Solving the Class of Nonsmooth Nonconvex Fuzzy Optimization
Problems via the Absolute Value Exact Fuzzy Penalty Function Method
TADEUSZ ANTCZAK
Faculty of Mathematics and Computer Science,
University of Lodz,
Banacha 22, 90-238 Lodz,
POLAND
Abstract: - In recent years, in optimization theory, there has been a growing use of optimization models of real
decision-making processes related to the activities of modern humans, in which the hypotheses are not
verifiable in a way typical of classical optimization. This increases the demand for tools that will enable the
effective solving of such more real optimization models. Fuzzy optimization problems were developed to
model real-world extremum problems with uncertainty, which means that they are not usually well-defined. In
this work, we investigate one of such tools, i.e. the absolute value exact fuzzy penalty function method which is
applied to solve invex nonsmooth minimization problems with fuzzy objective functions and inequality (crisp)
constraints. Namely, we analyze the exactness of the penalization which is the most important property of any
such method from a practical point of view. Further, the algorithm of the absolute value exact penalty function
method is presented in the context of finding weakly nondominated solutions of the analyzed nonsmooth fuzzy
optimization problem and, moreover, its convergence is proven in the considered fuzzy case. Finally, we also
simulate the choice of the penalty parameter in the aforesaid algorithm.
Key-Word: - fuzzy optimization, nondifferentiable optimization problem with the fuzzy objective function,
Clarke generalized gradient, Karush-Kuhn-Tucker optimality conditions, nondominated solution,
absolute value exact penalty function method, exactness of the penalization, invex fuzzy
function.
Received: September 9, 2023. Revised: April 13, 2024. Accepted: May 11, 2024. Published: June 27, 2024.
1 Introduction
Many real-world O.R. systems and processes cannot
be modeled easily in deterministic terms since they
involve imprecision of data. In fact, the data are
often uncertain in nondeterministic models of real-
world systems and processes due to, for example,
prediction and/or estimation errors, or lack of
information (e.g., some extremum problems that
arise in economics, industry, engineering
applications, commerce, sciences might involve
financial returns, differing costs, design parameters
of such systems in designing phase are usually under
uncertainties, future actions might be unknown at the
time of the decision). Hence, most of the real
research problems are subject to some form of
uncertainty. The reason for this is the fact that some
coefficients of the objective and/or the constraint
functions in such optimization problems cannot be
exactly assessed, due to the fact that they are
imprecise, unreliable vague, etc.
Fuzzy optimization is one of the useful and
efficient approaches for treating just such real-world
decision-making problems under uncertainty. The
basic concept of fuzzy decision-making was first
proposed by in the paper, [1]. Since then, many
authors studied extensively fuzzy mathematical
programming problems. Namely, the definition of a
convex fuzzy mapping was firstly introduced in the
paper, [2]. After that, the convexity notion for fuzzy
mapping has been widely used in fuzzy optimization
by several authors (see, for example, [3], [4], [5], [6],
[7], [8], [9], [10], [11], [12], and others). However,
the convexity notion is too restrictive in fuzzy
optimization, due to the fact that not all optimization
problems modeling real-world O.R. processes with
uncertain data are convex. Therefore, several authors
have defined and applied generalized convex fuzzy
mappings to fuzzy optimization (see, for example,
[13], [14], [15], [16], [17], [18], [19], [20], [21],
[22], [23], [24], and many others).
One of the well-known approaches in
optimization theory for looking for optimal solutions
in constrained mathematical programming problems
is exact penalty function methods. In the last few
decades, many researchers have been focused to find
optimal solutions in various types of extremum
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problems by using exact penalty function methods.
The idea behind the aforesaid methods is that, by
using chosen exact penalty function, the original
problem of a constrained extremum problem can be
reduced to an unconstrained optimization problem.
Thus, it is possible to avoid the difficulties, that take
place in other approaches, at least related to finding
feasible points and/or directions. Moreover, in this
way, to find optimal solutions of constrained
extremum problems the algorithms developed in
unconstrained optimization can be applied. The exact
penalty function that has been most frequently used
by many researchers to solve their constrained
optimization problems and, is, therefore, the most
popular exact penalty function, is the absolute value
exact penalty function, also called the l₁ exact
penalty function (see, for example, [25], [26], [27],
[28], [29], [30], [31], [32], [33], [34], [35], [36],
[37], [38], [39], [40], and others). In [25] and [26]
the most important property of the exact penalty
function method, that is, exactness of the
penalization, was analyzed for new classes of
nonconvex optimization problems. Whereas the
aforementioned property was investigated in the
paper [40], for the vector exact penalty function
method which they used to solve nondifferentiable
invex vector optimization problems. Recently, the
classical exact penalty function method was
applied in the paper [27] to solve a nonsmooth
constrained interval-valued optimization problem
with both equality and inequality constraints and the
property of exactness of the penalization was
analyzed when this method is applied to solve a
nondifferentiable interval-valued mathematical
programming problem.
According to the literature, only a few studies
have explored the methods for solving nonconvex
nondifferentiable fuzzy optimization problems so
far, and the present study is one of the first reports to
address this problem. In this article, therefore, we
use the absolute value exact penalty function method
to solve a nonsmooth optimization problem with
fuzzy objective function and inequality (crisp)
constraints. Then, for the considered fuzzy
minimization problem, we construct its associated
fuzzy penalized optimization problem with the
exact fuzzy penalty function. Further, in the fuzzy
context, we generalize the main property of all exact
penalty function methods, i.e. exactness of the
penalization. We analyze it, moreover, under
appropriate invexity hypotheses in the case when we
use the absolute value exact fuzzy penalty function
method for solving such nonsmooth fuzzy extremum
problems. Namely, we prove that a (weak) Karush-
Kuhn-Tucker point of the investigated
nondifferentiable fuzzy optimization problem is a
(weakly) nondominated solution of its associated
penalized fuzzy optimization problem with the fuzzy
exact penalty function for all penalty parameters
exceeding the given threshold. We also establish the
equivalence between a (weakly) nondominated
solution of the considered fuzzy optimization
problem and a (weakly) nondominated solution of its
associated fuzzy penalized optimization problem
with the exact fuzzy penalty function for
sufficiently large penalty parameters. Further, we
present an algorithm of the absolute value exact
penalty function method which is applied for finding
weakly nondominated solutions in the considered
nonsmooth optimization problem with fuzzy
objective function and inequality constraints. Its
convergence is also established in the considered
fuzzy case. After that, we analyze the strategy for
choosing the penalty parameter in the applied
absolute value exact fuzzy penalty function method
and we illustrate it by the appropriate examples of
constrained fuzzy minimization problems.
2 Notations and Preliminaries
We first present some preliminary notations and
present such definitions and results, which will be
used in this work. Throughout this paper, R is the set
of all real numbers, that is, endowed with the usual
topology. A fuzzy subset of R is a function
. We usually named this mapping a
membership function of a fuzzy number . We now
define the -level set for any fuzzy set (denoted by
) as follows
where is the closure of the support of , that is,
.
Definition 1. [7], [24] A fuzzy number in R is a
fuzzy set on R with the following properties: 1) is
normal, i.e. there exists such that ,
2) is quasi-concave, i.e.
for all
and any , 3) is upper semicontinuous, i.e.
is a closed subset of R for
each , 4) the 0-level set, i.e. , is a
compact subset of R.
Hence, if a fuzzy set is such that is a
singleton, then is called a fuzzy number, [5], [7].
Let us denote by the family of all fuzzy
numbers in R. Thus, for every , is a
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nonempty convex and compact subset of R for each
. Hence, the -levels of a fuzzy interval can
be described by , for all
. In particular, the fuzzy number is
given as follows if , and ,
otherwise. Also any can be regarded as a fuzzy
number defined by
Note that a fuzzy number is often defined in the
literature by the end points of the interval
[5], [6], [16], [24], [41] and many others.
Remark 1. The notation was introduced in the
paper [41], to represent the crisp number with the
value a. It is easy to see that
for all
Given two fuzzy numbers which are
represented by their -level sets as ,
for any , respectively, and
Then, we define the fuzzy addition and
the scalar multiplication as follows, [6], [7], [10],
[20]:
These operations on fuzzy numbers can be defined in
the equivalent way (see, [6], [7]). Namely, for every
,
Definition 2. A special type of a fuzzy number is a
triangular number which is described by three real
numbers as and its
definition is as follows:
The -level set of a triangular fuzzy number is
defined by
Definition 3. [41], Let and be two fuzzy
intervals. If there exists a unique such that
(note that the fuzzy addition is
commutative), then we call the Hukuhara
difference (H-difference, for short) of and and
we denote it by .
Proposition 1. [41] Let and be two fuzzy
intervals. If the Hukuhara difference
exists, then and for each
Throughout this paper, the following convention
for inequalities between two intervals
and in R are used : if and only if
and , and if and only if
and . The following two order relations on the
space are considered and used in this paper.
Let be given two fuzzy intervals
described by their -level sets and
for each , respectively.
Definition 4. [24] We write if and only if
for each , which is equivalent to
or or for all .
Definition 5. [24] We write that if and only if
for all , which is equivalent to
for all or for all
or for all .
3 Nondifferentiable Invex Crisp and
Fuzzy Functions
Now, we introduce some notations and recall some
basic definitions for nondifferentiable crisp
functions. It is well-known that a crisp mapping
is a locally Lipschitz function at a point
if there exist scalars and such
that the inequality holds
for all , where B is the open unit ball in
, so that is the open ball of radius about x.
We say that the mapping h is a locally Lipschitz
function (on ) if it is locally Lipschitz at any point
of .
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Definition 6. [42], Let be a locally
Lipschitz function at The Clarke generalized
directional derivative of h at in the direction
, which is denoted by , is defined by
.
Definition 7. [42], The Clarke generalized
subgradient of the locally Lipschitz crisp function
at , denoted by , is defined
by .
From the aforesaid definitions, it follows that, for
any , , [42].
We recall that the notion of a locally Lipschitz
invex function was introduced in [43].
Definition 8. Let be a locally Lipschitz
crisp function and be a given point. If there
exists a vector-valued function such
that the inequality
is fulfilled for all , then f is an invex
function (a strictly invex function) at on . If the
above inequality is satisfied at any point , then h is
an invex function (a strictly invex function) on . If
(5) is satisfied on a nonempty subset , then h
is a (strictly) invex function on S.
Proposition 2. [42], Let be a locally
Lipschitz function on a nonempty set ,
be
any scalar and be an arbitrary point of S. Then
.
Proposition 3. [42], Let be
locally Lipschitz crisp functions on a nonempty open
set and be an arbitrary point of . For
any scalars , one has
.
The following result is useful in proving one of
the main results in this paper.
Proposition 4. [40], Let be a locally
Lipschitz crisp function on S and . Further, let
be defined by . If
is an invex function at on S with respect to the
function , then is also a locally
Lipschitz invex function at on S with respect
to
.
Now, we re-call the definition of a fuzzy mapping
given, for example, in the paper, [6].
Definition 9. [6], Let S be a nonempty subset of .
Then is said to be a fuzzy mapping. For
each , we associate with the family of
interval-valued functions given by
. The -cut of at , which is a
bounded and closed interval for each , we
denote by
where and .
Thus, can be represented by two functions and
, which are functions from to the set R,
is a bounded increasing function of , is a
bounded decreasing function of and, moreover,
for all and each . Here,
the endpoint functions are called
left- and right-hand functions of , respectively.
Now, in a natural way, we generalize the
definition of a locally Lipschitz function to the case
of a fuzzy mapping.
Definition 10. A fuzzy mapping is said
to be locally Lipschitz at a given point if, for
each , its left- and right functions and
are locally Lipschitz at x.
We now give the definition of the Clarke
generalized derivative at of a locally Lipschitz
fuzzy function introduced in [44] as a pair of
Clarke generalized derivatives at of its left-
and right-hand functions and defined for
the fixed -cut .
Definition 11. The Clarke generalized directional -
derivative of a locally Lipschitz fuzzy function
(given by (6) at x for some -cut in
the direction d is defined as the Clarke generalized
directional -derivatives of its left- and right
functions and at x in the direction d as
follows:
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.
As it follows from the aforesaid definition, the
Clarke generalized derivative at of a locally
Lipschitz fuzzy function f does not represent an
interval.
Definition 12. It is said that a locally Lipschitz fuzzy
function is directionally differentiable
in the sense of Clarke at x if exists for each
direction d and for all -cuts.
Now, we give the definition of the Clarke
generalized gradient of a locally Lipschitz fuzzy
function introduced which was firstly introduced in
[44].
Definition 13. The Clarke generalized gradient of a
locally Lipschitz fuzzy function on the
-cut is defined as a pair of Clarke generalized
gradients of the left- and right-hand functions on this
-cut, that is, the pair ,
where
and
.
Remark 2. It follows by Definition 13 that, for each
-cut and any , we have
The the notion of invexity for a differentiable
fuzzy function was firstly introduced in [22]. This
definition was extended to the case of a locally
Lipschitz fuzzy function in [44]. Namely, the
concept of invexity for a locally Lipschitz fuzzy
function was defined via invexity of its left-hand
and right-hand functions and by using the
-cuts of given in [24], [41].
Definition 14. [44], Let be a locally
Lipschitz function and be a given point. If
there exists a vector-valued function
such that the following inequalities
are satisfied for any , then is an
invex fuzzy function (a strictly invex fuzzy function)
at u on . If (7) and (8) are satisfied at any point u,
then is an invex fuzzy function (a strictly invex
fuzzy function) on . If (7) and (8) are satisfied on a
nonempty subset S of , then is an invex fuzzy
function (a strictly invex fuzzy function) on S.
We now illustrate the concept of invexity for
locally Lipschitz fuzzy mappings and, therefore, we
present an example of a locally Lipschitz invex
fuzzy function.
Example 1. Let the fuzzy function be
defined by , where and are
continuous triangular fuzzy numbers which are
defined as triples and (see
Definition 2). Then, by (4), the -level sets of both
triangular fuzzy numbers are given by
and
respectively. Moreover, by (4) and (6), the -level
cut of the fuzzy function is given as follows
for each . Therefore, the left-hand side and
right-hand side functions and are given
by:
for each . Note that and are not
convex functions and so the fuzzy function is not
convex. Moreover, we note that and are
not differentiable at and, therefore, is not
a level-wise differentiable fuzzy function at x (see
Definition 4.2 [40]). The graphs of the left- and
right-hand functions of , , are given on
Figure 1.
It can be shown by definition that is a locally
Lipschitz strictly invex fuzzy function on with
respect to defined by
Since and are locally Lipschitz functions
for every , by Definition 10, is a locally
Lipschitz fuzzy ampping on . Further, note that,
for each , (7) and (8) are fulfilled for all
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with respect to defined above as strict
inequalities for the functions and .
Fig. 1: Graphs of the left- and right-hand functions of
, , .
Then, by Definition 14, is a strictly invex
fuzzy function on with respect to defined
above.
Remark 3. Note that there is, in general, more than
one vector-valued function with respect to which a
fuzzy function is invex. Indeed, if we consider again
the fuzzy function which is
defined in Example 1, then it can be shown that it is,
in fact, invex on also with other functions
. Let us define the vector-valued
function as follows
Thus, the functions and are strictly
invex on also with respect to the defined above
function . Hence, by Definition 14, the fuzzy
function is also strictly invex on R² with respect to
defined above.
In the work, we assume that only such fuzzy
mappings are considered for which
their left-hand side and right-hand side functions
and are locally Lipschitz at a given point
x of interest for all .
4 Nondifferentiable Invex Fuzzy
Optimization Problem and its
Optimality
In this work, we investigate the following
constrained optimization problem with a fuzzy-
valued objective function defined as follows:
where is a fuzzy function and
, are real-valued functions defined on
. Let be the set of
all feasible solutions of the problem (FO). Now, we
denote the set of active inequality constraints at a
point by . Throughout
the article, we shall assume that all functions
involved in the fuzzy optimization problem (FO)
given above, that is, its fuzzy objective function
and its constraint functions , are locally
Lipschitz on .
In this paper, we use the -cuts to describe the
objective fuzzy function, as it was done in the papers
[24] and [41]. Therefore, we shall assume that its
left- and right-hand side values of are given by the
functions and
for each , respectively.
Now, for the formulated above fuzzy
optimization problem (FO), we define its optimal
solutions as weakly nondominated and
nondominated solutions which have been introduced
in the paper [24].
Definition 15. [24], We say that is a weakly
nondominated solution in the considered fuzzy
optimization problem (FO) if there exists no other
such that . In other words, if
is a weakly nondominated solution in (FO), then, by
Definition 5, there exists no other such that
or
or . (9)
Definition 16. [24] We say that is a
nondominated solution in the considered fuzzy
optimization problem (FO) if there exists no other
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such that . This means that, if
is a nondominated solution in (FO), by
Definition 4, there exists no other such that
or
or .
(10)
Remark 4. If we denote by and the sets of
weakly nondominated and nondominated solutions
in (FO), respectively, then .
In [44], under invexity hypotheses, optimality
conditions of Karush-Kuhn-Tucker type were
established for a nonsmooth optimization problem
(FO) with fuzzy objective function. We now give the
aforesaid Karush-Kuhn-Tucker like optimality
conditions for to be a (weakly) nondominated
solution in the investigated nonsmooth fuzzy
optimization problem (FO).
Theorem 1. [44], Let be a feasible solution in the
investigated fuzzy optimization problem (FO).
Moreover, assume that there exist ,
and for each such that
the following Karush-Kuhn-Tucker like optimality
conditions
hold. Further, assume that the objective function is
an invex fuzzy mapping at on with respect to
and, moreover, each constraint , , is an
invex function at on with respect to the same
function , then is a nondominated solution in
(FO).
Theorem 2. [44], Let be a feasible solution in the
investigated fuzzy optimization problem (FO) and
there exist , such that
the weak Karush-Kuhn-Tucker like optimality
conditions hold.
Further, assume that and are invex at on
with respect to and, moreover, the functions ,
, are invex at on with respect to the
same function . Then is a weakly nondominated
solution of the fuzzy optimization problem (FO).
Now, we give the necessary optimality conditions
of Karush-Kuhn-Tucker type for the considered
invex fuzzy optimization problem (FO).
Theorem 3. [44], Let be a weakly
nondominated solution in the fuzzy optimization
problem (FO). Moreover, assume that the objective
function is an invex fuzzy function at on with
respect to each constraint , , is invex
at on with respect to the same function and the
Slater constraint qualification is satisfied for (FO).
Then, there exist , and
such that the Karush-Kuhn-
Tucker like optimality conditions hold at for (FO).
Corollary 1. [44] Let be a weakly
nondominated solution in the fuzzy optimization
problem (FO) and hypotheses of Theorem 3 be
fulfilled. Then, there exist ,
and satisfying (17)-
(18) such that
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Throughout this work, it is assumed that the
Slater constraint qualification, [28], is fulfilled at any
weakly nondominated solution in the investigated
fuzzy optimization problem (FO).
Definition 17. The point is said to be Karush-
Kuhn-Tucker point (a KKT point, for short) if, for
each , there are Lagrange multipliers
, and such that the
Karush-Kuhn-Tucker like optimality conditions
(11)-(13) are fulfilled at .
Definition 18. The point is said to be a weak
Karush-Kuhn-Tucker point (a weak KKT point, for
short) if, for some , there are Lagrange
multiplier such that the weak Karush-
Kuhn-Tucker like optimality conditions (14)-(16) are
fulfilled at .
5 Exactness Property of the Absolute
Value Exact Penalty Fuzzy function
Method for Fuzzy Optimization
Problem with Invex Functions
It is known in optimization theory that exact penalty
methods are one of approaches which can be applied
for finding optimal solutions in constrained
extremum problems. Their construction is based on
the so-called penalty function whose unconstrained
minimizing points are, at the same time, optimal
solutions of the constrained optimization problem for
all sufficiently large values of the penalty parameter.
Hence, an original constrained extremum problem is
transformed into a single unconstrained optimization
problem in each methods of such a type.
Therefore, if we use any exact penalty function
method to solve the given nonlinear constrained
optimization problem with a fuzzy objective
function, we have to construct in this approach its
corresponding unconstrained penalized fuzzy
optimization problem as follows
where is an fuzzy function, p is a
suitable penalty function,
is a penalty parameter
and is a crisp number with value (see
Remark 1). The aforesaid penalized fuzzy
optimization problem (FP(
)) is constructed in such
a way that its fuzzy objective function is the sum of a
certain fuzzy "merit" function (which is the
counterpart of the fuzzy objective function in the
original fuzzy extremum problem) and the penalty
term, which is the counterpart of the constraints
define its feasible set. The fuzzy merit function is
defined as the fuzzy original objective function of
the given constrained extremum problem and the
penalty term is formulated by multiplying a function
designed by the constraints of the aforesaid
optimization problem, by a positive parameter
. We
call the aforesaid parameter the penalty parameter.
Note that the objective function of the
unconstrained fuzzy penalized optimization problem
is a fuzzy mapping. Note that, by (6), for any
arbitrary fixed , we associate with the
family of interval-valued functions
given by
for any , where
are real-valued functions.
Therefore, for every fixed , the -cut of the
unconstrained fuzzy penalized optimization problem
(FP(
)) is defined by:
F
It is known from the optimization literature, that
the property of exactness of the penalization is the
most important property from a practical point of
view for each exact penalty function method. Now,
we extend and generalize in a natural way the
definition of this property given in the literature for
classical exact penalty function methods to the fuzzy
case.
Definition 19. If a threshold value exists such
that, for every ,
arg (weakly) nondominated
arg (weakly) nondominated ,
then is called a exact penalty fuzzy function
and, therefore, we call (FP(
)) the penalized fuzzy
optimization problem with exact penalty fuzzy
function.
Note that the function can be interpreted as
an exact penalty fuzzy function in such a way that a
constrained (weakly) nondominated solution in the
original fuzzy optimization problem (FO) can be
found by looking for unconstrained (weakly)
nondominated solutions of the aforesaid function
, for sufficiently large values of the penalty
parameter
.
The often used nondifferentiable exact penalty
function method to solve nonlinear extremum
problems is the absolute value penalty function
method, also called in optimization theory the
exact penalty function method. If the aforesaid exact
penalty function method is applied to solve (FO),
then its formulation is:
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(21)
where is a crisp number with the
value . We call (FP(
)) defined
above by (21) the fuzzy penalized optimization
problem with the exact fuzzy penalty function.
Hence, for any fixed , we define the -
levels of the exact fuzzy penalty function for the
original nonlinear fuzzy optimization problem (FO)
by
,
where left-hand side and right-hand side values of
are given
by and
,
respectively. Hence, we can re-write, for any fixed
, the unconstrained penalized fuzzy
optimization problem with the exact penalty fuzzy
function defined by (22) as follows
,
. (F (
))
(23)
Now, for any inequality constraint function
, we define the function
as follows
Note that the aforesaid function possesses the
suitable penalty features which depend on the single
inequality constraint function . If we use (24),
then, for any fixed , we can re-formulate the
definition of (FP(
)) as follows
,
. (F (
))
(25)
Now, we establish that a Karush-Kuhn-Tucker
point of (FO) is a nondominated solution in (FP(
))
for sufficiently large values of penalty parameters
greater than the threshold equal to the largest
Lagrange multiplier associated to some inequality
constraint.
Theorem 4. Let be a Karush-Kuhn-Tucker
point of the considered nonsmooth fuzzy
optimization problem (FO) and, for each ,
the Karush-Kuhn-Tucker like optimality conditions
(11)-(13) be fulfilled at with Lagrange multipliers
, and , .
Furthermore, assume that the objective function is
an invex fuzzy mapping at on with respect to
and each inequality constraint , , is an invex
function at on with respect to the same function
. If the penalty parameter is assumed to be
sufficiently large (namely, let us set the penalty
parameter to satisfy the condition
), then is a nondominated
solution in the penalized fuzzy optimization problem
(FP(
)) with the exact penalty fuzzy function.
Proof. Assume that is a Karush-Kuhn-
Tucker point in (FO) and, moreover, for each
the Karush-Kuhn-Tucker optimality
conditions (11)-(13) are satisfied at with Lagrange
multipliers , and ,
. By means of contradiction, we suppose that
is not a nondominated solution of (FP(
)). Thus, by
Definition 15, there exists such that
for all . Hence, by
Definition 5, the above relation implies
or or
.
By the definition of ( ()) (see (25)), it follows
that, for all ,
or
or
Multiplying the above inequalities by the
corresponding Lagrange multipliers ,
associated to the fuzzy objective function,
then adding the resulting inequalities and using
, we get
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Since , by (24), it follows that .
Thus, (26) gives
If we use the Karush-Kuhn-Tucker optimality
condition (12) together with , then we obtain
Since is assumed to be an invex fuzzy mapping at
on with respect to the function , therefore, by
Definition 14, for each , the inequalities
hold. Further, each inequality constraint function
, is invex at on with respect to the same
function . Then, by Definition 14, the following
inequalities
are satisfied. Multiplying each inequality (28) and
(29) by the corresponding Lagrange multiplier and
then adding both sides of the resulting inequalities
and (30), we get that the inequality
holds. Hence, from the Karush-Kuhn-Tucker
optimality condition (11) and Proposition 3, (31)
yields that the inequality
holds, contradicting (27). Thus, this completes the
proof of this theorem.
Corollary 2. Let be a nondominated solution
of (FO) and all the hypotheses of Theorem 4 be
satisfied. Moreover, if we assume that the penalty
parameter
is sufficiently large (namely, let us set
the penalty parameter
is assumed to satisfy the
condition ), then is also a
nondominated solution in each associated penalized
fuzzy optimization problem (FP(
)) with the exact
penalty fuzzy function.
Now, we show that a weak Karush-Kuhn-Tucker
point in the original fuzzy minimization problem
(FO) is also a weakly nondominated solution in its
associated penalized fuzzy optimization problem
(FP(
)) for sufficiently large
.
Theorem 5. Let be a weak Karush-Kuhn-
Tucker point of (FO) and the conditions (14)-(16) be
satisfied at with Lagrange multipliers ,
for some . Furthermore, assume that the
functions and are invex at on with respect
to and the constraints , , are also
invex at on with respect to the same function .
If we assume the penalty parameter
to be
sufficiently large (namely, let us set the penalty
parameter
to satisfy the condition
),), then is a weakly
nondominated solution of (FP(
)).
Proof. From the assumption, we have that the
weak Karush-Kuhn-Tucker optimality conditions
(14)-(16) are fulfilled at with Lagrange multipliers
, , for some . By means of
contradiction, suppose that is not a weakly
nondominated solution of (FP(
)). Therefore, by
Definition 15, there exists such that
. In particular, there exists
such that the system of inequalities
is satisfied for some . By (25), the above
inequalities yield, respectively,
By , the inequalities (32) imply, respectively,
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By assumption, ),) Thus, the
inequalities above give, respectively,
Using the feasibility of in (FO) and (24) together
with the Karush-Kuhn-Tucker optimality condition
(16), we get
or
By hypotheses, the functions and are invex at
on with respect to and, moreover, the
constraint functions , , are also invex at
on with respect to the same function . Hence,
by Definitions 14 and 8, respectively, the inequalities
hold.
Now if we multiply the inequalities (37)-(39) by
corresponding Lagrange multipliers and then adding
both sides of the resulting inequalities, then we
obtain that the inequalities
(40)
(41)
hold for any , ,
. Thus, by the Karush-Kuhn-Tucker
optimality conditions (14) and (15), (40)-(41) yield
that the inequalities
hold, which contradicting (35) or (36). Hence, the
proof of this theorem is completed.
Corollary 3. Let be a weakly nondominated
solution of (FO) and all the assumptions of Theorem
5 be satisfied. Then is also a weakly nondominated
solution of (FP(
)).
Now, we establish the converse results to those
ones established above. First, we derive some useful
results, which we use in proving them.
Proposition 5. Let be a nondominated solution
of (FP(
)). Then, there is no such that
. (42)
Proposition 6. Let be a weakly nondominated
solution of (FP(
)). Then, there is no such that
. (43)
Theorem 6. Let be a compact subset of and
be a (weakly) nondominated solution of the
fuzzy penalized optimization problem (FP(
)) with
the exact fuzzy penalty function. Further, assume
that the objective function f is an invex fuzzy
function at on with respect to , each inequality
constraint , , is invex at on with respect to
the same function . If the penalty parameter
is
sufficiently large, then is also a (weakly)
nondominated solution of the considered fuzzy
optimization problem (FO).
Proof. Let be a nondominated solution in the
fuzzy penalized optimization problem (FP(
)) with
the exact fuzzy penalty function.
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Firstly, we assume that . Hence, by
Proposition 5, it follows that that there does not exist
such that (42) is satisfied. Thus, by Definition
16, is a nondominated solution of the considered
fuzzy optimization problem (FO).
Now, under the assumptions of this theorem, we
shall prove that the case is impossible. By
means of contradiction, suppose that . Since
is a nondominated solution in the fuzzy penalized
optimization problem , there exist ,
, , , and
, such that
. Using the
definition of the absolute value exact fuzzy penalty
function, one has
.
Since the weights and are nonnegative
for each , therefore, equality holds in
Proposition 3. Thus, the above relation yields
.
Using together with Proposition 2,
we get
Thus, by Proposition 3, we have
By hypothesis, is an invex fuzzy function at on
(with respect to ). Then, by Definition 14, the
functions and are invex at on with respect
to for each Hence, for each , the
inequalities
hold for all . Further, since each constraint
function , , is invex at on with
respect to the same function , by Proposition 4, also
the functions , are invex on with respect
to the same function . Hence, by Definition 14, the
inequalities
hold for all . Multiplying (47) by , we
obtain, for any ,
Combining (45), (46) and (48), we have that the
inequalities
hold for all and any , ,
.
Now, if we multiply (49) and (50) by and
, respectively, and then we add both sides of
them, we get
Since , we have that, for all
and any , ,
,
.
Hence, by (44), (51) implies that the inequality
is satisfied for all . By (24), for each ,
one has . Hence, the above inequality
yields that the inequality
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is fulfilled for all . By assumption, is not a
feasible solution of the original fuzzy optimization
problem (FO). Hence, by (24), one has
. Then, by the foregoing inequality,
(52) gives
.
From the assumption, is sufficiently large. Now,
we suppose that, for each , is assumed to
satisfy
We now prove that, by (53), is a finite
nonnegative real number. Indeed, by assumption,
is a nondominated solution in the penalized fuzzy
optimization problem ( ) with the absolute
value exact penalty fuzzy function. Then, by
Definition 16, there does not exist such that
Hence, by (54), we have that . From the
assumption, is a compact subset of . This
implies that is a finite real number. Since the
inequality (54) contradicts the inequality (53), this
gives that the case is impossible. Thus, is
feasible in the oroginal fuzzy optimization problem
(FO). This means that, for any , , which is a
nondominated solution of (FP( )), is also a
nondominated solution of (FO). Thus, the conclusion
of the theorem follows from Proposition 5. The proof
of this theorem, in the case when is a weakly
nondominated solution of the fuzzy penalized
optimization problem (FP( )), is similar and the
conclusion theorem follows from Proposition 6 in
such a case. Thus, this completes the proof of this
theorem.
Now, we present one of the main results of this
work which follows directly from the results
established above.
Theorem 7. Let all the hypotheses of Corollary 2
(Corollary 3, respectively) and Theorem 6 be
satisfied. Then is a (weakly) nondominated of the
considered fuzzy optimization problem (FO) with
the fuzzy objective function if and only if is a
(weakly) nondominated of the penalized fuzzy
optimization problem (FP(
)) with the exact
penalty fuzzy function.
We now present the example of a nonlinear
nonconvex fuzzy optimization problem in which its
objective function is a nondifferentiable invex fuzzy
function and its constraints are invex crisp functions
with respect to the same function . Then, using the
exact penalty method analyzed in this paper, we
solve this nonsmooth fuzzy extremum problem in
order to illustrate the result formulated in Theorem 7.
Example 2. Consider the nonconvex nonsmooth
fuzzy optimization problem with the fuzzy-valued
objective function formulated as follows:
where and are continuous triangular fuzzy
numbers. These fuzzy numbers are defined as triples
and . Hence, by (4), the -
level sets of these triangular fuzzy numbers are
and , respectively.
Moreover, we notice that
is
the set of all feasible solutions of (FO1) and,
moreover, is a feasible solution of (FO1).
Further, by (1) and (2), the -level cut of the fuzzy
objective function is given by
for any . Clearly, the left- and right-hand
side functions and are not convex and so
is not convex. Since and are not
differentiable at , is not a level-wise
differentiable fuzzy function at (see Definition
4.2 [40]). The Karush-Kuhn-Tucker optimality
conditions (11)-(13) are fulfilled with Lagrange
multipliers , , , for
each . Moreover, all functions constituting
(FO1) are locally Lipschitz, that is, the objective
function is a locally Lipschitz fuzzy function by
Definition 10. Further, the functions involved in
(FO1) satisfy invexity hypotheses of Corollary 2.
Indeed, if we define by
, then the functions , ,
and are invex at on R with respect to .
Since we use the exact penalty function method in
solving (FO1), therefore, we have to construct its
associated penalized fuzzy optimization problem
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(FP1(
)) with the exact fuzzy penalty function
defined by:
(
Since all the assumptions of Corollary 2 are
satisfied, is a nondominated solution in
(FP1(
)) for any penalty parameter . Further,
all hypotheses Theorem 6 are also fulfilled. Thus, if
we assume that is a nondominated solution in
(FP1(
)), then it is also a nondominated solution of
(FO1). Hence, we have shown under invexity
hypotheses the equivalence between nondominated
solutions in fuzzy optimization problems (FO1) and
(FP1(
)) for any penalty parameter .
6 The Convergence of the Absolute
Value Exact Fuzzy Penalty Function
Method
In this section, we present an algorithm for solving
the investigated nondifferentiable fuzzy optimization
problem (FO) with fuzzy objective function and
crisp inequality constraints by using the exact
penalty fuzzy function method and we prove its
convergence in the considered fuzzy case.
Therefore, we create the following sequence of
the associated fuzzy penalized optimization
problems (FP( )) for the original nondifferentiable
fuzzy extremum problem (FO) as follows:
where is a sequence of penalty parameters with
and, moreover, .
Algorithm (l1EFPFM) of the exact fuzzy
penalty function method:
Given , tolerance and starting point ;
FOR
Starting at , solve to find a weakly
nondomi-nated solution ;
IF , THEN
STOP with an approximate weakly
nondominated solution ;
ELSE
a new penalty parameter should be
chosen;
a new starting point should be chosen;
END IF;
END FOR;
Before we establish the convergence of the
analyzed exact penalty fuzzy function method
which is used to solve the considered
nondifferentiable fuzzy optimization problem (FO),
we present and prove some useful results.
Lemma 1. i) If , then .
ii) If , then .
Proposition 7. Let be a weakly nondominated
solution of the penalized fuzzy optimization problem
(FP( )), generated by Algorithm
(l1EFPFM). If is a convergent subsequence of
and its limit point, i.e. is a feasible
solution of the original fuzzy optimization problem
(FO), then .
Proof. By means of contradiction, suppose that
Hence, (55) implies that there exists a convergent
subsequence of generated by Algorithm
(l1EFPFM) such that
, (56)
where is a nonnegative real number. Since is a
weakly nondominated solution of , by
Definition 15, there is no such that
(57)
Hence, (57) implies that there is no such that
for all
or for all
or for all .
Then, by (23), it follows that, for there is
no such that,
for all or
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for all or
for all .
By assumption, . Since the above
system of inequalities is not satisfied for any ,
therefore, it is not satisfied also for . Thus, we
get that the following system of inequalities
for all or
for all or
for all
is not satisfied. Therefore, for each point , we
have
or
for some ,
or
for some ,
or
for some .
Taking limit and using (56), we obtain
or
for some .
We have that . Since and are
continuous for each , the above system of
inequalities gives
or
for some .
Since , by Lemma 1, the above inequalities
reduce to the inequality , which contradicts the
fact that is a nonnegative real number. This
completes the proof of this proposition.
Theorem 8. Let be a weakly nondominated
solution in generated by
Algorithm (l1EFPFM). If is a convergent
subsequence of and is a feasible
point of (FO), then is its weakly nondominated
solution.
Proof. Let . It is known that is a
weakly nondominated solution of . Hence,
by Definition 15, there is no such that
.(58)
Hence, (58) gives
for all
or for all
or for all .
Then, by (23), it follows that, for
for all or
for all or
for all .
By assumption, . Thus, by Lemma 1
i), it follows that . Since ,
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are weakly nondominated solutions of
, generated by Algorithm
(l1EFPFM), by Proposition 6, we have that
. Taking limit in the
above system of inequalities, we get that there is no
such that
or
or .
Hence, by Definition 15, is a weakly
nondominated solution of (FO). This completes the
proof of this theorem.
7 The Simulation of the Choice of the
Penalty Parameter
One of the important factors that can ensure the
success of using the fuzzy exact penalty function l₁
method considered in the article for solving fuzzy
optimization problems is the strategy for appropriate
choosing the penalty parameter. Namely, if the initial
value of the penalty parameter is too small in the
algorithm, more cycles may be needed in the
aforesaid approach to determine its appropriate
value. Moreover, the choice of the initial value of
also affects the choice of the starting point . In the
next examples, we illustrate some difficulties that
can be caused by the choice of inappropriate values
of the penalty parameter
.
Example 3. Consider the following fuzzy
optimization problem:
where is a continuous triangular fuzzy number. It is
given as triple . Hence, by (4), the -level
set of this triangular fuzzy number is
. The set of all feasible
solutions is and is an
feasible solution in (FO2).
Now, we use the exact penalty fuzzy function
method in solving the fuzzy optimization problem
(F02) considered in this example. Then, by (22), we
construct the unconstrained fuzzy optimization
problem (FP2). Hence, by (25), for any fixed
the -levels of the fuzzy exact penalty
function are as follows:
where
Hence, one has
Now, we consider two cases:
1)
In this case, ,
where
The graphs of and for chosen
-cuts are presented on Figure 2.
2) .
In this case, , where
The graphs of and for chosen -
cuts are presented on Figure 3.
It is not difficult to note that there is the
significantly better case than the previous one.
Namely, if and the current iterate is any real
number (including the initial point ), then, for the
almost any -cuts , all implementations of
the absolute value exact penalty fuzzy function
method will
give a step that moves to the solution (maybe
except the 0-cut, namely if and
if .
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Fig. 2: The graphs of and
Fig. 3: The graphs of and .
As it follows even from these above cases, the
value of the penalty parameter is crucial to obtain a
correct solution in the applied exact penalty fuzzy
function method to solve fuzzy optimization
problems. However, also the value of is important
in the considered penalty method.
For example, for , for larger -cuts (that is,
), and any , the value of
in next iterates tends to a solution
(moreover, the method behaved similarly for even
greater values of the penalty parameter, for which
the functions , tends to a solution,
what is more, for all -cuts. Therefore, the initial
depends on the values of the penalty parameter and
also on -cuts.
From which it follows the aforesaid result? We
consider the Karush-Kuhn-Tucker necessary
optimality conditions at for the analyzed fuzzy
optimization problem. Then, we have the following
relation:
.
Note that Lagrange multipliers depend on the value
. In fact, we consider the following cases of -cuts
(where we normalize Lagrange to
satisfy ):
1) . Then, we have from the above equation:
2) . Then, we have from the above
equation:
3) . Then, we have from the above equation:
As it follows from the above, the maximum value
of Lagrange multiplier decreases for larger -
cuts. Hence, for all penalty parameters
greater than
the threshold equal to the largest Lagrange multiplier
associated to the constraint g, we can obtain the
solution (in such a case, threshold is equal to
since there is only one constraint in the analyzed
fuzzy optimization problem). Hence, we conclude
that that if the value of is larger then, in general,
the aforesaid threshold is smaller.
In the previous example, we considered such a
fuzzy optimization problem, for which there exists a
threshold of the penalty parameter such that, for
any penalty parameter greater than the aforesaid
threshold, all implementations of the exact fuzzy
penalty function method will give a step that moves
to the solution starting from any initial point (no
maybe, except for the case of 0-cut). Moreover, such
a behavior will be repeated in the algorithm and,
thus, it will produce increasingly better iterates, until
the penalty parameter is not decreased below some
threshold value. However, there are also such cases,
in which the aforesaid threshold of the penalty
parameter may not exist. So, there are such cases, in
which, even if we know an appropriate value of the
penalty parameter for a given solution , this value
may cause the appearance of iterations that move
away from the correct solution or it may be an
insufficient at the starting point. The next example of
a fuzzy optimization problem shows that it is not
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possible to prescribe in advance a value of the
penalty parameter that is adequate at every iteration.
Example 4. Consider the following fuzzy
optimization problem:
where is a continuous triangular fuzzy number,
which is defined as triple . Then, by
(4), the -level set of this triangular fuzzy number is
. The set of all feasible solutions
is and the solution .
We now apply the exact penalty fuzzy function
method in solving (FO3) considered in this example.
Then, by (22), we construct the unconstrained fuzzy
optimization problem (FP3(
)). Hence, by (25), for
any fixed , the -levels of the fuzzy exact
penalty function are as follows:
where
Hence, one has
Note that by the Karush-Kuhn-Tucker necessary
optimality conditions, one has
.
Then, if we normalize Lagrange multipliers
(to satisfy the condition
) associated to -cuts of the
objective function, then we note that
. Therefore, the threshold of
the penalty parameter
, for which there is the
equivalence between nondominated solutions in
(F03) and (FP3(
)) just for all penalty parameters,
satisfies the condition . Now, we
illustrate this result and, therefore, we consider two
sample values of
.
Fig. 4: The graphs of and dla ,
.
Fig. 5: The graphs of and dla , .
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Now, we consider two cases:
Firstly, we consider the case, in which the penalty
parameter .
Note that the graphs on Figure 4 confirm that if
is larger then is a local undominated solution
of provided that the penalty parameter
satisfies the condition given above. However,
and are unbounded below as
(Figure 4). Therefore, for all -cut and for
every penalty parameter
, there is a starting point
, such that there doesn't exist decreasing path in
both and from the aforesaid
starting point to the solution .
Now, we consider the case, in which the penalty
parameter .
Note that we know the value of the penalty
parameter
in the considered case, which is
analyzed for the given solution . However,
this value is inadequate at any starting point . In
fact, for any , the values of in next
iterates will tend to . In other words, each next
iterate moves away from the
solution . Analogously, if we take a
sufficiently small starting point , then the values of
in next iterates will tend to for each
. In other words, each next iterate
for some with , moves
away from the solution . As it follows from
the graphs on Figure 5, such a set is different for
various -cuts.
There is the following question why there is no
equivalence between nondominated solutions in
(FP3) and (FP3(
)), even if we take the penalty
parameter
greater than the threshold? This follows
from the fact that none of the functions and is
invex with respect to any function (see [45]).
Hence, the assumption that the functions and
are invex for all is not fulfilled. Therefore,
the objective fuzzy function is not invex with any
function . Thus, this example illustrates the
case in which not all the functions involved in the
investigated fuzzy optimization problem are invex.
In such a case, there is practically no starting point
at which the exact method of the fuzzy penalty
function could start successfully searching for the
correct solution (since next iterate may move to
and the functions and may tend also
to for all ).
8 Conclusions
We have mentioned in Introduction that there are
many works in the literature on fuzzy optimization
problems in which fundamental results from
optimization theory have been established for such
mathematical programming problems. However,
there are still open problems in the iterature
regarding the introduction of new methods for
solving such non-deterministic extreme problems in
optimization theory. In this work, the absolute value
exact penalty fuzzy function method has been
applied for the first time to solve a new class of
mathematical programming problems, which are
nonconvex and nonsmooth optimization problems
with fuzzy objective functions. Therefore, for the
considered minimization problem with fuzzy
objective function and inequality constraints, the
formulation of the corresponding fuzzy penalized
fuzzy optimization problem with the exact fuzzy
penalty function has been presented. Then, the main
from the practical point of view property of the
exact fuzzy penalty function method, i.e. exactness
of the penalization, has been defined and analyzed in
the considered fuzzy case. The equivalence between
(weakly) nondominated solutions in the analyzed
constrained fuzzy minimization problem and its
corresponding penalized fuzzy optimization problem
has been proven under appropriate invexity
hypotheses. Also the threshold of the penalty
parameter has been given. Then, it has been proven
that the aforesaid equivalence holds if the penalty
parameter in the penalized fuzzy optimization
problem exceeds this threshold. However, if the
functions constituting the considered fuzzy
optimization problems are not invex, this threshold
may not exist. This result has been investigated and
illustrated by an appropriate example of such fuzzy
optimization problems. In other words, the approach
for the choice of the penalty parameter has been
analyzed in the paper and the analysis has been made
both theoretically and practically. Thus, it has been
shown that the exact fuzzy penalty function
method is applicable also for solving a larger class of
nonsmooth optimization problems with fuzzy
objective functions than convex ones. Further, the
algorithm for the exact fuzzy penalty function
method, which is applied for finding weakly
nondominated solutions of the considered
nondifferentiable fuzzy optimization problem has
been given. Also the convergence results have been
obtained for the algorithm presented in this work.
Moreover, the simulation of the choice of the initial
penalty parameter in the aforesaid algorithm has
been performed. Hence, it can be concluded here that
the exact fuzzy penalty function method,
originally designed for deterministic constrained
extremum problems, can also be applied for solving
fuzzy optimization problems.
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Although we have focused on solving scalar
fuzzy extremum problems with fuzzy objective
functions and inequality constraints by applying the
absolute value exact penalty fuzzy function,
however, we believe that the established results are
also applicable for such not well-defined operations
research problems which are modeled by
nondeterministic optimization problems of other
types. Therefore, there remain some interesting
questions for further research. Namely, it would be
interesting to investigate whether it is possible to
prove analogous results for various types of fuzzy
extremum problems. This question will be
investigated in our subsequent works.
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