R E S E A R C H
Open Access
Optimality conditions of
E
-convex
programming for an
E
-differentiable function
Abd El-Monem A Megahed
1,2*, Hebaa G Gomma
3, Ebrahim A Youness
4and Abou-Zaid H El-Banna
4*Correspondence:
1Department of Mathematics,
Faculty of Computers and Informatics, Suez Canal University, Ismailia, Egypt
2Current address: Mathematics
Department, College of Science, Majmaah University, Majmaah, KSA Full list of author information is available at the end of the article
Abstract
In this paper we introduce a new definition of anE-differentiable convex function, which transforms a non-differentiable function to a differentiable function under an operatorE:Rn→Rn. 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; semiE-convex function;E-differentiable function
1 Introduction
The concepts ofE-convex sets and anE-convex function have been introduced by Youness in [, ], and they have some important applications in various branches of mathematical sciences. Youness in [] introduced a class of sets and functions which is calledE-convex sets andE-convex functions by relaxing the definition of convex sets and convex func-tions. This kind of generalized convexity is based on the effect of an operatorE:Rn→Rn
on the sets and the domain of the definition of functions. Also, in [] Youness discussed the optimality criteria ofE-convex programming. Xiusu Chen [] introduced a new con-cept of semiE-convex functions and discussed its properties. Yu-Ru Syan and Stanelty [] introduced some properties of anE-convex function, while Emam and Youness in [] in-troduced a new class ofE-convex sets andE-convex functions, which are called strongly
E-convex sets and stronglyE-convex functions, by taking the images of two pointsxandy
under an operatorE:Rn→Rnbesides the two points themselves. In [] Megahedet al. in-troduced a combined interactive approach for solvingE-convex multiobjective nonlinear programming. Also, in [, ] Iqbal andet al.introduced geodesicE-convex sets, geodesic
E-convex and some properties of geodesic semi-E-convex functions.
In this paper we present the concept of anE-differentiable convex function which trans-forms a non-differentiable convex function to a differentiable function under an operator
E:Rn→Rn, for which we can apply the Fritz-John and Kuhn-Tucker conditions [, ] to
find a solution of mathematical programming with a non-differentiable function. In the following, we present the definitions ofE-convex sets,E-convex functions, and semiE-convex functions.
Definition [] A setMis said to be anE-convex set with respect to an operatorE:Rn→
Rnif and only ifλE(x) + ( –λ)E(y)∈Mfor eachx,y∈Mandλ∈[, ].
Definition [] A functionf :Rn→Ris said to be anE-convex function with respect to
an operatorE:Rn→Rnon anE-convex setM⊆Rnif and only if
fλE(x) + ( –λ)E(y)≤λ(f ◦E)(x) + ( –λ)(f◦E)(y)
for eachx,y∈Mandλ∈[, ].
Definition [] A real-valued functionf:M⊆Rn→Ris said to be semiE-convex
func-tion with respect to an operatorE:Rn→RnonMifMis anE-convex set and
fλE(x) + ( –λ)E(y)≤λf(x) + ( –λ)f(y)
for eachx,y∈Mandλ∈[, ].
Proposition [] - Let a set M⊆Rnbe an E-convex set with respect to an operator E,
E:Rn→Rn,then E(M)⊆M.
- If E(M)is a convex set and E(M)⊆M,then M is an E-convex set.
- If Mand Mare E-convex sets with respect to E,then M∩Mis an E-convex set with respect to E.
Lemma [] Let M⊆Rnbe an E
- and E-convex set,then M is an(E◦E)- and(E◦E) -convex set.
Lemma [] Let E:Rn→Rnbe a linear map and let M
,M⊂Rnbe E-convex sets,then M+Mis an E-convex set.
Definition [] LetS⊂Rn×RandE:Rn→Rn, we say that the setSisE-convex if for
each (x,α), (y,β)∈Sand eachλ∈[, ], we have
λEx+ ( –λ)Ey,λα+ ( –λ)β∈S.
2 GeneralizedE-convex function
Definition [] LetM⊆Rnbe anE-convex set with respect to an operatorE:Rn→Rn. A functionf :M→Ris said to be a pseudoE-convex function if for eachx,x∈Mwith
∇(f ◦E)(x)(x–x)≥ impliesf(Ex)≥f(Ex) or for allx,x∈Mandf(Ex) <f(Ex)
implies∇(f ◦E)(x)(x–x) < .
Definition [] LetM⊆Rnbe anE-convex set with respect to an operatorE:Rn→Rn.
A functionf :M→Ris said to be a quasi-E-convex function if and only if
fλEx+ ( –λ)Ey≤max(f◦E)x, (f ◦E)y
3 E-differentiable function
Definition Letf :M⊆Rn→Rbe a non-differentiable function atxand letE:Rn→Rn
be an operator. A function f is said to beE-differentiable atxif and only if (f ◦E) is a differentiable function atxand
(f ◦E)(x) = (f ◦E)(x) +∇(f◦E)(x)(x–x) +x–xα(x,x–x),
α(x,x–x)→ asx→x.
Example Letf(x) =|x|be a non-differentiable function at the pointx= and letE:
R→Rbe an operator such thatE(x) =x, then the function (f ◦E)(x) =f(Ex) =x is a
differentiable function at the pointx= , and hencef is anE-differentiable function.
3.1 Problem formulation
Now, we formulate problemsPandPE, which have a non-differentiable function and an
E-differentiable function, respectively.
LetE:Rn→Rnbe an operator,Mbe anE-convex set andf be anE-differentiable
func-tion. The problemPis defined as
P ⎧ ⎨ ⎩
Minf(x),
subject toM={x:gi(x)≤,i= , , . . . ,m},
wheref is a non-differentiable function, and the problemPEis defined as
PE
⎧ ⎨ ⎩
Min(f ◦E)(x),
subject toM={x: (gi◦E)(x)≤,i= , , . . . ,m},
wheref is anE-differentiable function.
Now, we will discuss the relationship between the solutions of problemsPandPE. Lemma [] Let E:Rn→Rnbe a one-to-one and onto operator and let M={x: (g
i◦
E)(x)≤,i= , , . . . ,m}.Then E(M) =M,where M and Mare feasible regions of problems P and PE,respectively.
Theorem Let E :Rn→Rn 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∈Msuch that x=E(y)and y is an optimal solution of the problem PE.
Proof Letxbe an optimal solution of the problemP. From Lemma there existsy∈M
such thatx=E(y). Letybe a not optimal solution of the problemPE, then there isy∈M
such that (f◦E)(y)≤(f◦E)(y). Also, there existsx∈Msuch thatx=E(y) . Thenf(x) <f(x) contradicts the optimality ofxfor the problemP. Hence the proof is complete.
Theorem Let E:Rn→Rn be a one-to-one and onto operator, and let f be an
Proof Letxbe an optimal solution of the problemP. Then from Lemma there isy∈M
such thatx=E(y). Letybe a not optimal solution of the problemPE, then there isy∈M
and alsox∈M,x=E(y) such that (f ◦E)(y)≤(f ◦E)(y). Sincef is strictly quasi-E-convex function, then
fλE(y) + ( –λ)E(y)<max(f ◦E)(y), (f ◦E)(y) <maxf(x),f(x)
<f(x).
SinceMis anE-convex set andE(M)⊂M, thenλE(y) + ( –λ)E(y)∈Mcontradicts the assumption thatxis a solution of the problemP, then there existsy∈M, a solution of the problemPE, such thatx=E(y). Theorem Let M be an E-convex set,E:Rn→Rnbe a one-to-one and onto operator
and f :M⊆Rn→R be an E-differentiable function at x.If there is a vector d⊂Rnsuch
that∇(f ◦E)(x)d< ,then there existsδ> such that
(f ◦E)(x+λd) < (f ◦E)(x) for eachλ∈(,δ).
Proof Sincef is anE-differentiable function atx, then (f ◦E)(x+λd) = (f ◦E)(x) +λ∇(f ◦E)(x) +λdα(x,λd),
α(x,λd)→ asλ→.
Since∇(f◦E)(x)d< andα(x,λd)→ asλ→, then there existsδ> such that
∇(f ◦E)(x) +dα(x,λd) < for eachλ∈(,δ)
and thus (f ◦E)(x+λd) < (f◦E)(x).
Corollary Let M be an E-convex set,let E:Rn→Rnbe a one-to-one and onto operator,
and let f :M⊆Rn→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) = .
Proof Suppose that∇(f ◦E)(x)= and letd= –∇(f ◦E)(x), then∇(f◦E)(x)d= –∇(f◦ E)(x)< . By Theorem there existsδ> such that
(f ◦E)(x+λd) < (f ◦E)(x) for eachλ∈(,δ)
contradicting the assumption that x is a local minimum of (f ◦ E)(x), and thus
∇(f ◦E)(x) = .
Theorem Let M be an E-convex set,E:Rn→Rnbe a one-to-one and onto operator,
and f :M⊆Rn→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) = and the Hessian matrix H(x) =∇(f ◦E)(x)
Proof Suppose thatdis an arbitrary direction. Sincef is a twiceE-differentiable function atx, then
(f ◦E)(x+λd) = (f ◦E)(x) +λ∇(f ◦E)(x)d+ λ
dt∇(f◦E)(x)d
+λdα(x,λd), whereα(x,λd)→ asλ→.
From Corollary we have∇(f◦E)(x) = , and (f◦E)(x+λd) – (f◦E)(x)
λ =
d
t∇(f◦E)(x)d.
Sincexis a local minimum of (f◦E), then (f◦E)(x) < (f ◦E)(x+λd), and
dt∇(f ◦E)(x)d≥, i.e., H(x) =∇(f◦E)(x) is positive semidefinite.
Example Letf(x,y) =x+ y– x be a non-differentiable function at (,y), and let
E(x,y) = (x,y), then (f◦E)(x,y) =x+ y– x, and
∂(f◦E)
∂x = x
– = implies x=±
,
∂(f◦E)
∂y = y= implies y= ,
∂(f◦E)
∂x = x,
∂(f◦E)
∂y∂x = ,
∂(f◦E)
∂y = ,
∂(f ◦E)
∂x∂y = .
Then (x,y) = (
, ) and (x,y) = (–
, ) are extremum points of (f ◦E)(x,y), and
the Hessian matrixH(
, ) =
is positive definite. And thus the point (
, ) is a
local minimum of the function (f◦E)(x,y), but the Hessian matrixH(–
, ) = – is indefinite.
Theorem Let M be an E-convex set,let E:Rn→Rnbe a one-to-one and onto operator,
and let f :M⊆Rn→R be a twice E-differentiable and strictly E-convex function at x.If ∇(f ◦E)(x) = and the Hessian matrix H(x) =∇(f ◦E)(x)is positive definite,then x is a local minimum of(f◦E).
Proof Suppose thatxis not a local minimum of (f◦E)(x), and there exists a sequence{xk}
is converging toxsuch that (f◦E)(xk) < (f ◦E)(x) for eachk. Since∇(f ◦E)(x) = , andf
is twiceE-differentiable atx, then
(f ◦E)(xk) = (f ◦E)(x) +λ∇(f◦E)(x)(xk–x)
+ (xk–x)
t∇(f◦E)(x)(x
k–x) +(xk–x)
αx, (xk–x)
, whereα(x, (xk–x))→ ask→ ∞, and
(xk–x)
t∇(f ◦E)(x)(x
k–x) +(xk–x)
αx, (xk–x)
By dividing on(xk–x), and lettingdk=((xxk–x)
k–x), we get
d
t
k∇(f ◦E)(x)dk+α
x, (xk–x)
< for eachk.
Butdk= for eachk, and hence there exists an index setKsuch that{dk}K→d, where d= . Considering this subsequence and the fact that α(x, (xk–x))→ ask→ ∞,
thendt∇(f ◦E)(x)d< . This contradicts the assumption thatH(x) is positive definite.
Thereforexis indeed a local minimum.
Example Letf(x,y) =x +y– be a non-differentiable at the point (,y), and let
E(x,y) = (x,y), then (f◦E)(x,y) =x+y–
∂(f◦E)
∂x = x,
∂(f ◦E)
∂y∂x = ,
∂(f ◦E)
∂x = ,
∂(f◦E)
∂y = y,
∂(f◦E)
∂x∂y = ,
∂(f ◦E)
∂y = .
The necessary condition for xis a local minimum of (f ◦E) is∇(f ◦E)(x) = , then
x= (, ), and the Hessian matrixH(x)
H=
⎡ ⎣∂
(f◦E)
∂x
∂(f◦E)
∂y∂x
∂(f◦E)
∂x∂y
∂(f◦E)
∂y
⎤
⎦=
is positive definite.
Example Letf(x,y) =x+y– be non-differentiable at the point (,y), and letE(x,y) =
(x,y), then (f◦E)(x,y) =x+y– .
Now, letM={λ(, ) +λ(, ) +λ(, ) +λ(, )} ∪ {λ(, ) +λ(, –) +λ(, –) +
λ(, )},
i=λi= ,λi≥ be anE-convex set with respect to operatorE(the feasible
region is shown in Figure ) and
f(, ) = –, (f ◦E)(, ) = –, f(, –) = –, (f ◦E)(, ) = –,
f(, ) = , (f ◦E)(, ) = , f(, ) = , (f ◦E)(, ) = ,
f(, ) = , (f ◦E)(, ) = , f(, –) = –, (f ◦E)(, ) = –.
Thenx= (, –) is a solution of the problemPEandE(x) =E(, –) = (, –) is a solution
of the problemP.
Definition LetMbe a nonemptyE-convex set inRnand letE(x)∈clM. The cone of
feasible direction ofE(M) atE(x) denoted byDis given by
D=d:d= ,E(x) +λd∈Mfor eachλ∈[,δ],δ> .
Lemma Let M be an E-convex set with respect to an operator E:Rn→Rn,and let f :M⊆Rn→R be E-differentiable at x.If x is a local minimum of the problem P
Figure 1 The feasible region M.
F∩D=φ,where F={d:∇(f ◦E)(x)d< },and D is the cone of feasible direction of M at x.
Proof Suppose that there exists a vectord∈F∩D. Then by Theorem , there existsδ
such that
(f ◦E)(x+λd) < (f ◦E)(x) for eachλ∈(,δ). (.)
By the definition of the cone of feasible direction, there existsδsuch that
E(x) +λd∈M for eachλ∈(,δ). (.)
From . and . we have (f◦E)(x+λd) < (f◦E)(x) for eachλ∈(,δ), whereδ=min{δ,δ},
which contradicts the assumption thatxis a local optimal solution, thenF∩D=φ.
Lemma Let M be an open E-convex set with respect to an operator E:Rn→Rn,let f :M⊆Rn→R be E-differentiable at x and let g
i:Rn→R for i= , , . . . ,m.Let x be a
feasible solution of the problem PE and let I={i: (gi◦E)(x) = }.Furthermore, suppose
that gifor i∈I is E-differentiable at x and that gifor i∈/I is continuous at x.If x is a local
optimal solution,then F∩G=φ,where F=
d:∇(f◦E)(x)d< ,
G=
d:∇(gi◦E)(x)d< , for each i∈I
and E is one-to-one and onto.
Proof Letd∈G. SinceE(x)∈MandMis an openE-convex set, there exists aδ> such
that
E(x) +λd∈M forλ∈(,δ). (.)
Also, since (gi◦E)(x) < and sincegiis continuous atxfori∈/I, there exists aδ> such
that
Finally, sinced∈G,∇(gi◦E)(x)d< for eachi∈Iand by Theorem , there existsδ>
such that
(gi◦E)(x+λd) < (gi◦E)(x) forλ∈(,δ) andi∈I. (.)
From ., . and ., it is clear that points of the formE(x) +λdare feasible to the problem
PEfor eachλ∈(,δ), whereδ=min(δ,δ,δ). Thusd∈D, whereDis the cone of feasible
direction of the feasible region atx. We have shown that ford∈Gimplies thatd∈D,
and henceG⊂D. By Lemma , sincexis a local solution of the problemPE,F∩D=φ.
It follows thatF∩G=φ.
Theorem (Fritz-John optimality conditions) Let M be an open E-convex set with respect to the one-to-one and onto operator E : Rn →Rn, let f : M⊆ Rn→R be E-differentiable at x and let gi:Rn→R for i= , , . . . ,m.Let x be feasible solution of the
problem PEand let I={i: (gi◦E)(x) = }.Furthermore,suppose that gifor i∈I is
differen-tiable at x and that gifor i∈/I is continuous at x.If x is a local optimal solution,then there
exist scalars uand uifor i∈I such that
u◦∇(f ◦E)(x) +
i∈I
ui∇(gi◦E)(x) = , u◦,ui≥for i∈I,
(u◦,ui)= (, ) for i∈I
and E(x)is a local solution of the problem P.
Proof Letxbe a local solution of the problemPE, then there is no vectordsuch that∇(f◦
E)(x)d< and∇(gi◦E)(x)d< . LetAbe a matrix with rows∇(f ◦E)(x) and∇(gi◦E)(x).
From Gordon’s theorem [], we have the systemAd< is inconsistent, then there exists a vectorb≥ such thatAb= , whereb= (u·ui) for eachi∈I. And thus
u◦∇(f ◦E)(x) +
i∈I
ui∇(gi◦E)(x) =
holds andE(x) is a local solution of the problemP.
Theorem Let E:Rn→Rnbe a one-to-one and onto operator and let f:M⊆Rn→R be an E-differentiable function.If x is an optimal solution of the problem P, then there exists y∈Msuch that x=E(y)is an optimal solution of the problem PEand the Fritz-John
optimality condition of the problem PEis satisfied.
Proof Letxbe an optimal solution of the problem P. SinceE is one-to-one and onto, according to Theorem , there existsy∈M,x=E(y) is an optimal solution of the problem
PE. Hence there exist scalarsu.uisatisfying the Fritz-John optimality conditions of the
problemPE
u◦∇(f ◦E)(x) +
i∈I
ui∇(gi◦E)(x) = ,
(u,ui) = ,
Theorem (Kuhn-Tucker necessary condition) Let M be an open E-convex set with respect to the one-to-one and onto operator E : Rn →Rn, let f : M⊆ Rn→R be
E-differentiable and strictly E-convex at x and let gi:Rn→R for i= , , . . . ,m.Let y be a
feasible solution of the problem PEand let I={i: (gi◦E)(y) = }.Furthermore,suppose that
(gi◦E)is continuous at y for i∈/I and∇(gi◦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 PE,then there
exist scalars uifor i∈I such that
∇(f ◦E)(y) +
i∈I
ui∇(gi◦E)(y) = , ui≥for each i∈I.
Proof From the Fritz-John optimality condition theorem, there exist scalarsuanduifor
eachi∈Isuch that
u∇(f◦E)(y) +
i∈I
ui∇(gi◦E)(y) = , u,ui≥ for eachi∈I.
Ifu= , the assumption of linear independence of∇(gi◦E)(y) does not hold, thenu> .
By takingui=uui, then∇(f ◦E)(y) +
i∈Iui∇(gi◦E)(y) = ,ui≥ holds for eachi∈I.
From Theorem ,yis a local solution of the problemPE. Theorem Let M be an open E-convex set with respect to the one-to-one and onto oper-ator E:Rn→Rn,g
i:Rn→R for i= , , . . . ,m,and let f :M⊆Rn→R be E-differentiable
at x and strictly E-convex at x.Let x=E(y)be a feasible solution of the problem PEand
I={i: (gi◦E)(y) = }.Suppose that f is pseudo-E-convex at y and that giis quasi-E-convex
and differentiable at y for each i∈I.Furthermore,suppose that the Kuhn-Tucker condi-tions hold at y.Then y is a global optimal solution of the problem PEand hence x=E(y)is
a solution of the problem P.
Proof Letybe a feasible solution of the problemPE, then (gi◦E)(y)≤(gi◦E)(y) for each
i∈I. Since (gi◦E)(y)≤, (gi◦E)(y) = andgiis quasi-E-convex aty, then
(gi◦E)
y+λ(y–y)= (gi◦E)
λy+ ( –λ)y
≤max(gi◦E)(y), (gi◦E)(y)
= (gi◦E)(y).
This means that (gi◦E) does not increase by moving fromyalong the directiony–y. Then
we must have from Theorem that∇(gi◦E)(y–y)≤. Multiplying byuiand summing
overI, we get
i∈I
ui∇(gi◦E)(y)
(y–y)≤.
But since
∇(f ◦E)(y) +
i∈I
it follows that∇(f ◦E)(y)(y–y)≥. Sincef is pseudoE-convex aty, we get (f ◦E)(y)≥(f ◦E)(y).
Thenyis a global solution of the problemPE and from Theorem x=E(y) is a global
solution of the problemP.
Example Consider the following problem (problemP):
Minf(x,y) =x+y,
subject tox+y≤,
x+ y≤,
x,y≥.
The feasible region of this problem is shown in Figure . LetE(x,y) = (x,
y), then the problemPEis as follows:
min(f◦E)(x,y) = x
+
y
,
subject to x
+ y ≤, x +
y≤,
x,y≥.
We note thatE(M)⊂M, where
(√, )∈M implies E(√, ) =
√ ,
∈M,
(, )∈M implies E(, ) =
,
∈M,
(, )∈M implies E(, ) = (, )∈M, (, )∈M implies E(, ) =
,
∈M.
The Kuhn-Tucker conditions are as follows:
∇(f ◦E)(x,y) +u∇(g◦E)(x,y) +u∇(g◦E)(x,y) = , x y
+u x y
+u x = , u x + y – = , u x +
y–
Figure 2 The feasible region M.
The solution is{[x= .,u= .,u= .,y= .]},z= (, ), andx=E(z) = (, ) is a
solution of the problemP.
4 Conclusion
In this paper we introduced a new definition of anE-differentiable convex function, which transforms a non-differentiable function to a differentiable function under an operatorE:
Rn→Rn, 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.
Author details
1Department of Mathematics, Faculty of Computers and Informatics, Suez Canal University, Ismailia, Egypt.2Current
address: Mathematics Department, College of Science, Majmaah University, Majmaah, KSA.3Computer Science Institute
Suez City, Suez, Egypt.4Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt.
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.
Received: 3 May 2012 Accepted: 30 April 2013 Published: 16 May 2013
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doi:10.1186/1029-242X-2013-246
