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Using a New Projection Approach to Find the Optimal Solution for Nonlinear Systems of Monotone Equation

There are many algorithms are used to solve systems of nonlinear monotone equations with various advantages and disadvantages, including the line search algorithm, trust region algorithm, projection algorithm and others. In this paper we used a new projection algorithm to solve these systems. The projection methods are considered one of the effective free derivative methods to solve systems of nonlinear monotone equations. The framework of this method is that the current iterate is separated strictly from the solution set of the problem in each iteration by a suitable hyperplane which constructs by the new algorithm. Then, in order to determine the new approximation, the current iteration is projected on this hyperplane. The global convergence of the proposed algorithm is proven under standard assumptions. The numerical results showed that the suggested algorithm is very efficiency and promising.


Introduction
In sciences with different fields and aspects, nonlinear systems and their solutions are of great importance in them, as they are an important part of mathematics and physics (because most physical systems are nonlinear), as well as their importance in engineering, especially mechanical engineering, electricity, management, economics, population growth, weather and other natural phenomena. While it is possible to convert nonlinear problems into linear problems with multiple variables, it can be said that the study of nonlinear problems was side by side with the linear system. Consider the following nonlinear system of equations Where is a continuous and monotone function from , condition of monotony mean: Because of the difficulty of finding a solution to nonlinear system equations in many mathematical and engineering applications, we can only rely on iterative methods that use the iterative procedure to obtain approximate solutions. Newton's method may be one of the best numerical methods that use the iterative method to solve these systems, and it is considered a simple way to find approximate values of equations.
In recent years, many modifications have been made to the Newton method, these suggested methods may be equivalent to (or better than) the Newton method to solve the nonlinear system of equations. The line search method and the trust region method are the most important two methods to solve these systems.
The basic idea of the line search technique is how to find the step length within a specific direction, while the trust region technique is how to find an neighborhood to the current step , provided that the new frequency falls within the area that is the trust region determined by its radius, also, this technique is used to solve unconstrained Optimization [1].
Some methods proved ineffective for solving large-scale nonlinear system of equations as Newton method and quasi-Newton methods [2][3][4][5] because they need to solve the Jacobian matrix or an approximation of it in each iterative.
In our work, we used a new projection technique to solve large-scale systems of nonlinear equations. It can be said that the simplest idea of the projection method is interested in separating the present approximation from the result set of the problem (1) by appropriate hyperplane that is built in each iterate and then projecting this approximation on the same hyperplane to obtain the new approximation. Several authors use conjugate gradient approaches combining with projection techniques for solving (1) as well as optimization issues [6,7].
Solodov and Svaiter suggested the first projection approach in 1998 and it showed the totally convergent of solving nonlinear problems [8].
The authors worked in vary fields such as optimization, operation research and nonlinear systems, but in this work, see [9][10][11][12][13][14][15][16][17][18][19], we used a new algorithm to solve the nonlinear systems, we proved its global convergence . Then we compare with two famous methods, SBM method in [20] and DFPB1 in [1], the new algorithm will be more efficient.

The Framework
The projection technique is one of the ways that proved to be active in solving nonlinear problems and it is a suitable and applicable way to solve large-scale difficulties, these methods use a series of repetitions to arrive to the next iterate where is a step length and is the step direction, these processes are called an iterative procedures, so the projection techniques are called iterative methods . The projection approaches are family of derivative free. To define these effective methods we use the projection operator [. ].
Let  [. ]be a mapping from to , where is non-empty closed convex set [21]: The projection operator has interesting features [22] is non-expansive property: As a result produces, After a series of iterations, in every iteration, the present approximation is isolated from the result set of the problem by the hyperplane that is construction by using a line search technique .
By Solodov and Svaiteʼs suggestion [8], the following iterate xk+1 can be resolute by projection onto , where The approximation that is best among all result of system (1) can be determined by projection onto ,but ∉ . Then the following approximation, +1 , can be determined by The suggested method, built on the projection free-derivatives method for the system of nonlinear equations, determine a direction , a new direction has foreword as Generally used the direction which satisfies.
Our new algorithm will be state as below.

End while
End Remark (R1) [8]: from stage 3 of Algorithm 1, it is easy to note that the introduced direction satisfy the sufficient descent condition, and for any ,

Convergence possessions:
In this part, We need some interesting lemmas and assumptions in showing the global convergence of algorithm 1. Lemma (L1) [22] Let the set  be nonempty closed convex set and The projection operator  ( ) be the projection of x onto closed convex set . For any x , y  , The next statements hold: [8] Assume the assumption , and hold and the sequence{ } is generated via algorithm1. For any * such that ( * ) = 0, then And the sequence { } is bounded. Moreover, either the sequence { } is finite although the last iterate is a solution of (1) or the sequence { } is infinite and Proof Let *  be any point such that ( * ) = 0 by monotonicity of F F( ) , * −  ≤ 0.
The hyperplane separates k from * , it is easy to satisfy that k+1 is the projection of k onto the hyperplane H. Sine x * belongs to this hyperplane, from properties of the projection operator [22] we get [8] Assume that the assumption , and holds and the sequences { } and { } are generated by Algorithm 1, then  (14) , if k ≠  then ̂= −1 does not satisfy (14) this mean that By the Lipchitz continuity of F and (12) we get.
. the proof is complete and the equation (17) is correct. □ The results of Lemma L3 found that the line search of algorithm 1 is well defined.
Theorem (T1) Assume that ( )and( ) hold and the sequence { } is generated by Algorithm 1, then : Proof From (10) and (14) we get By lemma 3 from [1] , the sequence of direction { } that generated by algorithm 1 are bounded. there is a constant N > 0 such that And result that for all k ,there exists a constant L > 0 such that By the Lipschitz continuity of F, it can be concluded that: Now, by using Cauchy Schwartz inequality along with (12), we get From (24) and (26) we have

Numerical Experiment
In this section, we compare the performance of the new algorithm(NBM) with tow famous algorithms: SBM: This technique is taken from Yan, Q.R., et al [20] and it uses two modified HS approaches with the projection technique in Solodov and Svaiter (1998) [8].
We compared the numerical results of the three approaches with reference to (number of iterations), (number of function evaluations) and CPU time (time which required for each algorithm to reach the solution). For this purpose we used the test problems in [20] with 5000-50000 as dimensions for the taken primary points. The PC that used to run these algorithm was with 4 GB RAM and CPU 2.70 GHz. MATLAB R2014a programming environment is used to write all the codes. The terminate value is ‖ ‖ ≤ 10 −8 or ‖ ( )‖ ≤ 10 −4 , or the whole number of iterates surpasses 500000 for all algorithms to be terminated. The parameters are stated as: = 0.4 , β = 0.9 , = 0.1 , ϵ = 10 −8 . The numerical results are registered in tables 1 and 2. Where table 1 contains  and , while table 2   can see that the CPU time spent by the new technique (NBM) is lower than in the other two methods in most of problem, that indicated the efficiency and quality of our new method.

Conclusions
The current work suggests a new projection technique for solving a system of large-scale nonlinear monotone equations. The projection-based algorithms does not use any feature function or derivatives. It considered as a part of the class of derivative-free function-value based approaches. Likewise. For the new method we established the global convergence which is proved under classic assumptions. The numerical results showed that the suggested method is so efficient.