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Malaysian Journal of Computing and Applied Mathematics 2018, Vol 1(1): 11-19
©Universiti Sultan Zainal Abidin
(Online)
11
Solving Fuzzy Nonlinear Equations with a New Class of Conjugate
Gradient Method
*I. M. Sulaimana, M. Y. Wazirib, E. S. Olowoc, A. N. Talatd
a,cFederal College of Agricultural Produce Technology, Kano
bMathematics Department, Bayero University Kano
dApplied Science Department, Aqaba University College, Al-Balqa Applied University, Jordan
*Corresponding author: kademi4u@yahoo.com
Received: 06/05/2018, Accepted: 15/06/2018
Abstract
In this paper, we study the performance of a new conjugate gradient (CG) method for fuzzy nonlinear
equations. This method is simple and converges globally to the solution. The parameterized fuzzy
coefficients are transformed into unconstrained optimization problem (UOP) and the CG method under
exact line search was employed to solve the equivalent optimization problem. The method is discussed in
details followed by the simplification for easy analysis. Numerical result on some benchmark problems
illustrates the efficiency of the proposed method.
Keywords: Conjugate gradient; fuzzy nonlinear; parametric form; unconstrained optimization.
Introduction
Over the past decades, fuzzy nonlinear equations have been playing major role in medicine,
engineering, natural sciences, and many more. However, the main setback is that of using the
numerical method to obtain the solution of the problems. This is due to the fact that the standard
analytical techniques by Buckley and Qu (1990,1991) are only limited to solving the linear and
quadratic case of fuzzy equations. Recently, numerous researchers have proposed various
numerical methods for solving the fuzzy nonlinear equations. i.e. For nonlinear equation
(1)
whose parameters are all or partially represented by fuzzy numbers, Abbasbandy and Asady
(2004) investigated the performance of Newton’s method for obtaining the solution of the fuzzy
nonlinear equations and extended to systems of fuzzy nonlinear equations by Abbasbandy and
Ezzati (2006). Newton method converges rapidly if the initial guess is chosen close to the
solution point. The main drawback of Newton’s method is computing the Jacobian in every
iteration. One of the simplest variants of Newton’s method was considered by Waziri and Moye
(2016) for solving the dual fuzzy nonlinear equations. Another variant of Newton method known
Malaysian Journal of
Computing and Applied
Mathematics
Malaysian Journal of Computing and Applied Mathematics 2018, Vol 1(1): 11-19
©Universiti Sultan Zainal Abidin
(Online)
12
as Levenberg-Marquardt modification was use to solve fuzzy nonlinear equations by Ibrahim et
al. (2018). Also, Amirah et al. (2010) applied the Broyden’s method investigate the fuzzy
nonlinear equations. All these methods are Newton-like which requires the computation and
storage of either Jacobian or approximate Jacobian matrix at every iterative or after every few
iterations. Recently, a diagonal updating scheme for solution of fuzzy nonlinear equations was
proposed by Ibrahim et al. (2018). A gradient-based method by Abbasbandy and Jafarian
(2006) was employed obtaining the root of fuzzy nonlinear equations. This method is simple and
requires no Jacobian evaluation during computations. However, its convergence is linear and
very slow toward the solution (Chong and Zack, 2013). The steepest descent method is also
badly affected by ill-conditioning (Wenyu and Ya-Xiang, 2006). Lately, a derivative-free
approach by Sulaiman et al. (2016) was applied to obtain the solution of fuzzy nonlinear
equations. This bracketing method saves the computational cost of evaluating the derivate of a
function, and it is also bound to converge because it brackets the root any problem. On the other
hand, the convergence is very slow towards the solution due to lack of derivative information
(Touati-Ahmed and Storey, 1990). Motivated by this, we proposed a new CG coefficient and
applied it to solve fuzzy nonlinear equations. The conjugate gradient method is known to be
simple and very efficient in solving optimization problem (Ghani et al., 2016; Sulaiman et al.,
2015). The idea of this paper is to transform the parametric form of fuzzy nonlinear equation into
an unconstrained optimization problem before applying the new CG method to obtain the
solution.
This paper is structured as follows; some preliminaries are given in section 2. Section 3
presents a brief overview and the proposed CG method. The CG method for solving fuzzy
nonlinear equations is presented in section 4. Numerical experiments and implementation are in
section 5. Finally, in section 6, we give the conclusion.
Preliminaries
We present some useful definitions of fuzzy numbers as follows
Definition 1. (Zimmermann, 1991; Dubios and Prade, 1980). A fuzzy number is a set like
which satisfy the following
(1). is upper semi-continuous,
(2). outside some interval ,
(3). there are real numbers such that and
(3.1) is monotonic increasing on
(3.2) is monotonic deceasing on
(3.3) .
The set all these fuzzy numbers is denoted by . An equivalent parametric form is also given in
(Goetschel Jr and Voxman, 1986).
Definition 2 (Dubios and Prade, 1980) Fuzzy number in parametric form is a pair of
function which satisfies the following requirement:
(1) is a bounded monotonic increasing left continuous function,
(2) is a bounded monotonic decreasing left continuous function,
(3)
A popular fuzzy number is the Trapezoidal fuzzy number with interval
defuzzifier [ and left fuzziness and right fuzziness where the membership function is
Malaysian Journal of Computing and Applied Mathematics 2018, Vol 1(1): 11-19
©Universiti Sultan Zainal Abidin
(Online)
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Its parametric form is
New Conjugate Gradient method for Unconstrained Optimization
To overcome the computational burden of other iterative methods, the conjugate gradient
method was suggested as an alternative. This is due to its simplicity, low memory requirement,
and global convergence properties. The CG methods are very important for solving large-scale
unconstrained optimization problems (Mamat et al., 2010). Starting with an initial point , the
CG method compute through a search direction with a step size obtain by line search
procedure to obtain the next iterative given as
(2)
where
(3)
and is the conjugate gradient parameter that characterizes various CG methods. The
classical CG methods are Fletcher-Reeves (FR) (Fletcher and Reeves, 1964), Polak-Ribiere-
Polyak (PRP) (Polak and Ribière, 1969), Hestenes-Stiefel (HS) (Hestenes and Stiefel, 1952),
and a recent coefficient by Rivaie et al. (2014). These methods are defined as follows
,
,
,
.
The convergence of these methods under different line search techniques have been
discussed by Zoutendijk (1970), Al-Baali (1985), Touti-Ahmed and Storey (1990), Gilbert and
Nocedal (1992), and Rivaie et al. (2014). Studying the global convergence of the CG method
under exact line search technique would be very interesting. Hence, we proposed a new CG
coefficient known as
where SM denotes Sulaiman Mustafa and define as follows
(4)
For the convergence, we need to simplify (4) as follows
(5)
Malaysian Journal of Computing and Applied Mathematics 2018, Vol 1(1): 11-19
©Universiti Sultan Zainal Abidin
(Online)
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Since for exact line search,
(Rivaie et al., 2012). Also,
has been
proved to converge globally by Rivaie et al. (2014, 2012). Next, we apply this method to solve
fuzzy nonlinear equations.
New Conjugate Gradient Method for Solving Fuzzy Nonlinear Equations
Given a fuzzy nonlinear equation , and the parametric form defined as
(6)
The idea is to obtain the solution of (6) using conjugate gradient method. We need to
transform (6) into an Unconstrained optimization problem. We start by defining a function
as follows (Abbasbandy and Asady, 2004)
(7)
whose gradient at point is also define as
(8)
From the definition of in (7), then (6) can be transformed to the following
unconstrained optimization problem;
(9)
We define an appropriate CG method for as
(10)
where is obtained by exact line search produces, i.e.
(11)
and
(12)
From the above description, it can be observed that for the same parameters
the solution which satisfies is the same solution for (6) and vice versa.
For the initial point for , one can consider choosing the following fuzzy
number
(13)
whose parametric form is given as
Malaysian Journal of Computing and Applied Mathematics 2018, Vol 1(1): 11-19
©Universiti Sultan Zainal Abidin
(Online)
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, and
The algorithm for the CG method is as follows
Algorithm 1. CG Algorithm for Solving Fuzzy Nonlinear Equation
Step1. Transform the given parametric fuzzy nonlinear equation into UOP
Step2. Evaluate with the initial guess and compute .
Step3. Check if , terminate. Else let .
Step4. Compute step size by (11).
Step5. Update the new value
Step 6. Compute the CG parameter
by (4)
Step 7. Compute the search direction by (12)
Step 8. Repeat step 1 through 7 with until tolerance is satisfied.
Numerical Examples
In this section, we present the numerical solution of some examples using the CG method for
fuzzy nonlinear equations. This is to illustrate the efficiency of the method. All computations are
carried out on MATLAB 7.0 using a double precision computer. Also, details of the solutions are
presented in Figure 1 and Figure 2 respectively.
Example 1: Consider the fuzzy nonlinear equation (Buckley and Qu, 1990)
Without loss of generality, let’s purpose is positive, we give the parametric form of the above
equation as
Next, we solve the above parametric equation for and to obtain the initial guess.
for , we have
, and
for , we have
after computation, we have , , and . From (11), we
have . With the tolerance of , the solution was obtained after 6
iterations. See Fig. 1 for details of the solution.
Malaysian Journal of Computing and Applied Mathematics 2018, Vol 1(1): 11-19
©Universiti Sultan Zainal Abidin
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Figure 1: Solution of Conjugate gradient method for Example 1.
Example 2. Consider the fuzzy nonlinear equation
Without loss of generality, let’s assume is positive, the parametric form of the above equation
is as follows
We solve the parametric equation for and to obtain the initial guess. i.e.
for , we have
, and
for , we have
which implies , , and . By (11), we have
. The solution of the above problem was obtained after 6 iterations with
the tolerance error less than . Refer to Fig. 2 for details of the solution.
0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x1
x2
Malaysian Journal of Computing and Applied Mathematics 2018, Vol 1(1): 11-19
©Universiti Sultan Zainal Abidin
(Online)
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Figure 2: Solution of Conjugate gradient method for Example 2.
.
Conclusion
Recently, the area of fuzzy nonlinear equation has been enjoying a vivid growth with focus on
innovative numerical techniques for obtaining its solution. In this paper, we proposed a new
conjugate gradient method under exact line search for solving the fuzzy nonlinear equation. This
method is simple, requires less memory and hence reduces the computational cost during the
iteration process. The parameterized fuzzy quantities are transformed into unconstrained
optimization problem. Numerical results on some benchmark problems illustrate the efficiency of
the new method.
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(Online)
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