hi (¯x, y¯ , p¯ i ) – z¯ = ,
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i = , , . . . , k,
()
i=
ηT λ¯ T e – = ,
()
β ∈ NE (¯z),
()
γ ∈ D,
γ T x¯ = S(¯x|D),
()
(α, β, η) = .
()
Now, hypothesis (I) and () imply that λ¯ i β – λ¯ i α T e y¯ – αi p¯ i = ,
i = , , . . . , k.
()
Using () in (), we have k
k αi ∇yy fi (¯x, y¯ )¯pi = α T e λ¯ i ∇y fi (¯x, y¯ ) + ∇pi hi (¯x, y¯ , p¯ i )
i=
i=
–
k αi ∇y fi (¯x, y¯ ) + ∇y hi (¯x, y¯ , p¯ i ) ,
()
i=
which yields k
αi ∇yy fi (¯x, y¯ )¯pi ∈ span ∇y fi (¯x, y¯ ) + ∇y hi (¯x, y¯ , p¯ i ), ∇y fi (¯x, y¯ )
i=
+ ∇pi hi (¯x, y¯ , p¯ i ), i = , , . . . , k .
()
Now, we claim that p¯ i = for all i = , , . . . , k. On the contrary, suppose that for some i ∈ {, , . . . , k}, p¯ i = , then using hypothesis (II), we have k
αi ∇yy fi (¯x, y¯ )¯pi = .
()
i=
This contradicts hypothesis (III) (by () and ()). Hence, p¯ i = for all i = , , . . . , k.
()
Using () in (), we have λ¯ i β = λ¯ i (α T e)¯y, i = , , . . . , k. Since λ¯ ∈ int K + , λ¯ i = for at least one i, β = α T e y¯ .
()
It follows from () and () that ηei = , i = , , . . . , k, which from e ∈ int K implies η = . From (), (), and hypothesis (V), we get k
αi – α T e λ¯ i ∇y fi (¯x, y¯ ) = ,
i=
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which from hypothesis (IV) yields ¯ α = α T e λ.
()
Now, if α = , then α T e = . Therefore, from (), we get β = and hence, (α, β, η) = . This contradicts (). Thus α = . Since α ∈ K + and e ∈ int K , we have α T e > .
()
From () and (), we obtain y¯ =
β ∈ C . (α T e)
Further, using inequalities (), ()-() in (), we obtain
k λ¯ i ∇x fi (¯x, y¯ ) + ∇x hi (¯x, y¯ , p¯ i ) + γ
T (x – x¯ )
for all x ∈ C .
i=
For r¯ = , it follows from () and hypothesis (V) that
k λ¯ i ∇x fi (¯x, y¯ ) + ∇ri gi (¯x, y¯ , r¯i ) + γ
T (x – x¯ ) .
()
i=
Let x ∈ C . Then x¯ + x ∈ C and hence from (), we have
k λ¯ i ∇x fi (¯x, y¯ ) + ∇ri gi (¯x, y¯ , r¯i ) + γ
T x
for all x ∈ C .
i=
Therefore, [ ki= λ¯ i {∇x fi (¯x, y¯ ) + ∇ri gi (¯x, y¯ , r¯i )} + γ ] ∈ C+ . ¯ r¯ = ) is a feasible solution for the dual problem. Thus, (¯x, y¯ , λ¯ , γ = w, Consider x = and x = ¯x in (), we get
k λ¯ i ∇x fi (¯x, y¯ ) + ∇ri gi (¯x, y¯ , r¯i ) + γ
T x¯ = ,
i=
which implies that x¯ T
k λ¯ i ∇x fi (¯x, y¯ ) + ∇ri gi (¯x, y¯ , r¯i ) = –¯xT γ = –S(¯x|D).
()
i=
Now, () and () yield (α T e)¯y ∈ NE (¯z). Since α T e > , y¯ ∈ NE (¯z). Again as E is a compact convex set in Rm , y¯ T z¯ = S(¯y|E). Further, (), (), and () yield y¯ T
k λ¯ i ∇y fi (¯x, y¯ ) + ∇pi hi (¯x, y¯ , p¯ i ) = y¯ T z¯ = S(¯y|E). i=
()
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By hypothesis (V) for r¯ = , (), ()-(), we obtain k f (¯x, y¯ ) + h(¯x, y¯ , p¯ ) + S(¯x|D)e – p¯ ∇p h(¯x, y¯ , p¯ ) – y¯ λ¯ i ∇y fi (¯x, y¯ ) + ∇pi hi (¯x, y¯ , p¯ i ) e T
T
i=
= f (¯x, y¯ ) + g(¯x, y¯ , r¯ ) – S(¯y|E)e – r¯ T ∇r g(¯x, y¯ , r¯ ) – x¯ T
k λ¯ i ∇x fi (¯x, y¯ ) + ∇ri gi (¯x, y¯ , r¯i ) e. i=
Hence, the two objective values are equal. ¯ w, ¯ r¯ = ) be not an efficient solution of (WHD), then there exists a point Now, let (¯x, y¯ , λ, ¯ ˆ vˆ , λ, w, ˆ rˆ ) feasible for (WHD) such that (u,
k ˆ vˆ ) + g(u, ˆ vˆ , rˆ ) – S(ˆv|E)e – rˆ ∇r g(u, ˆ vˆ , rˆ ) – uˆ ˆ vˆ ) + ∇ri gi (u, ˆ vˆ , rˆi ) e λ¯ i ∇u fi (u, f (u, T
T
i=
– f (¯x, y¯ ) + g(¯x, y¯ , r¯ ) – S(¯y|E)e – r¯ T ∇r g(¯x, y¯ , r¯ ) k ¯ λi ∇x fi (¯x, y¯ ) + ∇ri gi (¯x, y¯ , r¯i ) e – x¯ T
i=
∈ K\{}. From (), (), and hypothesis (V) for r¯ = and p¯ = , we obtain k ˆ vˆ , rˆ ) – uˆ ˆ vˆ ) + ∇ri gi (u, ˆ vˆ , rˆi ) e ˆ vˆ ) + g(u, ˆ vˆ , rˆ ) – S(ˆv|E)e – rˆ ∇r g(u, λ¯ i ∇u fi (u, f (u,
T
T
i=
– f (¯x, y¯ ) + h(¯x, y¯ , p¯ ) + S(¯x|D)e – p¯ T ∇p h(¯x, y¯ , p¯ ) k ¯ λi ∇y fi (¯x, y¯ ) + ∇pi hi (¯x, y¯ , p¯ i ) e – y¯ T
i=
∈ K\{}, ¯ w, ¯ r¯ = ) is the efficient solution of which contradicts Theorem .. Hence, (¯x, y¯ , λ, (WHD). ¯ v¯ , λ¯ , w, ¯ r¯ ) be a weak efficient solution of (WHP). Theorem . (Converse duality) Let (u, Let ¯ v¯ , r¯i ) for all i = , , . . . , k be positive or negative definite; (I) the Hessian matrix ∇ri ri gi (u,
¯ v¯ )¯ri = for all ξ ∈ K + ; (II) r¯i = , for some i ∈ {, , . . . , k} implies that ki= ξi ∇uu fi (u,
k ¯ v¯ )¯ri ∈/ span{∇u fi (u, ¯ v¯ ) + ∇u gi (u, ¯ v¯ , r¯i ), ∇u fi (u, ¯ v¯ ) + ∇ri gi (u, ¯ v¯ , r¯i ), (III) i= ξi ∇uu fi (u, + i = , , . . . , k}\{}, for all ξ ∈ K ; ¯ v¯ ) : i = , , . . . , k} be linearly independent; (IV) the set of vectors {∇u fi (u, ¯ v¯ , ) = = ∇ri gi (u, ¯ v¯ , ), gi (u, ¯ v¯ , ) = hi (u, ¯ v¯ , ), ∇v gi (u, ¯ v¯ , ) = ∇pi hi (u, ¯ v¯ , ), for (V) ∇u gi (u, all i = {, , . . . , k}. Then
Debnath et al. Journal of Inequalities and Applications 2015, 2015:3 http://www.journalofinequalitiesandapplications.com/content/2015/1/3
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¯ z¯ , p¯ = ) is feasible for (WHP) and ¯ v¯ , λ, (I) there exists z¯ ∈ E such that (u, (II) the objective values of (WHP) and (WHD) are equal. Also, if the hypotheses of Theorem . are satisfied for all feasible solutions of (WHP) and ¯ z¯ , p¯ = ) is an efficient solution for (WHP). ¯ v¯ , λ, (WHD), then (u, Proof The proof follows along the lines of Theorem ..
Competing interests The authors declare that they have no competing interests. Authors’ contributions All the authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript. Author details 1 Department of Mathematics, Indian Institute of Technology, Roorkee, 247 667, India. 2 Faculty of Operations and Information Systems, Indian Institute of Management, Udaipur, 313 001, India. Acknowledgements The authors wish to thank the reviewer for her/his valuable and constructive suggestions, which have considerably improved the presentation of the paper. The first author is also grateful to the Ministry of Human Resource and Development, India for financial support to carry out this work. Received: 22 September 2014 Accepted: 3 December 2014 Published: 06 Jan 2015 References 1. Chen, X: Higher-order symmetric duality in nondifferentiable multiobjective programming problems. J. Math. Anal. Appl. 290, 423-435 (2004) 2. Agarwal, RP, Ahmad, I, Gupta, SK: A note on higher-order nondifferentiable symmetric duality in multiobjective programming. Appl. Math. Lett. 24, 1308-1311 (2011) 3. Khurana, S: Symmetric duality in multiobjective programming involving generalized cone-invex functions. Eur. J. Oper. Res. 165, 592-597 (2005) 4. Kim, MH, Kim, DS: Nondifferentiable symmetric duality for multiobjective programming with cone constraints. Eur. J. Oper. Res. 188, 652-661 (2008) 5. Gupta, SK, Jayswal, A: Multiobjective higher-order symmetric duality involving generalized cone-invex functions. Comput. Math. Appl. 60, 3187-3192 (2010) 6. Gulati, TR, Gupta, SK: Higher-order symmetric duality with cone constraints. Appl. Math. Lett. 22, 776-781 (2009) 7. Gulati, TR, Mehndiratta, G: Nondifferentiable multiobjective Mond-Weir type second-order symmetric duality over cones. Optim. Lett. 4, 293-309 (2010) 8. Agarwal, RP, Ahmad, I, Jayswal, A: Higher-order symmetric duality in nondifferentiable multiobjective programming problems involving generalized cone convex functions. Math. Comput. Model. 52, 1644-1650 (2010) 9. Suneja, SK, Louhan, P: Higher-order symmetric duality under cone-invexity and other related concepts. J. Comput. Appl. Math. 255, 825-836 (2014) 10. Gupta, SK, 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) 11. Suneja, SK, Aggarwal, S, Davar, S: Multiobjective symmetric duality involving cones. Eur. J. Oper. Res. 141, 471-479 (2002) 10.1186/1029-242X-2015-3 Cite this article as: Debnath et al.: Symmetric duality for a higher-order nondifferentiable multiobjective programming problem. Journal of Inequalities and Applications 2015, 2015:3
