An Alternative Modified Conjugate Gradient Coefficient for Solving Nonlinear System of Equations

Abstract In mathematical term, the method of solving models and finding the best alternatives is known as optimization. Conjugate gradient (CG) method is an evolution of computational method in solving optimization problems. In this article, an alternative modified conjugate gradient coefficient for solving large-scale nonlinear system of equations is presented. The method is an improved version of the Rivaie et el conjugate gradient method for unconstrained optimization problems. The new CG is tested on a set of test functions under exact line search. The approach is easy to implement due to its derivative-free nature and has been proven to be effective in solving real-life application. Under some mild assumptions, the global convergence of the proposed method is established. The new CG coefficient also retains the sufficient descent condition. The performance of the new method is compared to the well-known previous PRP CG methods based on number of iterations and CPU time. Numerical results using some benchmark problems show that the proposed method is promising and has the best efficiency amongst all the methods tested. ARTICLE INFORMATIONIn mathematical term, the method of solving models and finding the best alternatives is known as optimization. Conjugate gradient (CG) method is an evolution of computational method in solving optimization problems. In this article, an alternative modified conjugate gradient coefficient for solving large-scale nonlinear system of equations is presented. The method is an improved version of the Rivaie et el conjugate gradient method for unconstrained optimization problems. The new CG is tested on a set of test functions under exact line search. The approach is easy to implement due to its derivative-free nature and has been proven to be effective in solving real-life application. Under some mild assumptions, the global convergence of the proposed method is established. The new CG coefficient also retains the sufficient descent condition. The performance of the new method is compared to the well-known previous PRP CG methods based on number of iterations and CPU time. Numerical results using some benchmark problems show that the proposed method is promising and has the best efficiency amongst all the methods tested. ARTICLE INFORMATION Received: 22-Jul-2019 Revised: 12-Aug-2019 Accepted: 31-Aug-2019


INTRODUCTION
In this paper, the following nonlinear system of equations is considered.
( ) = 0, ∈ ; (Eq 1) where : → is continuously differentiable. Newton and quasi-Newton methods are the most widely used methods to solve such problems because they have very attractive convergence properties and practical application (see [1,2,3,4]). However, they are not usually suitable for large-scale nonlinear systems of equations because they require Jacobian matrix, or an approximation to it, at every iteration while solving optimization problems. The aim of this article is to proffer solution to these shortcomings.
The iterative method used to solve (Eq 1) is formed by +1 = + α , = 0,1,2,, (Eq 2) where α > 0 is the step-size, is the iterative point and is the search direction. In this CG method, the search direction is described by where : → is continuously differentiable, while the scalar parameter is the conjugate gradient coefficient. The following are example of .
In [12], a method for solving unconstrained optimization problems is discussed. This article gives an improved version of the method by extending it to nonlinear system of equations as follows.
Initially, a nonlinear conjugate gradient method for the unconstrained optimization problem is considered. min where the function is assumed to be continuously differentiable from into , and the gradient ∇ ( ) at point denoted as is available. The nonlinear conjugate gradient method generates a sequence { } by the recursive relation +1 = + α , = 0,1,2, ⋯, (Eq 5) where α is the step-length and the search direction d is updated by where , a scalar as is defined above [12] with ( + α ) = min ≥0 ( + α ). (Eq 8) Now the following section described the proposed method for solving large-scale nonlinear systems of equations (Eq 1).
The first section of this paper gives the introduction while the new CG method and its Algorithm are given in Section 2. In section 3, sufficient descent condition and global Convergence analysis are 6 presented. Implementation of new method and discussion on Numerical results are presented in the last section, conclusion was drawn based on the performance profile.

The new CG method and its Algorithm
This section presents the new modified conjugate gradient as an alternative method for solving nonlinear system of equations. The new CG coefficient is denoted as DSHM , where the acronym DSHM comes from the names of the researchers and stands for Dauda, Shehu, Hayatu and Mustafa respectively. The equations is represented as follows: where +1 = ( +1 )and ‖. ‖ indicates the Euclidian norm of vectors. Note that, for easy representation +1 = ( +1 ) and = ( ). This paper considered the parametrs 1 > 0, 2 > 0, ∈ (0,1) as constants and be a given positive sequence such that ∑ < ∞. and = {1, } that satisfy The line search (Eq 10) is used to calculate the stepsize. The following is the DSHM algorithm.
Step 5: Update the new point based on (Eq 2 or Eq 5).

Convergent Analysis
Every numerical analysis must satisfy convergence test, the new Modified CG coefficient is not an exception, because the algorithm satisfied the global convergence properties and the descent condition sufficiently.

Sufficient Descent Condition of DSHM Algorithm.
Consider the following theorem with exact line search as in (Eq 10). Theorem 1. Consider DSHM method with (Eq 3), DSHM is said satisfy descent condition sufficiently if ≤ − ‖ 2 ‖ for all ≥ 0, and ∈ ℜ.

Implementation of new Method and Numerical Results
This section presents the numerical results of the implementation of the modified conjugate gradient coefficient for solving nonlinear system of equations. By solving several benchmark problems with their respective initial points using five (5) different dimensions ranging from 10 to 5000, the numerical results of the comparison between the proposed method and the result in [10] is presented in table 1 below. The Table displayed the performances result of each method in terms of number of iteration and CPU time. The meaning of each column in the table are respectively stated "Prob": Benchmark problem; "Dim": Dimension of the test problems; "Iter": the total number of iterations; "CPU": the CPU time in seconds. In case there is no available number of iterations or CPU time, it is considered as failed, and is denoted as "-". Based on Table 1, DSHM's performance is more efficient than PRP. However, both methods failed at certain initial points as shown in the Table. Overall, DSHM manage to solve all given trigonometric functions with less time. Hence, DSHM can be defined as the best approximation method for solving large-scale systems of nonlinear equations.

List of Benchmark Test Problem Used
PROBLEM 1 [3] ( ) = ( ) 2 + ( − − 1); = 1,2,3, ⋯ , . 0 = (0.9,0.9,0.9, ⋯ ,0.9) and 0 = (0.7,0.7,0.7, ⋯ ,0.7) PROBLEM 2 [10] (1) = 1 −  The code for the proposed method and all the problems stated (Benchmark Test Problem) are computed using MATLAB 7.1, R2009b programming environment and run on a personal computer 2.4GHz, Intel (R) Core (TM) i7-5500U CPU processor, 4GB RAM memory and on windows XP operator. Both the methods was implemented with the following parameters: The search is terminated if: (i) ‖ ‖ ≤ 10 −6 or (ii) The total number of iteration exceeds 2000. In particular problem , the DSHM performs better if the number of iteration or the CPU (Time) of DSHM is less than the number of iteration or the CPU time corresponding to the performance in PRP method respectively. To further evaluate the performance of DSHM method relative to other CG method, the results are compiled in two graphs using performance profile based on Dolan and More [13]. Let be the set of benchmark problems and let be the set of algorithms. We define , to be the number of iterations (or the CPU time in seconds) required to solve the problem ∈ by algorithm ∈ . The comparison of each of the two measures is based on the performance ratio given by ; =∶= , / { , }. Then the performance profile is defined by for all ∈ where ( ) is the probability for solver ∈ that a performance ratio , is within a factor ∈ of the best possible ratio, where is the number of benchmark problems.    Figure 2 presents the graphical results of problems 1-4 relative to CPU time. The right side represents the test problems that are successfully solved by each method while the left side of the figure represents the method which is fastest in solving test problems. The top curve is the method that performs better in a time that was within a factor of the best time. Therefore, from Figure 1, the proposed DSHM methods relative to the number of iteration, performs relatively better. Figure 2 gives the performance of DSHM methods relative to CPU time which outperforms PRP.

Discussion and Conclusion
Considering the benchmark problems above with their respective initial points, the new CG, DSHM has been proven to be the best method when compared to standard PRP CG methods. It also manages to solve all of the tested problems within the shortest possible time. The proposed methods are indeed capable for solving large-scale systems of nonlinear equations.