Open Access

Optimality conditions of E-convex programming for an E-differentiable function

  • Abd El-Monem A Megahed1, 2Email author,
  • Hebaa G Gomma3,
  • Ebrahim A Youness4 and
  • Abou-Zaid H El-Banna4
Journal of Inequalities and Applications20132013:246

https://doi.org/10.1186/1029-242X-2013-246

Received: 3 May 2012

Accepted: 30 April 2013

Published: 16 May 2013

Abstract

In this paper we introduce a new definition of an E-differentiable convex function, which transforms a non-differentiable function to a differentiable function under an operator E : R n R n . By this definition, we can apply Kuhn-Tucker and Fritz-John conditions for obtaining the optimal solution of mathematical programming with a non-differentiable function.

Keywords

E-convex set E-convex function semi E-convex function E-differentiable function

1 Introduction

The concepts of E-convex sets and an E-convex function have been introduced by Youness in [1, 2], and they have some important applications in various branches of mathematical sciences. Youness in [1] introduced a class of sets and functions which is called E-convex sets and E-convex functions by relaxing the definition of convex sets and convex functions. This kind of generalized convexity is based on the effect of an operator E : R n R n on the sets and the domain of the definition of functions. Also, in [2] Youness discussed the optimality criteria of E-convex programming. Xiusu Chen [3] introduced a new concept of semi E-convex functions and discussed its properties. Yu-Ru Syan and Stanelty [4] introduced some properties of an E-convex function, while Emam and Youness in [5] introduced a new class of E-convex sets and E-convex functions, which are called strongly E-convex sets and strongly E-convex functions, by taking the images of two points x and y under an operator E : R n R n besides the two points themselves. In [6] Megahed et al. introduced a combined interactive approach for solving E-convex multiobjective nonlinear programming. Also, in [7, 8] Iqbal and et al. introduced geodesic E-convex sets, geodesic E-convex and some properties of geodesic semi-E-convex functions.

In this paper we present the concept of an E-differentiable convex function which transforms a non-differentiable convex function to a differentiable function under an operator E : R n R n , for which we can apply the Fritz-John and Kuhn-Tucker conditions [9, 10] to find a solution of mathematical programming with a non-differentiable function.

In the following, we present the definitions of E-convex sets, E-convex functions, and semi E-convex functions.

Definition 1[1]

A set M is said to be an E-convex set with respect to an operator E : R n R n if and only if λ E ( x ) + ( 1 λ ) E ( y ) M for each x , y M and λ [ 0 , 1 ] .

Definition 2[1]

A function f : R n R is said to be an E-convex function with respect to an operator E : R n R n on an E-convex set M R n if and only if
f ( λ E ( x ) + ( 1 λ ) E ( y ) ) λ ( f E ) ( x ) + ( 1 λ ) ( f E ) ( y )

for each x , y M and λ [ 0 , 1 ] .

Definition 3[3]

A real-valued function f : M R n R is said to be semi E-convex function with respect to an operator E : R n R n on M if M is an E-convex set and
f ( λ E ( x ) + ( 1 λ ) E ( y ) ) λ f ( x ) + ( 1 λ ) f ( y )

for each x , y M and λ [ 0 , 1 ] .

Proposition 4[1]

1- Let a set M R n be an E-convex set with respect to an operator E, E : R n R n , then E ( M ) M .

2- If E ( M ) is a convex set and E ( M ) M , then M is an E-convex set.

3- If M 1 and M 2 are E-convex sets with respect to E, then M 1 M 2 is an E-convex set with respect to E.

Lemma 5[1]

Let M R n be an E 1 - and E 2 -convex set, then M is an ( E 1 E 2 ) - and ( E 2 E 1 ) -convex set.

Lemma 6[1]

Let E : R n R n be a linear map and let M 1 , M 2 R n be E-convex sets, then M 1 + M 2 is an E-convex set.

Definition 7[1]

Let S R n × R and E : R n R n , we say that the set S is E-convex if for each ( x , α ) , ( y , β ) S and each λ [ 0 , 1 ] , we have
( λ E x + ( 1 λ ) E y , λ α + ( 1 λ ) β ) S .

2 Generalized E-convex function

Definition 8[1]

Let M R n be an E-convex set with respect to an operator E : R n R n . A function f : M R is said to be a pseudo E-convex function if for each x 1 , x 2 M with ( f E ) ( x 1 ) ( x 2 x 1 ) 0 implies f ( E x 2 ) f ( E x 1 ) or for all x 1 , x 2 M and f ( E x 2 ) < f ( E x 1 ) implies ( f E ) ( x 1 ) ( x 2 x 1 ) < 0 .

Definition 9[1]

Let M R n be an E-convex set with respect to an operator E : R n R n . A function f : M R is said to be a quasi-E-convex function if and only if
f ( λ E x + ( 1 λ ) E y ) max { ( f E ) x , ( f E ) y }

for each x , y M and λ [ 0 , 1 ] .

3 E-differentiable function

Definition 10 Let f : M R n R be a non-differentiable function at x ¯ and let E : R n R n be an operator. A function f is said to be E-differentiable at x ¯ if and only if ( f E ) is a differentiable function at x ¯ and
( f E ) ( x ) = ( f E ) ( x ¯ ) + ( f E ) ( x ¯ ) ( x x ¯ ) + x x ¯ α ( x ¯ , x x ¯ ) , α ( x ¯ , x x ¯ ) 0 as  x x . ¯

Example 11 Let f ( x ) = | x | be a non-differentiable function at the point x = 0 and let E : R R be an operator such that E ( x ) = x 2 , then the function ( f E ) ( x ) = f ( E x ) = x 2 is a differentiable function at the point x = 0 , and hence f is an E-differentiable function.

3.1 Problem formulation

Now, we formulate problems P and P E , which have a non-differentiable function and an E-differentiable function, respectively.

Let E : R n R n be an operator, M be an E-convex set and f be an E-differentiable function. The problem P is defined as
P { Min f ( x ) , subject to  M = { x : g i ( x ) 0 , i = 1 , 2 , , m } ,
where f is a non-differentiable function, and the problem P E is defined as
P E { Min ( f E ) ( x ) , subject to  M = { x : ( g i E ) ( x ) 0 , i = 1 , 2 , , m } ,

where f is an E-differentiable function.

Now, we will discuss the relationship between the solutions of problems P and P E .

Lemma 12[11]

Let E : R n R n be a one-to-one and onto operator and let M = { x : ( g i E ) ( x ) 0 , i = 1 , 2 , , m } . Then E ( M ) = M , where M and M are feasible regions of problems P and P E , respectively.

Theorem 13 Let E : R n R n be a one-to-one and onto operator and let f be an E-differentiable function. If f is non-differentiable at x ¯ , and x ¯ is an optimal solution of the problem P, then there exists y ¯ M such that x ¯ = E ( y ¯ ) and y ¯ is an optimal solution of the problem P E .

Proof Let x ¯ be an optimal solution of the problem P. From Lemma 12 there exists y ¯ M such that x ¯ = E ( y ¯ ) . Let y ¯ be a not optimal solution of the problem P E , then there is y ˆ M such that ( f E ) ( y ˆ ) ( f E ) ( y ¯ ) . Also, there exists x ˆ M such that x ˆ = E ( y ˆ ) . Then f ( x ˆ ) < f ( x ¯ ) contradicts the optimality of x ¯ for the problem P. Hence the proof is complete. □

Theorem 14 Let E : R n R n be a one-to-one and onto operator, and let f be an E-differentiable function and strictly quasi-E-convex. If x ¯ is an optimal solution of the problem P, then there exists y ¯ M such that x ¯ = E ( y ¯ ) and y ¯ is an optimal solution of the problem  P E .

Proof Let x ¯ be an optimal solution of the problem P. Then from Lemma 12 there is y ¯ M such that x ¯ = E ( y ¯ ) . Let y ¯ be a not optimal solution of the problem P E , then there is y ˆ M and also x ˆ M , x ˆ = E ( y ˆ ) such that ( f E ) ( y ˆ ) ( f E ) ( y ¯ ) . Since f is strictly quasi-E-convex function, then
f ( λ E ( y ¯ ) + ( 1 λ ) E ( y ˆ ) ) < max { ( f E ) ( y ¯ ) , ( f E ) ( y ˆ ) } < max { f ( x ¯ ) , f ( x ˆ ) } < f ( x ¯ ) .

Since M is an E-convex set and E ( M ) M , then λ E ( y ¯ ) + ( 1 λ ) E ( y ˆ ) M contradicts the assumption that x ¯ is a solution of the problem P, then there exists y ¯ M , a solution of the problem P E , such that x ¯ = E ( y ¯ ) . □

Theorem 15 Let M be an E-convex set, E : R n R n be a one-to-one and onto operator and f : M R n R be an E-differentiable function at x ¯ . If there is a vector d R n such that ( f E ) ( x ¯ ) d < 0 , then there exists δ > 0 such that
( f E ) ( x ¯ + λ d ) < ( f E ) ( x ¯ ) for each λ ( 0 , δ ) .
Proof Since f is an E-differentiable function at x ¯ , then
( f E ) ( x ¯ + λ d ) = ( f E ) ( x ¯ ) + λ ( f E ) ( x ¯ ) + λ d α ( x ¯ , λ d ) , α ( x ¯ , λ d ) 0 as  λ 0 .
Since ( f E ) ( x ¯ ) d < 0 and α ( x ¯ , λ d ) 0 as λ 0 , then there exists δ > 0 such that
( f E ) ( x ¯ ) + d α ( x ¯ , λ d ) < 0 for each  λ ( 0 , δ )

and thus ( f E ) ( x ¯ + λ d ) < ( f E ) ( x ¯ ) . □

Corollary 16 Let M be an E-convex set, let E : R n R n be a one-to-one and onto operator, and let f : M R n R be an E-differentiable and strictly E-convex function at x ¯ . If x ¯ is a local minimum of the function ( f E ) , then ( f E ) ( x ¯ ) = 0 .

Proof Suppose that ( f E ) ( x ¯ ) 0 and let d = ( f E ) ( x ¯ ) , then ( f E ) ( x ¯ ) d = ( f E ) ( x ¯ ) 2 < 0 . By Theorem 15 there exists δ > 0 such that
( f E ) ( x ¯ + λ d ) < ( f E ) ( x ¯ ) for each  λ ( 0 , δ )

contradicting the assumption that x ¯ is a local minimum of ( f E ) ( x ) , and thus ( f E ) ( x ¯ ) = 0 . □

Theorem 17 Let M be an E-convex set, E : R n R n be a one-to-one and onto operator, and f : M R n R be twice E-differentiable and strictly E-convex function at x ¯ . If x ¯ is a local minimum of ( f E ) , then ( f E ) ( x ¯ ) = 0 and the Hessian matrix H ( x ¯ ) = 2 ( f E ) ( x ¯ ) is positive semidefinite.

Proof Suppose that d is an arbitrary direction. Since f is a twice E-differentiable function at x ¯ , then
( f E ) ( x ¯ + λ d ) = ( f E ) ( x ¯ ) + λ ( f E ) ( x ¯ ) d + 1 2 λ 2 d t 2 ( f E ) ( x ¯ ) d + λ 2 d 2 α ( x ¯ , λ d ) ,

where α ( x ¯ , λ d ) 0 as λ 0 .

From Corollary 16 we have ( f E ) ( x ¯ ) = 0 , and
( f E ) ( x ¯ + λ d ) ( f E ) ( x ¯ ) λ 2 = 1 2 d t 2 ( f E ) ( x ¯ ) d .
Since x ¯ is a local minimum of ( f E ) , then ( f E ) ( x ¯ ) < ( f E ) ( x ¯ + λ d ) , and
d t 2 ( f E ) ( x ¯ ) d 0 , i.e. , H ( x ¯ ) = 2 ( f E ) ( x ¯ ) is positive semidefinite .

 □

Example 18 Let f ( x , y ) = x + 2 y 2 2 x 1 3 be a non-differentiable function at ( 0 , y ) , and let E ( x , y ) = ( x 3 , y ) , then ( f E ) ( x , y ) = x 3 + 2 y 2 2 x , and
( f E ) x = 3 x 2 2 = 0 implies x = ± 2 3 , ( f E ) y = 4 y = 0 implies y = 0 , 2 ( f E ) x 2 = 6 x , 2 ( f E ) y x = 0 , 2 ( f E ) y 2 = 4 , 2 ( f E ) x y = 0 .

Then ( x 1 , y 1 ) = ( 2 3 , 0 ) and ( x 2 , y 2 ) = ( 2 3 , 0 ) are extremum points of ( f E ) ( x , y ) , and the Hessian matrix H ( 2 3 , 0 ) = [ 6 2 3 0 0 4 ] is positive definite. And thus the point ( 2 3 , 0 ) is a local minimum of the function ( f E ) ( x , y ) , but the Hessian matrix H ( 2 3 , 0 ) = [ 6 2 3 0 0 4 ] is indefinite.

Theorem 19 Let M be an E-convex set, let E : R n R n be a one-to-one and onto operator, and let f : M R n R be a twice E-differentiable and strictly E-convex function at x ¯ . If ( f E ) ( x ¯ ) = 0 and the Hessian matrix H ( x ¯ ) = 2 ( f E ) ( x ¯ ) is positive definite, then x ¯ is a local minimum of ( f E ) .

Proof Suppose that x ¯ is not a local minimum of ( f E ) ( x ) , and there exists a sequence { x k } is converging to x ¯ such that ( f E ) ( x k ) < ( f E ) ( x ¯ ) for each k. Since ( f E ) ( x ¯ ) = 0 , and f is twice E-differentiable at x ¯ , then
( f E ) ( x k ) = ( f E ) ( x ¯ ) + λ ( f E ) ( x ¯ ) ( x k x ¯ ) + 1 2 ( x k x ¯ ) t 2 ( f E ) ( x ¯ ) ( x k x ¯ ) + ( x k x ¯ ) 2 α ( x ¯ , ( x k x ¯ ) ) ,
where α ( x ¯ , ( x k x ¯ ) ) 0 as k , and
1 2 ( x k x ¯ ) t 2 ( f E ) ( x ¯ ) ( x k x ¯ ) + ( x k x ¯ ) 2 α ( x ¯ , ( x k x ¯ ) ) < 0 for each  k .
By dividing on ( x k x ¯ ) 2 , and letting d k = ( x k x ¯ ) ( x k x ¯ ) , we get
1 2 d k t 2 ( f E ) ( x ¯ ) d k + α ( x ¯ , ( x k x ¯ ) ) < 0 for each  k .

But d k = 1 for each k, and hence there exists an index set K such that { d k } K d , where d = 1 . Considering this subsequence and the fact that α ( x ¯ , ( x k x ¯ ) ) 0 as k , then d t 2 ( f E ) ( x ¯ ) d < 0 . This contradicts the assumption that H ( x ¯ ) is positive definite. Therefore x ¯ is indeed a local minimum. □

Example 20 Let f ( x , y ) = x 2 3 + y 2 1 be a non-differentiable at the point ( 0 , y ) , and let E ( x , y ) = ( x 3 , y ) , then ( f E ) ( x , y ) = x 2 + y 2 1
( f E ) x = 2 x , 2 ( f E ) y x = 0 , 2 ( f E ) x 2 = 2 , ( f E ) y = 2 y , 2 ( f E ) x y = 0 , 2 ( f E ) y 2 = 2 .
The necessary condition for x ¯ is a local minimum of ( f E ) is ( f E ) ( x ¯ ) = 0 , then x ¯ = ( 0 , 0 ) , and the Hessian matrix H ( x ¯ )
H = [ 2 ( f E ) x 2 2 ( f E ) y x 2 ( f E ) x y 2 ( f E ) y 2 ] = [ 2 0 0 2 ]

is positive definite.

Example 21 Let f ( x , y ) = x 1 3 + y 1 be non-differentiable at the point ( 0 , y ) , and let E ( x , y ) = ( x 3 , y ) , then ( f E ) ( x , y ) = x + y 1 .

Now, let M = { λ 1 ( 0 , 0 ) + λ 2 ( 0 , 3 ) + λ 3 ( 1 , 2 ) + λ 4 ( 1 , 0 ) } { λ 1 ( 0 , 0 ) + λ 2 ( 0 , 3 ) + λ 3 ( 1 , 2 ) + λ 4 ( 1 , 0 ) } , i = 1 4 λ i = 1 , λ i 0 be an E-convex set with respect to operator E (the feasible region is shown in Figure 1) and
f ( 0 , 0 ) = 1 , ( f E ) ( 0 , 0 ) = 1 , f ( 0 , 3 ) = 4 , ( f E ) ( 0 , 3 ) = 4 , f ( 1 , 2 ) = 2 , ( f E ) ( 1 , 2 ) = 2 , f ( 1 , 0 ) = 0 , ( f E ) ( 1 , 0 ) = 0 , f ( 0 , 3 ) = 2 , ( f E ) ( 0 , 3 ) = 2 , f ( 1 , 2 ) = 2 , ( f E ) ( 1 , 2 ) = 2 .
Figure 1

The feasible region M.

Then x ¯ = ( 0 , 3 ) is a solution of the problem P E and E ( x ¯ ) = E ( 0 , 3 ) = ( 0 , 3 ) is a solution of the problem P.

Definition 22 Let M be a nonempty E-convex set in R n and let E ( x ¯ ) c l M . The cone of feasible direction of E ( M ) at E ( x ¯ ) denoted by D is given by
D = { d : d 0 , E ( x ¯ ) + λ d M  for each  λ [ 0 , δ ] , δ > 0 } .

Lemma 23 Let M be an E-convex set with respect to an operator E : R n R n , and let f : M R n R be E-differentiable at x ¯ . If x ¯ is a local minimum of the problem P E , then F 0 D = ϕ , where F 0 = { d : ( f E ) ( x ¯ ) d < 0 } , and D is the cone of feasible direction of M at  x ¯ .

Proof Suppose that there exists a vector d F 0 D . Then by Theorem 15, there exists δ 1 such that
( f E ) ( x ¯ + λ d ) < ( f E ) ( x ¯ ) for each  λ ( 0 , δ 1 ) .
(3.1)
By the definition of the cone of feasible direction, there exists δ 2 such that
E ( x ¯ ) + λ d M for each  λ ( 0 , δ 2 ) .
(3.2)

From 3.1 and 3.2 we have ( f E ) ( x ¯ + λ d ) < ( f E ) ( x ¯ ) for each λ ( 0 , δ ) , where δ = min { δ 1 , δ 2 } , which contradicts the assumption that x ¯ is a local optimal solution, then F 0 D = ϕ . □

Lemma 24 Let M be an open E-convex set with respect to an operator E : R n R n , let f : M R n R be E-differentiable at x ¯ and let g i : R n R for i = 1 , 2 , , m . Let x ¯ be a feasible solution of the problem P E and let I = { i : ( g i E ) ( x ¯ ) = 0 } . Furthermore, suppose that g i for i I is E-differentiable at x ¯ and that g i for i I is continuous at x ¯ . If x ¯ is a local optimal solution, then F 0 G 0 = ϕ , where
F 0 = { d : ( f E ) ( x ¯ ) d < 0 } , G 0 = { d : ( g i E ) ( x ¯ ) d < 0 , for each i I }

and E is one-to-one and onto.

Proof Let d G 0 . Since E ( x ¯ ) M and M is an open E-convex set, there exists a δ 1 > 0 such that
E ( x ¯ ) + λ d M for  λ ( 0 , δ 1 ) .
(3.3)
Also, since ( g i E ) ( x ¯ ) < 0 and since g i is continuous at x ¯ for i I , there exists a δ 2 > 0 such that
( g i E ) ( x ¯ + λ d ) < 0 for  λ ( 0 , δ 2 )  and for  i I .
(3.4)
Finally, since d G 0 , ( g i E ) ( x ¯ ) d < 0 for each i I and by Theorem 15, there exists δ 3 > 0 such that
( g i E ) ( x ¯ + λ d ) < ( g i E ) ( x ¯ ) for  λ ( 0 , δ 3 )  and  i I .
(3.5)

From 3.3, 3.4 and 3.5, it is clear that points of the form E ( x ¯ ) + λ d are feasible to the problem P E for each λ ( 0 , δ ) , where δ = min ( δ 1 , δ 2 , δ 3 ) . Thus d D , where D is the cone of feasible direction of the feasible region at x ¯ . We have shown that for d G 0 implies that d D , and hence G 0 D . By Lemma 23, since x ¯ is a local solution of the problem P E , F 0 D = ϕ . It follows that F 0 G 0 = ϕ . □

Theorem 25 (Fritz-John optimality conditions)

Let M be an open E-convex set with respect to the one-to-one and onto operator E : R n R n , let f : M R n R be E-differentiable at x ¯ and let g i : R n R for i = 1 , 2 , , m . Let x ¯ be feasible solution of the problem P E and let I = { i : ( g i E ) ( x ¯ ) = 0 } . Furthermore, suppose that g i for i I is differentiable at x ¯ and that g i for i I is continuous at x ¯ . If x ¯ is a local optimal solution, then there exist scalars u 0 and u i for i I such that

and E ( x ¯ ) is a local solution of the problem P.

Proof Let x ¯ be a local solution of the problem P E , then there is no vector d such that ( f E ) ( x ¯ ) d < 0 and ( g i E ) ( x ¯ ) d < 0 . Let A be a matrix with rows ( f E ) ( x ¯ ) and ( g i E ) ( x ¯ ) . From Gordon’s theorem [10], we have the system A d < 0 is inconsistent, then there exists a vector b 0 such that A b = 0 , where b = ( u 0 u i ) for each i I . And thus
u ( f E ) ( x ¯ ) + i I u i ( g i E ) ( x ¯ ) = 0

holds and E ( x ¯ ) is a local solution of the problem P. □

Theorem 26 Let E : R n R n be a one-to-one and onto operator and let f : M R n R be an E-differentiable function. If x ¯ is an optimal solution of the problem P, then there exists y ¯ M such that x ¯ = E ( y ¯ ) is an optimal solution of the problem P E and the Fritz-John optimality condition of the problem P E is satisfied.

Proof Let x ¯ be an optimal solution of the problem P. Since E is one-to-one and onto, according to Theorem 13, there exists y ¯ M , x ¯ = E ( y ¯ ) is an optimal solution of the problem P E . Hence there exist scalars u 0 . u i satisfying the Fritz-John optimality conditions of the problem P E
u ( f E ) ( x ¯ ) + i I u i ( g i E ) ( x ¯ ) = 0 , ( u 0 , u i ) = 0 , u 0 , u i 0 .

 □

Theorem 27 (Kuhn-Tucker necessary condition)

Let M be an open E-convex set with respect to the one-to-one and onto operator E : R n R n , let f : M R n R be E-differentiable and strictly E-convex at x ¯ and let g i : R n R for i = 1 , 2 , , m . Let y ¯ be a feasible solution of the problem P E and let I = { i : ( g i E ) ( y ¯ ) = 0 } . Furthermore, suppose that ( g i E ) is continuous at y ¯ for i I and ( g i E ) ( y ¯ ) for i I are linearly independent. If x ¯ is a solution of the problem P, x ¯ = E ( y ¯ ) and y ¯ is a local solution of the problem P E , then there exist scalars u i for i I such that
( f E ) ( y ¯ ) + i I u i ( g i E ) ( y ¯ ) = 0 , u i 0 for each i I .
Proof From the Fritz-John optimality condition theorem, there exist scalars u 0 and u i for each i I such that
u 0 ( f E ) ( y ¯ ) + i I u ˆ i ( g i E ) ( y ¯ ) = 0 , u 0 , u ˆ i 0  for each  i I .

If u 0 = 0 , the assumption of linear independence of ( g i E ) ( y ¯ ) does not hold, then u 0 > 0 . By taking u i = u ˆ i u 0 , then ( f E ) ( y ¯ ) + i I u i ( g i E ) ( y ¯ ) = 0 , u i 0 holds for each i I . From Theorem 26, y ¯ is a local solution of the problem P E . □

Theorem 28 Let M be an open E-convex set with respect to the one-to-one and onto operator E : R n R n , g i : R n R for i = 1 , 2 , , m , and let f : M R n R be E-differentiable at x ¯ and strictly E-convex at x ¯ . Let x ¯ = E ( y ¯ ) be a feasible solution of the problem P E and I = { i : ( g i E ) ( y ¯ ) = 0 } . Suppose that f is pseudo-E-convex at y ¯ and that g i is quasi-E-convex and differentiable at y ¯ for each i I . Furthermore, suppose that the Kuhn-Tucker conditions hold at y ¯ . Then y ¯ is a global optimal solution of the problem P E and hence x ¯ = E ( y ¯ ) is a solution of the problem P.

Proof Let y ˆ be a feasible solution of the problem P E , then ( g i E ) ( y ˆ ) ( g i E ) ( y ¯ ) for each i I . Since ( g i E ) ( y ˆ ) 0 , ( g i E ) ( y ¯ ) = 0 and g i is quasi-E-convex at y ¯ , then
( g i E ) ( y ¯ + λ ( y ˆ y ¯ ) ) = ( g i E ) ( λ y ˆ + ( 1 λ ) y ¯ ) max { ( g i E ) ( y ˆ ) , ( g i E ) ( y ¯ ) } = ( g i E ) ( y ¯ ) .
This means that ( g i E ) does not increase by moving from y ¯ along the direction y ˆ y ¯ . Then we must have from Theorem 15 that ( g i E ) ( y y ¯ ) 0 . Multiplying by u i and summing over I, we get
[ i I u i ( g i E ) ( y ¯ ) ] ( y y ¯ ) 0 .
But since
( f E ) ( y ¯ ) + i I u i ( g i E ) ( y ¯ ) = 0 ,
it follows that ( f E ) ( y ¯ ) ( y y ¯ ) 0 . Since f is pseudo E-convex at y ¯ , we get
( f E ) ( y ) ( f E ) ( y ¯ ) .

Then y ¯ is a global solution of the problem P E and from Theorem 13 x ¯ = E ( y ¯ ) is a global solution of the problem P. □

Example 29 Consider the following problem (problem P):
Min f ( x , y ) = x 2 3 + y 2 , subject to  x 2 + y 2 5 , x + 2 y 4 , x , y 0 .
The feasible region of this problem is shown in Figure 2.
Figure 2

The feasible region M.

Let E ( x , y ) = ( 1 8 x 3 , 1 3 y ) , then the problem P E is as follows:
min ( f E ) ( x , y ) = 1 4 x 2 + 1 9 y 2 , subject to  x 6 64 + y 2 9 5 , 1 8 x 3 + 2 3 y 4 , x , y 0 .
We note that E ( M ) M , where
( 5 , 0 ) M implies E ( 5 , 0 ) = ( 5 5 8 , 0 ) M , ( 0 , 2 ) M implies E ( 0 , 2 ) = ( 0 , 2 3 ) M , ( 0 , 0 ) M implies E ( 0 , 0 ) = ( 0 , 0 ) M , ( 2 , 1 ) M implies E ( 2 , 1 ) = ( 1 , 1 3 ) M .
The Kuhn-Tucker conditions are as follows:
( f E ) ( x , y ) + u 1 ( g 1 E ) ( x , y ) + u 2 ( g 2 E ) ( x , y ) = 0 , [ 1 2 x 2 9 y ] + u 1 [ 6 64 x 5 2 9 y ] + u 2 [ 3 8 x 2 2 3 ] = 0 , u 1 [ x 6 64 + y 2 9 5 ] = 0 , u 2 [ 1 8 x 3 + 2 3 y 4 ] = 0 .

The solution is { [ x = 0.0 , u 1 = 0.0 , u 2 = 0.0 , y = 0.0 ] } , z ¯ = ( 0 , 0 ) , and x ¯ = E ( z ¯ ) = ( 0 , 0 ) is a solution of the problem P.

4 Conclusion

In this paper we introduced a new definition of an E-differentiable convex function, which transforms a non-differentiable function to a differentiable function under an operator E : R n R n , and we studied Kuhn-Tucker and Fritz-John conditions for obtaining an optimal solution of mathematical programming with a non-differentiable function. At the end, some examples have been presented to clarify the results.

Declarations

Acknowledgements

The authors express their deep thanks and their respect to the referees and the Journal for these valuable comments in the evaluation of this paper.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Computers and Informatics, Suez Canal University
(2)
Mathematics Department, College of Science, Majmaah University
(3)
Computer Science Institute Suez City
(4)
Department of Mathematics, Faculty of Science, Tanta University

References

  1. Youness EA: E -convex sets, E -convex functions and E -convex programming. J. Optim. Theory Appl. 1999, 102(3):439–450.MathSciNetView ArticleGoogle Scholar
  2. Youness EA: Optimality criteria in E -convex programming. Chaos Solitons Fractals 2001, 12: 1737–1745. 10.1016/S0960-0779(00)00036-9MathSciNetView ArticleGoogle Scholar
  3. Chen X: Some properties of semi- E -convex functions. J. Math. Anal. Appl. 2002, 275: 251–262. 10.1016/S0022-247X(02)00325-6MathSciNetView ArticleGoogle Scholar
  4. Syau Y-R, Lee ES: Some properties of E -convex functions. Appl. Math. Lett. 2005, 18: 1074–1080. 10.1016/j.aml.2004.09.018MathSciNetView ArticleGoogle Scholar
  5. Emam T, Youness EA: Semi strongly E -convex function. J. Math. Stat. 2005, 1(1):51–57.MathSciNetView ArticleGoogle Scholar
  6. Megahed AA, Gomma HG, Youness EA, El-Banna AH: A combined interactive approach for solving E -convex multi- objective nonlinear programming. Appl. Math. Comput. 2011, 217: 6777–6784. 10.1016/j.amc.2010.12.086MathSciNetView ArticleGoogle Scholar
  7. Iqbal A, Ahmad I, Ali S: Some properties of geodesic semi- E -convex functions. Nonlinear Anal., Theory Methods Appl. 2011, 74: 6805–6813. 10.1016/j.na.2011.07.005MathSciNetView ArticleGoogle Scholar
  8. Iqbal, A, Ali, S, Ahmad, I: On geodesic E-convex sets, geodesic E-convex functions and E-epigraphs. J. Optim. Theory Appl. (2012), (Article Available online)Google Scholar
  9. Mangasarian OL: Nonlinear Programming. Mcgraw-Hill, New York; 1969.Google Scholar
  10. Bazaraa MS, Shetty CM: Nonlinear Programming Theory and Algorithms. Wiley, New York; 1979.Google Scholar
  11. Youness EA: Characterization of efficient solution of multiobjective E -convex programming problems. Appl. Math. Comput. 2004, 151(3):755–761. 10.1016/S0096-3003(03)00526-5MathSciNetView ArticleGoogle Scholar

Copyright

© Megahed et al.; licensee Springer 2013

This article is published under license to BioMed Central Ltd. 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.