New Investigation for the Liu-Story Scaled Conjugate Gradient Method for Nonlinear Optimization

+is article considers modified formulas for the standard conjugate gradient (CG) technique that is planned by Li and Fukushima. A new scalar parameter θNew k for this CG technique of unconstrained optimization is planned. +e descent condition and global convergent property are established below using strong Wolfe conditions. Our numerical experiments show that the new proposed algorithms are more stable and economic as compared to some well-known standard CG methods.


Introduction
Conjugate gradient (CG) strategies consists of a category of nonlinear optimization algorithms, which needs low memory and powerful local and global convergence properties [1,2]. Typically, a CG method is meant to resolve massive scaled nonlinear optimization problem: On the understanding that the function is defined in the form f: R n ⟶ R is smooth nonlinear function. e repetitive formula is in the form (2) e most important component of this formula is α k step-size, and the search direction d k consists of whereas g k � g(x k ) denotes ∇f(x k ) and β k denotes a positive scalar. e step-size α k is sometimes chosen to satisfy bound line search condition [3]. Among these search direction conditions, the strong Wolfe line search condition is sometimes outlined as follows: and 0 < δ < σ < 1. ere are many different formulas for conjugate coefficients as in the following sources, e.g., Hestenes and Stiefel, HS [4]; Fletcher and Reeves, FR [5]; Polak and Ribière, PR [6]; Conjugate Descent, CD [7]; Li and Fukushima, LF [1]; and Liu and Story, LS [8], correspond to different choice for the scalar parameter β k .

A New Scalar Formula for the Parameter θ New k
Here in this part of this article, we proposed a new version for the parameter θ k by relying on the modified BFGS method proposed by Li and Fukushima [1]. In the BFGS method, the matrix B k+1 is updated to the following formula [9]: where y k � g k+1 − g k and s k � x k+1 − x k . In addition, the normal secant relation is outlined consistent with the subsequent formula: e researchers Li and Fukushima presented an appropriate modified BFGS technique which is globally and super-linearly convergent, even though while not requiring convex objective functions. e subsequent modified secant equation is outlined consistent with the subsequent formula as follows: where and r > 0; h k > 0, h k is a parameter defined as Specifically, take value c is constant, and it is greater than zero.
ere are three different cases for the term s T k y k : Case 1: if s T k y k ≤ 0, in this case we have the problem of the nonpositive definite matrix, so Li and Fukushima proposed y k formula as in (9) and developed the corresponding BFGS formula as follows: Moreover, the form of h k in (10), when max is used so that the value (0; zero), is not selected in this case. rough this formula, the researchers proved that the modified symmetric matrix is positive definite [10]. Case 2: if s T k y k > 0, in this case, we can say surely that the BFGS update matrix is symmetric and positive definite when applied within this formula (in other words, when applying the inequality s T k y k > 0 in the formula h k , max � 0) [11].
If we use β LS k of Liu and Story (LS), we use any scalar θ k ; then, (3) becomes where When any positive value to k is greater than one, the new parallel search direction d k+1 provided in equation (12) is the Newton direction. Hence, Newton's direction is Hence, Using equation (8), the new scalar θ New By substituting equations (9) and (10) and by taking max � 0 (because we use the strong Wolfe line search in equation (10), yields h k � c). erefore, the new scalar within the new search direction is Hence, we conclude from equation (17) that the new parameter θ New k is best because it is up to date to find the value of y, and also we find different forms when changing the value of c as we will notice in the section of numerical results.

New Theorem (Sufficient Descent Direction)
If we presume that the line search satisfies conditions (4) and (5), then the new search direction which is generated from equations (12) and (17) could be a sufficient descent direction.

Journal of Mathematics
Proof. From equations (12) and (17) we obtained By using Powell restart equation (i.e., |g T k g k+1 | ≥ 0.2g 2 k+1 ), If s T k g k (0.2 ‖g k+1 ‖ 2 + c‖g k ‖ r s T k y k ) > 0, the next inequality is true: (20) Using strong Wolfe line search condition (5a) yields Journal of Mathematics is latter equation implies that us, our requirement is complete.

Outlines of the New CG-Algorithms
Step 1: select the initial point x 0 ∈ R n , ∈ > 0, and select some positive values for δ and σ. en, set d 0 � − ∇f(x 0 ) and set k � 0.
Step 2: test for stopping criterion. If satisfied, then stop; otherwise, continue.
Step 5: calculate the scalar parameter θ New k from equation (17).
Step 8: set the next iteration k � k + 1, and go to Step 2.

Convergence Analysis for the New Proposed Algorithm
In the following parts, we have a tendency to discuss the convergence analysis property for the new algorithm thoroughly. First, we offer an assumption for the convergence analysis property for the new algorithms. en, we offer another well-known lemma needed within the study of convergence analysis property. Finally, we have a tendency to set new theorems aboard their proofs that area unit associated with the convergence analysis for the new algorithm.
(ii) In neighbourhood N of S, f is continuously differentiable, and its gradient is Lipschitz continuous, that is, there exists a constant L > 0, such as From the assumptions (i) and (ii) on f, we are able to deduce that there exists c > 0 such as Lemma. If we suppose that [3,13] (1) Assumption holds.
(2) Search direction d k+1 in the standard CG method is a descent direction. (3) Optimal step α k is calculated by equations (4) and (5). en,

New Theorem (Uniformly Convex Function)
If we suppose that (1) Assumption holds.
(2) e new search direction d k+1 defined by equations (12) and (17) is a descent direction. (3) e optimal step α k is calculated by equations (4) and (5). (4) e objective function f is uniformly convex; then, 4 Journal of Mathematics Proof. Consider the new direction in equation (12) and the parameter of equation (17) satisfy the next absolute value condition: Moreover, by combining the results, we obtained We got the required proof. We put similar points to the previous hypotheses, but there are some variations in the formulas.

New Theorem (General Function)
If we suppose that (1) Assumption holds.
(4) e objective function f is general function; then, Proof. Using the same proof style of the previous theorem with the difference in the fact that the functions of the algorithm are general functions, Since en, we obtain erefore, the proof of the new theorems in regards to the convergence analyses of the proposed algorithms is complete.

Numerical Experiments
In this section, we have reported some numerical experiments that are performed on a set of (60) unconstrained optimization test problems to analyse the efficiency of θ New k . Detail of these test problems, with their given initial points, can be found in [14,15]. We handled each of these (60) test functions by adding 1000 for each n to arrive at maximum number of n which is equal to 10000. e termination criterion used in our experiments is ‖g k ‖ ≤ 10 − 6 , where δ � 0.01 and σ � 0.1.
In our comparisons below, we employ the following algorithms: In Tables 1 and 2, we numerically compare the proposed new CG algorithms against other well-known CG algorithms to verify their performance using the known comparison tools for such algorithms which are as follows:  NOI � the total number of calculated iterative iterations NOFG � the total number of function and gradient calculations TIME � the total CPU time required for the processor to execute the CG algorithm and reach the minimum value of the required function minimization erefore, among these CG algorithms, the new algorithm appears to generate the best search direction. In Table 3    All these comparisons were made using the performance profile of Dolan and Moré [16], and we can conclude that (1) Figure 1

Conclusions
In this study, we have submitted two proposed new CG methods (by changing the value of c). A crucial property of proposed CG methods is that it secures sufficient descent directions. Under mild conditions, we have demonstrated that the new algorithms are globally convergent for each uniformly convex and general functions using the strong Wolfe line search conditions. e preliminary numerical results show that if we decide a good value of parameter c, the new algorithms perform very well. However, an optimal value of the parameter c can be handled theoretically (in future research studies) to achieve more best numerical results.

Data Availability
e data used the support the findings of this study are available from the corresponding author upon request.