A new projection technique for developing a Liu-Storey method to solve nonlinear systems of monotone equations

The projection technique is a very important method and efficient for solving unconstraint optimization and nonlinear equations. In this study, we developed a Liu-Story (LS) algorithm for solving monotone equations of nonlinear systems. The new algorithm satisfies the sufficient descent condition and it’s a suitable method of large scale equations for its limited memory. We established a global convergence of suggest method under the mild conditions. Numerical results proved that the new algorithm works well and promising.


Introduction
As we know, the projection approach is a very simple iterative method to find a solution vector * of nonlinear systems: ( ) = 0, ∈ , (1.1) s.t. : → is continuous mapping and monotonicity condition hold, i.e. 〈 ( ) − ( ), − 〉 ≥ 0, ∀ , ∈ . This problem arises in various applications in applied mathematics, power engineering, economics and chemical systems. For example, the variational inequality [1], the problems in proximal algorithm with Bregman distances [2], the problems of economic equilibrium [3] can be reformulated as (1.1). For solving systems of equation, there are many numerical methods including the Newton method [4], derivative-free method [5] and projection technique based gradient direction [6]. Many from that approaches are iterative process begin with , the next iterate is = + , ∈ , where is called search direction while denote step length. We mixed the conjugate gradient method with projection algorithm to be a suitable to deal with large-scale equations. In 1964 Goldstein [7] introduced the first projection technique for convex programming in Hilbert spaces. It was then Solodov and Svaiters [8] Where Ω ⊆ . be a closed convex .
Recently, the Liu-Storey (LS) conjugate gradient formula [9] is Dependent on the last method, we will suggest the developed (LS) form for solving nonlinear systems (1.1) such that where , , ℶ and ∈ (0,1). We win adopt the projection based technique to present a developed (LS) gradient method with projection approach to solve monotone equations of nonlinear systems. In our work, we discuss a developed (LS) projection algorithm for nonlinear systems (1.1). In the next section, we introduce the new algorithm with some assumptions and analysis it's the global convergence. Finally, some numerical experiments and conclusion are presented in the last section.

Projection Based Method
Now, projection gradient technique is another efficient algorithm to solve large scale unconstrained minimization problem: min ∈ ( ), (2.1) where : → is smooth nonlinear function, because of its simplicity and low storage requirements. The steps of a new algorithm are stated as follows:

Global Convergence Test
In this part, we investigate the global convergence of the offer approach and we need some necessary assumptions: Thus (3.2) hold for all ≥ 1, and And the inequality follows form max ‖ ‖‖ ‖, , − ( ) ≥ ‖ ‖‖ ‖. Then (3.3) is hold. □ Now, we derive some properties of algorithm (2.1) and show that the line search is well defined.
This impels contradicts with definition of * . So the line search (1.4) can hold a nonnegative step length in a limited number of backtracking steps its well defined. □ Now, the next lemma like to lemma (3.1) in Solodov and Svaiter [8], so we omit the proof. Where = Therefore, taking the lim as → ∞ in both sides of (3.11) for all ∈ generates ( ̅ ) ̅ > 0. In the other term, by making the lim as → ∞ in both terms of (3.2) for all ∈ , such that ( ̅ ) ̅ ≤ 0. Which generates a contradiction. □

Numerical Experiments
In the present section, we shown the our results of numerical experiments to analyze the performance of (MOH2) and compare it with the three famous methods, a scaled derivative-free projection method (UU) [10], a modified Liu-Story conjugate projection method (LS) [11] and a projection based technique (DFBP) [12].
In the suggested algorithm, we used the parameters: = 0.7, = 0.1, = 0.4, = 2, = 0.5 and = 10 . The parameters in the LS, PDFB and UU come from [10], [11] and [12] respectively. We take on the similar finish condition for each the forth algorithms i.e. we break them when the upper number of approximation override 500000 or the inequality ‖F(x )‖ ≤ 10 is satisfied. All algorithms written in MATLAB program R2014a and turn on a PC (win8) , CPU 2.30 GHz and 4 GB RAM where all these method applied in the same computer. We solved 7 constraint test problem see Awwal et.al. [3] by using 8 initial different starting point similar to the problems in [13,14,15]  It can be observed from the tables that our proposed MOH2 method wins higher percentage of the numerical experiments. Numerical results listed in tables (4.1) and (4.2) show that the new method is efficient for solving problem (1.1).

Conclusions
In the present paper, we introduce a developed Liu-Story (LS) projection type based gradient algorithm to solve the nonlinear systems of monotone equations. The new algorithm is a suitable method of large scale equations due to its low memory requirements. The proposed method satisfies the sufficient descent condition and the global convergence with some suitable assumptions. The numerical experiments indicate that the proposed technique is efficient and very competitive to solve nonlinear systems of monotone equations