7_619_saini.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 4, No. 3, 2011, 266-275

ISSN 1307-5543 – www.ejpam.com

Higher-order (F,α,β ,ρ, d)-Convexity and its Application in Frac-

tional Programming

T. R. Gulati1,∗, Himani Saini2

1 Department of Mathematics, Professor, Indian Institute of Technology, Roorkee-247 667, India.
2 Applied Mathematics Division, Scientist, Vikram Sarabhai Space Centre, Indian Space Research

Organisation, Thiruvananthapuram-695 022, India.

Abstract. In this paper we introduce the concept of higher-order (F,α,β ,ρ, d)-convexity with respect

to a differentiable function K . Based on this generalized convexity, sufficient optimality conditions

for a nonlinear programming problem (NP) are obtained. Duality relations for Mond-Weir and Wolfe

duals of (NP) have also been discussed. These duality results are then applied to nonlinear fractional

programming problems.

2000 Mathematics Subject Classifications: 90C30, 90C32, 90C46.

Key Words and Phrases: Higher-order (F,α,β ,ρ, d)-convexity; Sufficiency; Optimality conditions;

Duality; Fractional programming.

1. Introduction

Optimality conditions and duality in nonlinear programming were first investigated under

convexity assumptions. As they have played an important role in the development of mathe-

matical programming, several authors have generalized the concept of convexity under which

sufficient optimality conditions and duality theorems holds. Hanson [2] defined invex func-

tions.

The concept of (F,ρ)-convexity was introduced by Pareda [7] as an extension of F -

convexity [3] and ρ-convexity [8]. Liang et al. [4] introduced a unified formulation of

generalized convexity called (F,α,ρ, d)-convexity and obtained some optimality conditions

and duality results for nonlinear fractional programming problems.

Recently, Yuan et al. [9] introduced the concept of (C ,α,ρ, d)-convexity which is the gen-

eralization of (F,α,ρ, d)-convexity, and proved optimality conditions and duality theorems for

∗Corresponding author.

Email addresses: trgmaiitr�rediffmail.
om (T. Gulati), himanisaini.iitr�gmail.
om (H. Saini)

http://www.ejpam.com 266 c© 2011 EJPAM All rights reserved.



T. Gulati, H. Saini / Eur. J. Pure Appl. Math, 4 (2011), 266-275 267

non-differentiable minimax fractional programming problems.

This paper is organized as follows. In Section 2, we define higher-order (F,α,β ,ρ, d)-

convex functions. Under this generalized convexity, we obtain sufficient optimality conditions

for a nonlinear programming problem (NP) in Section 3. In Section 4 we establish weak and

strong duality for Mond-Weir dual program for (NP). An application for a fractional program-

ming problem (FP) has been discussed in Section 5. In the last Section we present Wolfe

duality for (NP) and (FP).

2. Definitions and Preliminaries

We consider the following nonlinear programming problem:

(NP) Minimize φ(x),

Subject to h(x)≦ 0, x ∈ X ,

where X is an open subset of Rn and the functionsφ : X 7→ R and h= (h1,h2, . . . ,hm) : X 7→ Rm

are differentiable on X . Let S = {x ∈ X : h(x)≦ 0} denote the set of all feasible solutions for

(NP).

Definition 1. A functional F : X × X ×Rn 7→ R is said to be sublinear in the third variable, if for

all x , x̄ ∈ X ,

(i) F(x , x̄ ;ξ1 + ξ2)≦ F(x , x̄;ξ1) + F(x , x̄ ;ξ2), for all ξ1,ξ2 ∈ Rn ; and

(ii) F(x , x̄ ;αa) = αF(x , x̄ ; a) for all α ∈ R+, and a ∈ Rn.

From (ii), it is clear that F(x , x̄ ; 0) = 0.

Based on the concept of the sublinear functional, we now introduce the class of higher-

order (F,α,β ,ρ, d)-convex functions as follows:

Let X ⊆ Rn be an open set. Let φ : X 7→ R, K : X × Rn 7→ R be differentiable functions,

F : X × X ×Rn 7→ R be a sublinear functional in the third variable and d : X × X 7→ R. Further,

let α, β : X × X 7→ R+ \ {0} and ρ ∈ R.

Definition 2. The function φ is said to be higher-order (F,α,β ,ρ, d)-convex at x̄ with respect to

K , if for all x ∈ X and p ∈ Rn,

φ(x)−φ( x̄) ≧ F(x , x̄ ;α(x , x̄){∇φ( x̄) +∇pK( x̄ , p)})

+ β(x , x̄){K( x̄ , p)− pT∇pK( x̄ , p)}+ρd2(x , x̄).

Remark 1. Let K( x̄ , p) = 0.

(i) Then the above definition becomes that of (F,α,ρ, d)-convex function introduced by Liang

et al. [4].



T. Gulati, H. Saini / Eur. J. Pure Appl. Math, 4 (2011), 266-275 268

(ii) If α(x , x̄) = 1, we obtain the definition of (F,ρ)-convex function given by Pareda [7].

(iii) If α(x , x̄) = 1,ρ = 0 and F(x , x̄ ;∇φ( x̄)) = ηT (x , x̄)∇φ( x̄) for a certain map

η : X × X −→ Rn, then (F,α,β ,ρ, d)-convexity reduces to the invexity in Hanson [2].

(iv) If F is convex with respect to the third argument, then we obtain the definition of (F,α,ρ, d)-

convex function introduced by Yuan et al. [9].

Remark 2. Let β(x , x̄) = 1.

(i) If K( x̄ , p) = 1

2
pT∇2φ( x̄)p, then the above inequality reduces to the definition of second

order (F,α,ρ, d)-convex function given by Ahmad and Husain [1].

(ii) If α(x , x̄) = 1,ρ = 0, K( x̄ , p) = 1

2
pT∇2φ( x̄)p and F(x , x̄; a) = ηT (x , x̄)a, where

η : X × X −→ Rn, the above definition becomes that of η-bonvexity introduced by Pandey

[6].

Proposition 1 (Kuhn-Tucker Necessary Optimality Conditions [see 5]). Let x̄ ∈ S be an opti-

mal solution of (NP) and let h satisfy a constraint qualification [Theorem 7.3.7 in 5]. Then there

exists a v̄ ∈ Rm such that

∇φ( x̄) +∇h( x̄)v̄ = 0, (1)

v̄T h( x̄) = 0, (2)

v̄ ≧ 0, h( x̄)≦ 0, (3)

where ∇h( x̄) denotes the n×m matrix [∇h1( x̄),∇h2( x̄), . . . ,∇hm( x̄)].

3. Sufficient Optimality Conditions

In this section, we establish Kuhn-Tucker sufficient optimality conditions for (NP) under

(F,α,β ,ρ, d)-convexity assumptions.

Theorem 1. Let x̄ ∈ S and v̄ ∈ Rm satisfy (1)-(3). If

(i) φ is higher-order (F,α,β ,ρ1, d)-convex at x̄ with respect to K ,

(ii) v̄T h is higher-order (F,α,β ,ρ2, d)-convex at x̄ with respect to −K, and

(iii) ρ1 +ρ2 ≧ 0,

then x̄ is an optimal solution of the problem (NP).

Proof. Let x̄ ∈ S. Since φ is higher-order (F,α,β ,ρ1, d)-convex at x̄ with respect to K , for

all x ∈ S, we have

φ(x)−φ( x̄) ≧ F
�

x , x̄;α(x , x̄)[∇φ( x̄) +∇pK( x̄ , p)]
�

+ β(x , x̄)
�

K( x̄ , p)− pT∇pK( x̄ , p)
�

+ρ1d2(x , x̄). (4)



T. Gulati, H. Saini / Eur. J. Pure Appl. Math, 4 (2011), 266-275 269

Using (1) , we get

φ(x)−φ( x̄) ≧ F
�

x , x̄;α(x , x̄)[−∇h( x̄)v̄+∇pK( x̄ , p)]
�

+ β(x , x̄)
�

K( x̄ , p)− pT∇pK( x̄ , p)
�

+ρ1d2(x , x̄). (5)

Also, v̄T h is higher-order (F,α,β ,ρ2, d)-convex at x̄ with respect to −K . Therefore

v̄T h(x)− v̄T h( x̄) ≧ F
�

x , x̄;α(x , x̄)[∇v̄T h( x̄)−∇pK( x̄ , p)]
�

− β(x , x̄)
�

K( x̄ , p)− pT∇pK( x̄ , p)
�

+ρ2d2(x , x̄). (6)

Since v̄T h( x̄) = 0, v̄ ≧ 0 and h(x)≦ 0, we get

0 ≧ F
�

x , x̄;α(x , x̄)[∇v̄T h( x̄)−∇pK( x̄ , p)]
�

− β(x , x̄)
�

K( x̄ , p)− pT∇pK( x̄ , p)
�

+ρ2d2(x , x̄). (7)

Adding the inequalities (5) and (7) , we obtain

φ(x)−φ( x̄)≧ (ρ1+ρ2)d
2(x , x̄),

which by hypothesis (iii) implies,

φ(x)≧ φ( x̄).

Hence x̄ is an optimal solution of the problem (NP).

4. Mond Weir Duality

In this section, we establish weak and strong duality theorems for the following Mond

Weir dual (MD) for (NP):

(MD) Maximize φ(u),

Subject to ∇φ(u) +∇h(u)v = 0, (8)

vT h(u)≧ 0, (9)

u ∈ X , v ≧ 0, v ∈ Rm. (10)

Theorem 2 (Weak Duality). Let x and (u, v) be feasible solutions of (NP) and (MD) respectively.

Let

(i) φ be higher-order (F,α,β ,ρ1, d)-convex at u with respect to K,

(ii) vT h be higher-order (F,α,β ,ρ2, d)-convex at u with respect to −K, and

(iii) ρ1 +ρ2 ≧ 0.

Then

φ(x)≧ φ(u).



T. Gulati, H. Saini / Eur. J. Pure Appl. Math, 4 (2011), 266-275 270

Proof. By hypothesis (i), we have

φ(x)−φ(u) ≧ F
�

x ,u;α(x ,u)[∇φ(u) +∇pK(u, p)]
�

+ β(x ,u)
�

K(u, p)− pT∇pK(u, p)
�

+ρ1d2(x ,u). (11)

Also hypothesis (ii) yields

vT h(x)− vT h(u) ≧ F
�

x ,u;α(x ,u)[∇vTh(u)−∇pK(u, p)]
�

− β(x ,u)
�

K(u, p)− pT∇pK(u, p)
�

+ρ2d2(x ,u).

By (9), (10) and h(x)≦ 0, it follows that

0 ≧ F
�

x ,u;α(x ,u)[∇vT h(u)−∇pK(u, p)]
�

− β(x ,u)
�

K(u, p)− pT∇pK(u, p)
�

+ρ2d2(x ,u). (12)

Adding the inequalities (11), (12) and applying the properties of sublinear functional, we

obtain

φ(x)−φ(u) ≧ F
�

x ,u;α(x ,u)[∇φ(u) +∇vT h(u)]
�

+ρ1d2(x ,u) +ρ2d2(x ,u)

which in view of (8) implies

φ(x)−φ(u) ≧ (ρ1+ρ2)d
2(x ,u).

Using hypothesis (iii) in the above inequality, we get

φ(x)≧ φ(u).

Remark 3. A constraint qualification is not required to establish weak duality. It has been

erroneously assumed in Theorem 3.4 in [4].

Theorem 3 (Strong Duality). Let x̄ be an optimal solution of the problem (NP) and let h satisfy

a constraint qualification. Further, let Theorem 2 hold for the feasible solution x̄ of (NP) and

all feasible solutions (u, v) of (MD). Then there exists a v̄ ∈ Rm
+ such that ( x̄ , v̄) is an optimal

solution of (MD).

Proof. Since x̄ is an optimal solution for the problem (NP) and h satisfies a constraint

qualification, by Proposition 1 there exists a v̄ ∈ Rm
+ such that the Kuhn-Tucker conditions, (1)

- (3) hold. Hence ( x̄ , v̄) is feasible for (MD).

Now let (u, v) be any feasible solution of (MD). Then by weak duality (Theorem 2), we

have

φ( x̄)≧ φ(u).

Therefore ( x̄ , v̄) in an optimal solution of (MD).



T. Gulati, H. Saini / Eur. J. Pure Appl. Math, 4 (2011), 266-275 271

5. Application in Fractional Programming

If φ : X → R is defined by

φ(x) =
f (x)

g(x)
,

where f , g : X → R, f (x) ≧ 0 and g(x) > 0 on X , then the nonlinear programming problem

(NP) becomes the following fractional programming problem (FP):

(FP) Minimize
f (x)

g(x)

Subject to h(x)≦ 0, x ∈ X .

We now prove the following result, which gives higher-order (F, ᾱ, β̄ ,ρ, d̄)-convexity of the

ratio function f (x)/g(x).

Theorem 4. Let f (x) and −g(x) be higher-order (F,α,β ,ρ, d)-convex at x̄ with respect to the

same function K. Then the fractional function
f (x)

g(x)
is higher-order (F, ᾱ, β̄ ,ρ, d̄)-convex at x̄ with

respect to K̄, where

ᾱ(x , x̄) = α(x , x̄)
g( x̄)

g(x)

β̄(x , x̄) = β(x , x̄)
g( x̄)

g(x)
,

K̄( x̄ , p) =

�

1

g( x̄)
+

f ( x̄)

g2( x̄)

�

K( x̄ , p),

d̄(x , x̄) =

�

1

g(x)
+

f ( x̄)

g(x)g( x̄)

�
1

2

d(x , x̄).

Proof. Since f (x) and −g(x) are higher-order (F,α,β ,ρ, d)-convex at x̄ with respect to

the same function K , we have

f (x)− f ( x̄) ≧ F(x , x̄;α(x , x̄){∇ f ( x̄) +∇pK( x̄ , p)})

+ β(x , x̄){K( x̄ , p)− pT∇pK( x̄ , p)}+ρd2(x , x̄)

and

−g(x)+ g( x̄)≧ F(x , x̄ ;α(x , x̄){−∇g( x̄) +∇pK( x̄ , p)})

+ β(x , x̄){K( x̄ , p)− pT∇pK( x̄ , p)}+ρd2(x , x̄).

Also
f (x)

g(x)
−

f ( x̄)

g( x̄)
=

1

g(x)

�

f (x)− f ( x̄)
�

+
f ( x̄)

g(x)g( x̄)

�

−g(x)+ g( x̄)
�

.



T. Gulati, H. Saini / Eur. J. Pure Appl. Math, 4 (2011), 266-275 272

Using the above inequalities and sublinearity of F , we get

f (x)

g(x)
−

f ( x̄)

g( x̄)
≧

1

g(x)
F(x , x̄;α(x , x̄){∇ f ( x̄)+∇pK( x̄ , p)})

+
1

g(x)

�

β(x , x̄){K( x̄ , p)− pT∇pK( x̄ , p)}+ρd2(x , x̄)
�

+
f ( x̄)

g(x)g( x̄)
F(x , x̄ ;α(x , x̄){−∇g( x̄) +∇pK( x̄ , p)})

+
f ( x̄)

g(x)g( x̄)

�

β(x , x̄){K( x̄ , p)− pT∇pK( x̄ , p)}+ρd2(x , x̄)
�

.

= F(x , x̄ ;
α(x , x̄)

g(x)
{∇ f ( x̄) +∇pK( x̄ , p)})

+F(x , x̄;α(x , x̄)
f ( x̄)

g(x)g( x̄)
{−∇g( x̄) +∇pK( x̄ , p)})

+β(x , x̄)

�

1

g(x)
+

f ( x̄)

g(x)g( x̄)

�

{K( x̄ , p)− pT∇pK( x̄ , p)}

+ρ

�

1

g(x)
+

f ( x̄)

g(x)g( x̄)

�

d2(x , x̄).

= F(x , x̄ ;α(x , x̄)
g( x̄)

g(x)
{∇

f ( x̄)

g( x̄)
+

�

1

g( x̄)
+

f ( x̄)

g2( x̄)

�

∇pK( x̄ , p)})

+β(x , x̄)
g( x̄)

g(x)

�

1

g( x̄)
+

f ( x̄)

g2( x̄)

�

{K( x̄ , p)− pT∇pK( x̄ , p)}

+ρ

�

1

g(x)
+

f ( x̄)

g(x)g( x̄)

�

d2(x , x̄).

Therefore,

f (x)

g(x)
−

f ( x̄)

g( x̄)
≧ F

�

x , x̄; ᾱ(x , x̄)

�

∇
f ( x̄)

g( x̄)
+∇pK̄( x̄ , p)

��

+ β̄(x , x̄){K̄( x̄ , p)− pT∇pK̄( x̄ , p)}+ρd̄2(x , x̄),

i.e.,
f (x)

g(x)
is higher-order (F, ᾱ, β̄ ,ρ, d̄)-convex at x̄ with respect to K̄ .

In view of Theorem 4, the results of Section 4 lead to the following duality relations

between (FP) and its Mond-Weir dual (MFD).

(MFD) Maximize
f (u)

g(u)

Subject to ∇
�

f (u)

g(u)

�

+∇h(u)v = 0,

vT h(u)≧ 0,

u ∈ X , v ≧ 0, v ∈ Rm.



T. Gulati, H. Saini / Eur. J. Pure Appl. Math, 4 (2011), 266-275 273

Theorem 5 (Weak Duality). Let x and (u, v) be feasible solutions of (FP) and (MFD) respectively.

Let

(i) f and −g be higher-order (F,α,β ,ρ1, d)-convex at u with respect to K,

(ii) vT h be higher-order (F, ᾱ, β̄ ,ρ2, d̄)-convex at u with respect to −K̄, where ᾱ, β̄ , K̄ and d̄

are as given in Theorem 4, and

(iii) ρ1 +ρ2 ≧ 0.

Then
f (x)

g(x)
≧

f (u)

g(u)
.

Theorem 6 (Strong Duality). Let x̄ be an optimal solution of the problem (FP) and let h satisfy

a constraint qualification. Further, let Theorem 5 hold for the feasible solution x̄ of (FP) and

all feasible solutions (u, v) of (MFD). Then there exists a v̄ ∈ Rm
+ such that ( x̄ , v̄) is an optimal

solution of (MFD).

6. Wolfe Duality

The Wolfe dual of (NP) and (FP) are respectively

(WD) Maximize φ(u) + vT h(u)

Subject to ∇φ(u) +∇h(u)v = 0,

u ∈ X , v ≧ 0, v ∈ Rm,

(WFD) Maximize
f (u)

g(u)
+ vT h(u)

Subject to ∇
�

f (u)

g(u)

�

+∇h(u)v = 0,

u ∈ X , v ≧ 0, v ∈ Rm.

Now we state duality relations for the primal problems (NP) and (FP) and their Wolfe duals

(WD) and (WFD) respectively. Their proofs follow as in Section 4.

Theorem 7 (Weak Duality). Let x and (u, v) be feasible solutions of (NP) and (WD) respectively.

Let

(i) φ be higher-order (F,α,β ,ρ1, d)-convex at u with respect to K,

(ii) vT h be higher-order (F,α,β ,ρ2, d)-convex at u with respect to −K, and

(iii) ρ1 +ρ2 ≧ 0.

Then

φ(x)≧ φ(u) + vT h(u).



REFERENCES 274

Theorem 8 (Strong Duality). Let x̄ be an optimal solution of the problem (NP) and let h satisfy

a constraint qualification. Further, let Theorem 7 hold for the feasible solution x̄ of (NP) and

all feasible solutions (u, v) of (WD). Then there exists a v̄ ∈ Rm
+ such that ( x̄ , v̄) is an optimal

solution of (WD) and the optimal objective function values of (NP) and (WD) are equal.

Theorem 9 (Weak Duality). Let x and (u, v) be feasible solutions of (FP) and (WFD) respectively.

Let

(i) f and −g be higher-order (F,α,β ,ρ1, d)-convex at u with respect to K,

(ii) vT h be higher-order (F, ᾱ, β̄ ,ρ2, d̄)-convex at u with respect to −K̄, where ᾱ, β̄ , K̄ and d̄

are as given in Theorem 4, and

(iii) ρ1 +ρ2 ≧ 0.

Then
f (x)

g(x)
≧

f (u)

g(u)
+ vT h(u).

Theorem 10 (Strong Duality). Let x̄ be an optimal solution of the problem (FP) and let h satisfy

a constraint qualification. Further, let Theorem 9 hold for the feasible solution x̄ of (FP) and all

feasible solutions (u, v) of (WFD). Then there exists a v̄ ∈ Rm
+ such that ( x̄ , v̄) is an optimal

solution of (WFD) and the optimal objective function values of (FP) and (WFD) are equal.

7. Conclusion

In this paper a new concept of generalized convexity has been introduced. Under this

generalized convexity we establish sufficient optimality conditions and duality results for a

nonlinear programming problem.These duality relations lead to duality in nonlinear fractional

programming.

ACKNOWLEDGEMENTS The second author is thankful to the University Grants Commis-

sion, New Delhi (India) for providing financial support during this work.

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