Megahed Journal of Inequalities and Applications 2014, 2014:457
http://www.journalofinequalitiesandapplications.com/content/2014/1/457
RESEARCH
Open Access
A fuzzy semi-infinite optimization problem
Abd El-Monem A Megahed*
*
Correspondence:
a_megahed15@yahoo.com
Permanent address: Department of
Basic Science, Faculty of Computers
and Informatics, Suez Canal
University, Ismalia, Egypt
Present address: Department of
Mathematics, College Of Science,
Majmaah University, Al-Zulfi, KSA
Abstract
In this paper, we present a fuzzy semi-infinite optimization problem. Moreover, we
will deduce the Fritz-John and Kuhn-Tucker necessary conditions of this problem.
Finally, a numerical example is given to illustrate the results.
MSC: 90C34; 90C05; 90C70; 90C30; 90C46
Keywords: fuzzy; semi-infinite; optimization; necessary conditions
1 Introduction
In many practical problems, we might have information containing some uncertainty,
which is treated in this paper as fuzzy information; the considered semi-infinite optimization problem with fuzzy information is a fuzzy semi-infinite optimization problem.
Fuzzy set theory was introduced into conventional linear programming by Zimmermann [], fuzzy mathematical programming was presented in [], and fuzzy programming
and linear programming with several objective functions were presented by [].
Optimality conditions of a nonlinear programming problem with fuzzy parameters are
established, and a fuzzy function is defined, with its differentiability, convexity, and some
important properties being studied in []; the fuzzy solution of optimization problems
and the incentive solution of the optimization problems are presented which explain that
the solution of optimization problems is a generalization of the solutions in the case of
crisp optimization problems []. As regards fuzzy mathematical programming: theory,
application, and an extension are presented in [].
This paper is organized as follows: In Section , the formulation of the problem of semiinfinite optimization is considered. In Section , the main section of the paper, we will
study a fuzzy semi-infinite optimization problem, and the Fritz-John and Kuhn-Tucker
necessary conditions. Finally, the conclusion is drawn in Section .
2 A semi-infinite programming problem
A semi-infinite programming problem is an optimization problem in which finitely many
variables appear with infinitely many constraints [, ], and we consider a generalized
semi-infinite optimization problem (GSIP) [, ] of the form
⎫
min f (x) s.t. x ∈ M,⎪
⎪
x
⎪
⎪
⎬
x ∈ Rn /hi (x) = , i ∈ I, G(x, y) ≥ 
,
M=
⎪
for all y ∈ Y (x)
⎪
⎪
⎪
⎭
r
where Y (x) = y ∈ R /uk (x, y) = , k ∈ K, vl (x, y) ≥ , l ∈ L .
(.)
©2014 Megahed; 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.
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I, K , and L are finite index sets with |l| < n and |K| < r (where | · | denotes the cardinality),
all appearing functions are real valued and continuously differentiable, and the set Y (x) is
compact for each x ∈ Rn , and the set-valued mapping Y : x ∈ Rn → Y (x) ⊂ Rn is upper
semi-continuous at each x ∈ Rn .
For the special case that the set Y (x) = Y does not depend on the variable x, this problem is a common semi-infinite problem (SIP). The generalized semi-infinite and bi-level
optimization problem are presented by Stein and Still []. Bi-level problems are of the
following form.
(BL):
min f (x) s.t. x ∈ M and y is a solution of
x
(.)
min G(x, y)
y
s.t. y ∈ Y (x).
The generalized semi-infinite programming on generic local minimizers was introduced
by Gunzel et al. []. The feasible set in generalized semi-infinite optimization is presented
by Jongen et al. in []. Furthermore, the linear and linearized generalized semi-infinite
optimization problems were introduced by Rukmann []. In [], a first-order optimality condition in generalized semi-infinite programming is introduced. Still discussed the
optimality conditions for generalized semi-infinite programming problems in [].
3 A fuzzy semi-infinite programming problem
3.1 Problem formulation
A fuzzy semi-infinite programming problem is defined as
⎫
f (x) s.t. x ∈ M,⎪
min
⎪
x
⎪
⎪
⎬
n
x ∈ R /hi (x) = , i ∈ I, G(x, y) ≥ 
,
where M =
⎪
for all y ∈ Y (x)
⎪
⎪
⎪
⎭
r
Y (x) = y ∈ R /uk (x, y) = , k ∈ K, vl (x, y) ≥ , l ∈ L .
(.)
All functions of this problem have the same properties as the problem (.). For x ∈ M
denote the index sets of the active inequality constraints by
Y (x) = y ∈ Y (x)/G(x, y) =  ,
L (x, y) = l ∈ L/vl (x, y) =  for y ∈ Y (x) .
(.)
3.2 Lower level problem
Consider the following lower level problem:
⎫
G(x, y) ⎬
min
y
s.t. y ∈ Y (x).⎭
(.)
The fuzzy requirements of the lower level problem (.) can be quantified by electing
a membership function μ(G(x, y)) (Figure ) which is differentiable in the open interval
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Figure 1 The membership function μ(G(x, y)).
G(x, y ) < G(x, y) < G(x, y ), where μ(G(x, y)) is defined by
μ G(x, y) =
⎧
⎪
⎨,
G(x,y)–G(x,y )
G(x,y )–G(x,y )
⎪
⎩
,
G(x, y) ≤ G(x, y ),
,
G(x, y ) ≤ G(x, y) ≤ G(x, y ),
(.)
G(x, y) ≥ G(x, y ),

where G(x, y ) and G(x, y ) denote the values of the objective function of the lower level
problem (.) with the degree of the membership function  and , respectively, i.e.,
G(x, y ) is an undesirable value and G(x, y ) is a desirable value of the objective function
G(x, y).
Definition  The α-level set of the fuzzy goal G(x, y) is defined as the ordinary set
Lα (G(x, y)); for the value of G(x, y) the degree of its membership function exceeds the
level α, i.e.
Lα G(x, y) = G(x, y)/μ G(x, y) ≥ α, α ∈ (, ) ,
where α is the least acceptable degree of the required value. For certain degrees α the
problem (.) can be transformed into the following equivalent form:
⎫
⎪
min G(x, y) ⎪
⎬
y
s.t. y ∈ Y (x), ⎪
⎪
μ(G(x, y)) ≥ α.⎭
(.)
If x ∈ M and y ∈ Y (x), then y is a minimizer of the problem (.).
By the Fritz-John conditions there exist coefficients λ, β = (β k , k ∈ K), γ = (γ l , l ∈
L (x, y)), and ν satisfying
Dy L(x,y) (x, y, λ, β, γ , ν) = 
and λ + ν +
k∈K
|β k | +
γ l = ,
(.)
l∈L (x,y)
λ ≥ , γ l ≥ , ν ≥ , l ∈ L (x, y),
where
L(x,y) (x, y, λ, β, γ , ν) = λG(x, y) – v μ G(x, y) – α –
βi uk (x, y) –
γl vl .
k∈K
l∈L (x,y)
In other words, for x ∈ M and y ∈ Y (x) the set F(x, y) = {(λ, β, γ , v) ∈ R × R|k| × R|L (x,y)| ×
R/(λ, β, γ , v) satisfies (.)} is nonempty and, furthermore, is also compact.
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4 Fritz-John conditions
Firstly, we will give some lemmas, definitions, and a proposition which will be used in the
proof of the Fritz-John conditions.
Lemma  [] Let xt ∈ M, yt ∈ Y (xt ) and xt → x. Then the set {yt , t ∈ N} is bounded, and
y ∈ Y (x) whenever yt → y.
Lemma  [] Let l = φ, and for x ∈ M let Y (x) = φ. Then x is an interior point of M.
Definition  For x ∈ M define
Dx L(x,y) (x, y, λ, β, γ , ν)/(α, β, γ ) ∈ F(x, y) .
V (x) =
(.)
y∈Y (x)
Lemma  [] Let x ∈ M. Then the set V (x) is compact.
Definition  [] For a set V , we define Conv(V ) to denote the convex hull of V , i.e.
x ∈ Conv(V ) if and only if
x=
n
λi x i ,
xi ∈ V ,
i=
n
λi = , λi ≥ ,
(.)
i=
i.e. Conv(V ) consists of all finite convex combinations of the elements of V .
Lemma  [] Let V ⊂ Rn be a nonempty compact set. Then there exists a ξ ∈ Rn with
sT ξ >  for all s ∈ V if and only if  ∈/ Conv(V ).
Lemma  [] Let I , I be finite index sets and si ∈ Rn , i ∈ I , and zj ∈ Rn , j ∈ I . Then
either (i) or (ii) holds.
(i) There are real numbers ai , i ∈ I , bj ≥ , j ∈ I , satisfying
ai si +
i∈I
i∈I
bj zj = ,
j∈I
|ai | +
(.)
bj = .
j∈I
(ii) The set {si , i ∈ I } is linearly independent and there exists a ξ ∈ Rn with
i T
s ξ = ,
i ∈ I ,
j T
z ξ > ,
j ∈ I .
Proposition  [] For t ∈ R and y ∈ Rr , we define the following functions:
uk (t, y) = uk (x + tξ , y),
vl (t, y) = vl (x + tξ , y),
k ∈ K,
l ∈ L,
G(t, y) = G(x + tξ , y).
Then the following conditions are satisfied.
(.)
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Figure 2 The membership function μ(f (x)).
(i) The set {Duk (, y), k ∈ K} (= {[Dx uk (x, y)ξ , Dy uk (x, y)], k ∈ K}) is linearly
independent.
(ii) There is a w ∈ Rr+ satisfying
⎫
DG(, y)w > ,⎪
⎬
Duk (, y)w = , k ∈ K,
⎪
⎭
Dvk (, y)w < , l ∈ L (Ex, y).
(.)
The fuzzy requirements of the problem (.) can be quantified by electing a membership
function μ(f (x)) (Figure ) which is differentiable in the open interval f (x ) < f (x) < f (x )
where μ(f (x)) is defined by
⎧
f (x) ≤ f (x ),
⎪,
⎨ f (x)–f (x )
μ f (x) = f (x )–f (x ) , f (x ) ≤ f (x) ≤ f (x ),
⎪
⎩
,
f (x) ≥ f (x ),
(.)
where f (x ) and f (x ) denote the values of the objective function of the problem (.) with
the degree of membership function  and , respectively, i.e., f (x ) is an undesirable value
and f (x ) is a desirable value of the objective function f (x). For a certain degree α the
defuzzification of the problem (.) is
min f (x) s.t. x ∈ M,
x
x ∈ Rn /hi (x) = , i = , . . . , m, G(x, y) ≥ 
,
where M =
for all y ∈ Y (x)
Y (x) = y ∈ Rr /uk (x, y) = , k ∈ K, vl (x, y) ≥ , l ∈ L ,
μ f (x) ≥ α , α ∈ (, ),
μ G(x, y) ≥ α, α ∈ (, ).
(.)
Theorem  Let x be a local minimizer of the problem (.). Then either Y (x) = φ and
there exist
⎫
yj ∈ Y (x), j = , . . . , p,⎪
⎬
(λj , β j , γ j , vj ) ∈ F(x, yj ), j = , . . . , p,
⎪
⎭
k ≥ , η ≥ , ρi , i ∈ I, ζj ≥ , j = , . . . , p,
(.)
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satisfying
⎫
> ⎪
⎪
⎪
as well as⎬
kDf (x) – ηD(μ(f (x) – α )) – i∈I ρi Dhi (x)⎪
⎪
⎪
p
⎭
j
– j= ζj Dx L(x,y ) (x, yj , λj , β j , γ j , ν j ) = ,
k+η+
i∈I
|ρi | +
p
j= ζj
(.)
or Y (x) = φ and there exist k ≥ , η ≥ , ρi , i ∈ I, satisfying
k + η + i∈I |ρi | > ,
kDf (x) – ηD(μ(f (x) – α )) – i∈I ρi Dhi (x) = .
(.)
Proof Let x be a local minimizer of the problem (.). We distinguish three cases.
Case : The set {Dhi (x), i ∈ I} is linearly dependent. Then we are done by choosing k = ,
η = , ζ =  in (.) (if Y (x) = φ), or k = , η =  in (.) (if Y (x) = φ) as well as a linear
combination i∈I ρi Dhi (x) =  with i∈I |ρi | > .
Case : There exists a y ∈ Y (x) and the set {Duk (x, y) = , k ∈ K} is linearly dependent.
Then there exists a linear combination k∈K β k Duk (x, y) = , k∈K |β k | =  and therefore
we have k∈K β k Dy uk (x, y) = , (, β, , ) ∈ F(x, y) (with β = (β k , k ∈ K)), and
Dx L(x,y) (x, y, , , β, ) =
β k Duk (x, y) = .
(.)
k∈K
Case : Neither Case  nor Case  holds. Then the proof is similar to Theorem . in
[].
5 A constraint qualification
Definition  The Mangasarian-Fromovitz constraint qualification (MFCQ) of the problem (.) is said to hold at x if:
() the set {Dhi (x), i ∈ I} is linearly independent and
() there exists a ∈ Rn such that
⎫
for all i ∈ I : Dhi (x) = ,⎪
⎬
for all y ∈ Y (x) : Dx L(x,y) (x, y, λ, β, γ , ν) > ,
⎪
⎭
for all (λ, β, γ , v) ∈ F(x, y).
(.)
Theorem  Let x be a local minimizer of the problem (.) and (MFCQ) be satisfied.
Then either Y (x) = φ and there exist
⎫
yj ∈ Y (x), j = , . . . , p,⎪
⎬
(λj , β j , γ j , vj ) ∈ F(x, yj ), j = , . . . , p,
⎪
⎭
k ≥ , η ≥ , ρi , i ∈ I, ζj ≥ , j = , . . . , p,
(.)
satisfying
⎫
> ⎪
⎪
⎪
as well as⎬
⎪
kDf (x) – ηD(μ(f (x) – α )) – i∈I ρi Dhi (x)⎪
⎪
p
⎭
j
– j= ζj Dx L(x,y ) (x, yj , λj , β j , γ j , ν j ) = ,
k+η+
i∈I
|ρi | +
p
j= ζj
(.)
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or Y (x) = φ and there exist k ≥ , η ≥ , ρi , i ∈ I, satisfying the set
k + η + i∈I |ρi | > ,
kDf (x) – ηD(μ(f (x) – α )) – i∈I ρi Dhi (x) = .
(.)
The proof is similar to the proof of Theorem  if we choose k = .
Example

f (x) = x –  + x ,
min

subject to G(x, y) = y + x ,
V (x, y) = x – y .
The lower level problem is
G(x, y) = y + x ,
min
subject to V (x, y) = x – y .
The optimal solution of the crisp problem is (x , x ) = (  , ), y = , and (λ, γ ) = (, ); and
G (x, y) = .
Let f (x) =  and G (x, y) =  be undesirable values of the problem, the membership
functions of f and G are defined by
f  (  , ) = ,

μ(f ) =
 
f –f

=
–
x
–
+
x
–

,


f –f

μ(G) =
G – G
=  – y – x .
G – G
The new problem is
 
+ x ,
min f (x) = x –

subject to G(x, y) = y + x ,
V (x, y) = x – y ,
 
+ x –  ≥ α ,
– x –

 – y – x ≥ α  ,
α ∈ (, ).
The lower level problem is
min G(x, y) = y + x ,
V (x, y) = x – y ,
 – y – x ≥ α ,
α ∈ (, ),
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then
L(x, y, α , γ , γ ) = λ (y + x ) – γ x – y – γ (–y – x +  – α ),
Dy L(x, y, α , γ, γ ) = λ + yγ + γ = ,
then y = –
λ  + γ
, λ + γ + γ = .
γ
Since
kDf (x) – γ D μ f (x) – α – β Dx L(x, y, λ , γ, γ ) = ,
we have


+ γ x –
+ β γ = ,
k x –


kx + γ x – β (λ + γ ) = ,
x =
x =
β γ

–
,
 k + γ
β (λ + γ )
;
k + γ
the solution is
⎫
⎧⎡
⎤
β = ., k = ., γ = ., λ = .,
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎨⎢
⎥
γ = ., γ = .,
⎢
⎥
⎢
⎥ , f = ., G = . .
⎪
⎪
⎣α = ., α = ., x = .,⎦
⎪
⎪
⎪
⎪
⎭
⎩
x = ., y = –.
6 Conclusion
In this work, we discussed a fuzzy semi-infinite optimization problem, by considering that
The Fritz-John conditions and the
the minimum of the objective function is fuzzy (min).
constraint qualification are discussed for this problem. Finally, an illustrative example is
given to clarify the results.
Competing interests
The author declares that they have no competing interests.
Acknowledgements
The author wants to express his deep thanks and his respect to his faculty, colleagues, the Journal, and Prof. Dr. Rachel M
Bernales of the Journal of Editorial office. Also the author wants to express his thanks and respect to Jane Doe who
provided medical writing services on behalf of XYZ pharmaceuticals Ltd.
Received: 11 July 2014 Accepted: 5 November 2014 Published: 19 Nov 2014
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10.1186/1029-242X-2014-457
Cite this article as: Megahed: A fuzzy semi-infinite optimization problem. Journal of Inequalities and Applications
2014, 2014:457
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