Modified spectral conjugate gradient iterative scheme for unconstrained optimization problems with application on COVID-19 model

1. Introduction

The coronavirus disease, often called COVID-19, is an acute vector infectious disease that emerged in 2019. This disease is caused by the newly discovered coronavirus (SARS-CoV-2) and can be transmitted through droplets produced when an infected person exhales, sneezes, or coughs. Most people infected with the virus will experience mild to moderate symptoms, such as low-grade fever, runny nose, and difficulty breathing, and recover without special treatment [1].

Clinically, as of December 19, 2021, a total of 4,260,544 confirmed cases of COVID-19, with 4,111,619 recoveries and 144,002 deaths, were recorded from all regions in Indonesia since the disease was first reported in Wuhan, China [2]. To date, many studies have been carried out to model various aspects related to the coronavirus outbreak, and several researchers have also applied numerical methods to several COVID-19 models. For instance, Aggarwal et al. [3] proposed a partial differential equation model to calculate the number of COVID-19 cases in Punjab by using the modified cubic B-spline function and differential quadrature method. Other numerical methods which are applied to solve the COVID-19 model were proposed by Amar et al. [4] and Sulaiman et al. [5]. Amar et al. used various statistics and machine learning modeling approaches to forecast the COVID-19 spread in Egypt. Meanwhile, Sulaiman et al. proposed a new three-term conjugate gradient optimization method for the data from the global confirmed cases of COVID-19 from January to September 2020.

The conjugate gradient (CG) method plays an important role in solving large-scale optimization models because it uses low memory and good convergence properties. This method was first introduced by Hestenes and Stiefel [26] and is used to solve a system of linear equations. After that, in 1964, Fletcher and Reeves extended the form of the conjugate gradient method to solve large-scale nonlinear systems of equations and optimization problems without constraints. The results of the expansion carried out by Fletcher and Reeves prompted researchers to propose a new conjugate gradient method to improve computational performance and the level of convergence [6]. In 2020, Jian et al. proposed a conjugate gradient method with a spectral conjugate gradient type named the JYJLL method which is a modification of the Fletcher-Reeves (FR) and conjugate descent (CD) methods [7]. The author has determined the convergence analysis of the JYJLL method which resulted in an efficient computational performance. In addition, Zheng and Shi [8] also proposed a modification of the conjugate gradient method with a three-term type symbolized by ZPRP. This ZPRP method is an extension of the Polak-Ribiére-Polyak (PRP) method [9, 10] in which modifications are made by changing the denominator of the parameters in the PRP method. The computational performance resulting from this method is very efficient when compared to the CG-Descent method [11]. Several CG methods that have been proposed can be seen in literature [12–17]. Besides the CG method, the class of accelerated gradient descent schemes of Quasi-Newton type also contains very efficient and robust methods and can be considered for solving optimization problems. The accelerated parameters highlights can be seen in other studies [18–22]. However, in this paper we restrict the discussion to the CG method.

The CG method has recently been used to solve various problems related to optimization. For example, image reconstruction [23–25], compressed sensing [26], signal processing [27], robotic motion control [5, 15, 16, 28, 29], portfolio selection [5, 13, 14, 29–31], regression analysis [5, 32] and many more.

In this paper, we consider the general unconstrained optimization problems as follows:

$\begin{array}{ll}{\displaystyle \underset{\text{r}\in {ℝ}^n}{min}}f(\text{r}),& (1)\end{array}$

where f : ℝⁿ → ℝ is the continuously differentiable function and its gradient is written by h(r) = ∇f(r). The iterative formula of the standard CG method can be formulated as

$\begin{array}{ll}{\text{r}}_{k+1}={\text{r}}_k+{\alpha }_k{\text{z}}_k,k=0,1,2,...& (2)\end{array}$

and

$\begin{array}{ll}{z}_k:=\{\begin{array}{ll}-h_k,\hfill & \text{if }k=0,\hfill \\ -h_k+{\beta }_k{z}_{k-1},\hfill & \text{if }k>0,\hfill \end{array}& (3)\end{array}$

where r_k is the current iteration, h_k is the gradient value of h at r_k, z_k is the search direction, β_k is the conjugate parameter and α_k > 0 is the step size to be obtained by some line search techniques. To calculate the step size α_k > 0, we can use exact line search, weak Wolfe line search, or strong Wolfe line search. The exact line search is computed such that α_k satisfy

$\begin{array}{l}f({\text{r}}_k+{\alpha }_k{\text{z}}_k)=minf({\text{r}}_k+\alpha {\text{z}}_k),\alpha >0.\end{array}$

The weak Wolfe line search is computed such that α_k satisfy

$\begin{array}{ll}f({\text{r}}_k+{\alpha }_k{\text{z}}_k)\le f({\text{r}}_k)+\delta {\alpha }_k{\text{h}}_k^T{\text{z}}_k,& (4)\end{array}$

$\begin{array}{ll}\text{h}{({\text{r}}_k+{\alpha }_k{\text{z}}_k)}^T{\text{z}}_k\ge \sigma {\text{h}}_k^T{\text{z}}_k,& (5)\end{array}$

and the strong Wolfe line search is computed such that α_k satisfy

$\begin{array}{l}f({\text{r}}_k+{\alpha }_k{\text{z}}_k)\le f({\text{r}}_k)+\delta {\alpha }_k{\text{h}}_k^T{\text{z}}_k,\\ \hfill \left|\text{h}{({\text{r}}_k+{\alpha }_k{\text{z}}_k)}^T{\text{z}}_k\right|\le -\sigma {\text{h}}_k^T{\text{z}}_k,\end{array}$

where 0 < δ < σ < 1.

The most well-known standard CG methods are the Hestenes-Stiefel (HS) method [33], the Fletcher-Reeves (FR) method [34], the Polak-Ribiére-Polyak (PRP) method [9, 10], the Conjugate-Descent (CD) method [35], the Dai-Yuan (DY) method [36], the Liu-Storey (LS) method [37], and the Rivaie-Mustafa-Ismail-Leong (RMIL) method [38] and their β_k parameters are

$\begin{array}{l}\hfill {\beta }_k^{HS}=\frac{{\text{h}}_k^T{\text{q}}_{k-1}}{{\text{z}}_{k-1}^T{\text{q}}_{k-1}},{\beta }_k^{FR}=\frac{||{\text{h}}_k|{|}^2}{||{\text{h}}_{k-1}|{|}^2},\hfill \\ \hfill {\beta }_k^{PRP}=\frac{{\text{h}}_k^T{\text{q}}_{k-1}}{||{\text{h}}_{k-1}|{|}^2},{\beta }_k^{CD}=\frac{||{\text{h}}_k|{|}^2}{-{\text{z}}_{k-1}^T{\text{h}}_{k-1}},\hfill \\ \hfill {\beta }_k^{DY}=\frac{||{\text{h}}_k|{|}^2}{{\text{z}}_{k-1}^T{\text{q}}_{k-1}},{\beta }_k^{LS}=\frac{{\text{h}}_k^T{\text{q}}_{k-1}}{-{\text{h}}_{k-1}^T{\text{z}}_{k-1}},{\beta }_k^{RMIL}=\frac{{\text{h}}_k^T{\text{q}}_{k-1}}{||{\text{z}}_{k-1}|{|}^2},\hfill \end{array}$

respectively, where q_k−1: = h_k − h_k−1 and ||.|| is a symbol for Euclidean norm on ℝⁿ.

The z_k in formula (2) is the search direction used as a guide to move to the next point and must satisfy the descent direction property

$\begin{array}{ll}{\text{h}}_k^T{\text{z}}_k<0,\forall k.& (6)\end{array}$

It should be noted that formula (6) is an important property for the CG method to be globally convergent.

Inspired by the JYJLL method, in this study we propose a modification of the new CG method to improve the computational performance. In addition, in this study, we also apply the new method for solving a model of COVID-19 in Indonesia in which the data is taken from March 2020 (the month of the first recorded case) until May 2022.

The paper is structured as follows. In Section 2, we describe the proposed method, algorithm, and convergence analysis. In Section 3, we present the numerical experiments to show the efficiency of our new method. Finally, the application of regression models of COVID-19 using the new method is illustrated in Section 4.

2. Proposed method, algorithm and convergence analysis

Recently, Jian et al. [7] proposed a new spectral JYJLL CG method where the method satisfies the descent condition without depending on any line search. The JYJLL method is globally convergent under a weak Wolfe line search and the numerical result is efficient compared with HZ [39], KD [40], AN1 [41], and LPZ [42] methods. This new method has search direction as follows:

$z_k:=\{\begin{array}{ll}-h_k,\hfill & \text{if }k=0,\hfill \\ -{\theta }_k^{JYJLL}{h}_k+{\beta }_k^{JYJLL}{z}_{k-1},\hfill & \text{if }k>0,\hfill \end{array}$

where ${\theta }_k^{JYJLL}$ is the spectral parameter defined as

$\begin{array}{ll}{\theta }_k^{JYJLL}=1+\frac{|{\text{h}}_k^T{\text{z}}_{k-1}|}{-{\text{h}}_{k-1}^T{\text{z}}_{k-1}},& (7)\end{array}$

and ${\beta }_k^{JYJLL}$ is formulated as

$\begin{array}{l}{\beta }_k^{JYJLL}=\frac{||{\text{h}}_k|{|}^2-\frac{{\left({\text{h}}_k^T{\text{z}}_{k-1}\right)}^2}{||{\text{z}}_{k-1}|{|}^2}}{max\left\{||{\text{h}}_{k-1}|{|}^2,{\text{z}}_{k-1}^T({\text{h}}_k-{\text{h}}_{k-1})\right\}}.\end{array}$

Additionally, Zheng and Shi [8] proposed a modified three-term HS method by taking a modification to the denominator of the HS formula. The new method is named ZHS where the search direction is defined as follows:

$z_k:=\{\begin{array}{ll}-h_k,\hfill & \text{if }k=0,\hfill \\ -h_k+{\beta }_k^{ZHS}{z}_{k-1}-{\beta }_k^{ZHS}\frac{h_k^T{z}_{k-1}}{h_k^T{q}_{k-1}}{q}_{k-1},\hfill & \text{if }k>0,\hfill \end{array}$

and

$\begin{array}{l}{\beta }_k^{ZHS}=\frac{{\text{h}}_k^T{\text{q}}_{k-1}}{max\left\{\mu ||{\text{z}}_{k-1}||||{\text{q}}_{k-1}||,{\text{z}}_{k-1}^T{\text{q}}_{k-1}\right\}},\mu >0.\end{array}$

The ZHS method satisfies the sufficient descent condition without relying on a certain line search. Under some conditions, the ZHS method fulfills global convergence properties under a weak Wolfe line search and the numerical results are better than the CG-DESCENT method [39].

Motivated by the JYJLL and ZHS parameters, in this paper, the new conjugate parameter is proposed in the form as follows:

$\begin{array}{ll}{\beta }_k^{FMSD}=\frac{||{\text{h}}_k|{|}^2-\frac{{\left({\text{h}}_k^T{\text{z}}_{k-1}\right)}^2}{||{\text{z}}_{k-1}|{|}^2}}{max\left\{\mu ||{\text{z}}_{k-1}||||{\text{q}}_{k-1}||,{\text{z}}_{k-1}^T{\text{q}}_{k-1}\right\}},\mu >0,& (8)\end{array}$

that is, replacing the JYJLL denominator with the ZHS denominator and retaining the JYJLL numerator. In addition, we retain the same formula of the spectral parameters ${\theta }_k^{FMSD}$ by the JYJLL method as in formula (7). So, the search direction of our proposed method is defined as follows:

$\begin{array}{ll}{z}_k:=\{\begin{array}{ll}-h_k,\hfill & \text{if }k=1,\hfill \\ -{\theta }_k^{FMSD}{h}_k+{\beta }_k^{FMSD}{z}_{k-1},\hfill & \text{if }k>1.\hfill \end{array}& (9)\end{array}$

Our proposed method is called the spectral FMSD (Fevi-Malik-Sulaiman-Dipo) method.

Next, we give the algorithm of our proposed method below.

ALGORITHM 1

Algorithm 1. Spectral FMSD method.

The following lemma shows that the spectral FMSD always satisfies the descent direction condition regardless of any line search.

Lemma 2.1. Suppose that z_k is generated by formula (9), then

1. the search direction z_k satisfies the descent direction property, that is, ${\text{h}}_k^T{\text{z}}_k<0$ for k ≥ 1.

2. $0\le {\beta }_k^{FMSD}\le \frac{{\text{h}}_k^T{\text{z}}_k}{{\text{h}}_{k-1}^T{\text{z}}_{k-1}}$.

Proof: We will prove the theorem by induction. For k = 1, it is true, i.e., ${\text{h}}_1^T{\text{z}}_1=\left|\right|{\text{h}}_1|{|}^2$. Now, assume that ${\text{h}}_{k-1}^T{\text{z}}_{k-1}<0$ is true for k − 1, thus we prove ${\text{h}}_k^T{\text{z}}_k<0$ is true for k. With regard to formula (8), the proof is divided into two cases, as presented below:

• Case 1: if ${\text{z}}_{k-1}^T{\text{q}}_{k-1}\le \mu \left|\right|{\text{z}}_{k-1}\left|\right|\left|\right|{\text{q}}_{k-1}\left|\right|$ and μ > 0, then

$\begin{array}{l}{\text{z}}_{k-1}^T{\text{q}}_{k-1}={\text{z}}_{k-1}^T({\text{h}}_k-{\text{h}}_{k-1})\\ ={\text{h}}_k^T{\text{z}}_{k-1}-{\text{h}}_{k-1}^T{\text{z}}_{k-1}\le \mu ||{\text{z}}_{k-1}||||{\text{q}}_{k-1}||,\end{array}$

it implies

$\begin{array}{ll}{\text{h}}_k^T{\text{z}}_{k-1}\le \mu ||{\text{z}}_{k-1}||||{\text{q}}_{k-1}||+{\text{h}}_{k-1}^T{\text{z}}_{k-1}.& (10)\end{array}$

Let θ_k is angle between h_k and z_k−1, then

$\begin{array}{ll}cos{\theta }_k=\frac{{\text{h}}_k^T{\text{z}}_{k-1}}{||{\text{h}}_k||||{\text{z}}_{k-1}||}.& (11)\end{array}$

From formulas (8), (7), (9), (10) and (11), we have

$\begin{array}{l}{\text{h}}_k^T{\text{z}}_k\\ ={\text{h}}_k^T(-{\theta }_k^{FMSD}{\text{h}}_k+{\beta }_k^{FMSD}{\text{z}}_{k-1})\\ =-{\theta }_k^{FMSD}||{\text{h}}_k|{|}^2+{\beta }_k^{FMSD}{\text{h}}_k^T{\text{z}}_{k-1}\\ =-\left[1-\frac{|{\text{h}}_k^T{\text{z}}_{k-1}|}{{\text{h}}_{k-1}^T{\text{z}}_{k-1}}\right]||{\text{h}}_k|{|}^2+\frac{||{\text{h}}_k|{|}^2-\frac{{({\text{h}}_k^T{\text{z}}_{k-1})}^2}{||{\text{z}}_{k-1}|{|}^2}}{\mu ||{\text{z}}_{k-1}||||{\text{q}}_{k-1}||}{\text{h}}_k^T{\text{z}}_{k-1}\\ =-||{\text{h}}_k|{|}^2+\frac{|{\text{h}}_k^T{\text{z}}_{k-1}|}{{\text{h}}_{k-1}^T{\text{z}}_{k-1}}||{\text{h}}_k|{|}^2+\frac{||{\text{h}}_k|{|}^2-||{\text{h}}_k|{|}^2{cos}^2{\theta }_k}{\mu ||{\text{z}}_{k-1}||||{\text{q}}_{k-1}||}{\text{h}}_k^T{\text{z}}_{k-1}\\ \le -||{\text{h}}_k|{|}^2+\frac{|{\text{h}}_k^T{\text{z}}_{k-1}|}{{\text{h}}_{k-1}^T{\text{z}}_{k-1}}||{\text{h}}_k|{|}^2+\frac{||{\text{h}}_k|{|}^2-||{\text{h}}_k|{|}^2{cos}^2{\theta }_k}{\mu ||{\text{z}}_{k-1}||||{\text{q}}_{k-1}||}\\ \times (\mu ||{\text{z}}_{k-1}||||{\text{q}}_{k-1}||+{\text{h}}_{k-1}^T{\text{z}}_{k-1})\\ =\frac{|{\text{h}}_k^T{\text{z}}_{k-1}|}{{\text{h}}_{k-1}^T{\text{z}}_{k-1}}||{\text{h}}_k|{|}^2+\frac{||{\text{h}}_k|{|}^2(1-{cos}^2{\theta }_k)}{\mu ||{\text{z}}_{k-1}||||{\text{q}}_{k-1}||}{\text{h}}_{k-1}^T{\text{z}}_{k-1}\\ -||{\text{h}}_k|{|}^2{cos}^2{\theta }_k\le \frac{|{\text{h}}_k^T{\text{z}}_{k-1}|}{{\text{h}}_{k-1}^T{\text{z}}_{k-1}}||{\text{h}}_k|{|}^2+\frac{||{\text{h}}_k|{|}^2{sin}^2{\theta }_k}{\mu ||{\text{z}}_{k-1}||||{\text{q}}_{k-1}||}\\ {\text{h}}_{k-1}^T{\text{z}}_{k-1}<0.& (12)\end{array}$

• Case 2: if ${\text{z}}_{k-1}^T{\text{q}}_{k-1}>\mu \left|\right|{\text{z}}_{k-1}\left|\right|\left|\right|{\text{q}}_{k-1}\left|\right|$ and μ > 0, then ${\text{z}}_{k-1}^T{\text{q}}_{k-1}>0$. Using formulas (8), (7), (9), and (11), we get

$\begin{array}{l}{\text{h}}_k^T{\text{z}}_k={\text{h}}_k^T(-{\theta }_k^{FMSD}{\text{h}}_k+{\beta }_k^{FMSD}{\text{z}}_{k-1})\\ =-{\theta }_k^{FMSD}||{\text{h}}_k|{|}^2+{\beta }_k^{FMSD}{\text{h}}_k^T{\text{z}}_{k-1}\\ =-\left[1-\frac{|{\text{h}}_k^T{\text{z}}_{k-1}|}{{\text{h}}_{k-1}^T{\text{z}}_{k-1}}\right]||{\text{h}}_k|{|}^2\\ +\frac{||{\text{h}}_k|{|}^2-\frac{{({\text{h}}_k^T{\text{z}}_{k-1})}^2}{||{\text{z}}_{k-1}|{|}^2}}{{\text{z}}_{k-1}^T{\text{q}}_{k-1}}{\text{h}}_k^T{\text{z}}_{k-1}\\ =-||{\text{h}}_k|{|}^2+\frac{|{\text{h}}_k^T{\text{z}}_{k-1}|}{{\text{h}}_{k-1}^T{\text{z}}_{k-1}}||{\text{h}}_k|{|}^2+\frac{||{\text{h}}_k|{|}^2-||{\text{h}}_k|{|}^2{cos}^2{\theta }_k}{{\text{z}}_{k-1}^T{\text{q}}_{k-1}}\\ \times {\text{h}}_k^T{\text{z}}_{k-1}\\ =-||{\text{h}}_k|{|}^2+\frac{|{\text{h}}_k^T{\text{z}}_{k-1}|}{{\text{h}}_{k-1}^T{\text{z}}_{k-1}}||{\text{h}}_k|{|}^2+\frac{||{\text{h}}_k|{|}^2-||{\text{h}}_k|{|}^2{cos}^2{\theta }_k}{{\text{z}}_{k-1}^T{\text{q}}_{k-1}}\\ \times ({\text{h}}_k^T{\text{z}}_{k-1}-{\text{h}}_{k-1}^T{\text{z}}_{k-1}+{\text{h}}_{k-1}^T{\text{z}}_{k-1})\\ =-||{\text{h}}_k|{|}^2+\frac{|{\text{h}}_k^T{\text{z}}_{k-1}|}{{\text{h}}_{k-1}^T{\text{z}}_{k-1}}||{\text{h}}_k|{|}^2+\frac{||{\text{h}}_k|{|}^2-||{\text{h}}_k|{|}^2{cos}^2{\theta }_k}{{\text{z}}_{k-1}^T{\text{q}}_{k-1}}\\ \times ({\text{z}}_{k-1}^T{\text{q}}_{k-1}+{\text{h}}_{k-1}^T{\text{z}}_{k-1})\\ =\frac{|{\text{h}}_k^T{\text{z}}_{k-1}|}{{\text{h}}_{k-1}^T{\text{z}}_{k-1}}||{\text{h}}_k|{|}^2+\frac{||{\text{h}}_k|{|}^2(1-{cos}^2{\theta }_k)}{{\text{z}}_{k-1}^T{\text{q}}_{k-1}}{\text{h}}_{k-1}^T{\text{z}}_{k-1}\\ -||{\text{h}}_k|{|}^2{cos}^2{\theta }_k\\ =\frac{|{\text{h}}_k^T{\text{z}}_{k-1}|}{{\text{h}}_{k-1}^T{\text{z}}_{k-1}}||{\text{h}}_k|{|}^2+\frac{||{\text{h}}_k|{|}^2{sin}^2{\theta }_k}{{\text{z}}_{k-1}^T{\text{q}}_{k-1}}{\text{h}}_{k-1}^T{\text{z}}_{k-1}\\ -||{\text{h}}_k|{|}^2{cos}^2{\theta }_k\\ \le \frac{|{\text{h}}_k^T{\text{z}}_{k-1}|}{{\text{h}}_{k-1}^T{\text{z}}_{k-1}}||{\text{h}}_k|{|}^2+\frac{||{\text{h}}_k|{|}^2{sin}^2{\theta }_k}{{\text{z}}_{k-1}^T{\text{q}}_{k-1}}{\text{h}}_{k-1}^T{\text{z}}_{k-1}<0.& (13)\end{array}$

Hence, ${\text{h}}_k^T{\text{z}}_k<0$ is satisfied for k ≥ 1.

Next, we will prove the interval of ${\beta }_k^{FMSD}$. From formulas (12 and (13, we obtain the relation ${\text{h}}_k^T{\text{z}}_k\le {\beta }_k^{FMSD}{\text{h}}_{k-1}^T{\text{z}}_{k-1}$. Furthermore, since ${\text{h}}_k^T{\text{z}}_k<0$, we have ${\beta }_k^{FMSD}\le \frac{{\text{h}}_k^T{\text{z}}_k}{{\text{h}}_{k-1}^T{\text{z}}_{k-1}}$.

Now, from formulas (8) and (7), we get

$\begin{array}{l}{\beta }_k^{FMSD}=\frac{||{\text{h}}_k|{|}^2-\frac{{({\text{h}}_k^T{\text{z}}_{k-1})}^2}{||{\text{z}}_{k-1}|{|}^2}}{max\left\{\mu ||{\text{z}}_{k-1}||||{\text{q}}_{k-1}||,{\text{z}}_{k-1}^T{\text{q}}_{k-1}\right\}}\\ =\frac{||{\text{h}}_k|{|}^2-||{\text{h}}_k|{|}^2{cos}^2{\theta }_k}{max\left\{\mu ||{\text{z}}_{k-1}||||{\text{q}}_{k-1}||,{\text{z}}_{k-1}^T{\text{q}}_{k-1}\right\}}\\ =\frac{||{\text{h}}_k|{|}^2{sin}^2{\theta }_k}{max\left\{\mu ||{\text{z}}_{k-1}||||{\text{q}}_{k-1}||,{\text{z}}_{k-1}^T{\text{q}}_{k-1}\right\}}\ge 0.\end{array}$

Thus, $0\le {\beta }_k^{FMSD}\le \frac{{\text{h}}_k^T{\text{z}}_k}{{\text{h}}_{k-1}^T{\text{z}}_{k-1}}$ holds. The proof is complete.

In the analysis below, we establish the global convergence properties of the spectral FMSD method. First, we need the following assumption, proposition, and Zoutendijk conditions.

Assumption 2.2. (A1) The level set ${B}:=\left\{\text{r}\in {ℝ}^n:f(\text{r})\le f({\text{r}}_0)\right\}$ is bounded where r₀ is the starting point; (A2) In a neighborhood ${L}$ of ${B}$ the function f is continuously differentiable and its gradient Lipschitz continuous on ${H}$. That is, we can find L > 0 such that

$\begin{array}{l}||\text{h}(\text{r})-\text{h}(\text{s})||\le L||\text{r}-\text{s}||,\forall \text{r},\text{s}\in {L}.\end{array}$

Proposition 2.3. Suppose that z_k is generated by formula (9) and Assumption 2.2 holds. If the step length α_k is calculated by weak Wolfe line search (4) and (5), then

$\begin{array}{ll}{\alpha }_k\ge \frac{(\sigma -1){\text{h}}_k^T{\text{z}}_k}{L||{\text{z}}_k|{|}^2},& (14)\end{array}$

where σ and L are positive constant in Assumption 2.2 and formula (5) respectively.

Proof: Both sides of formula (5) are subtracted by ${\text{h}}_k^T{\text{z}}_k$, we get

$\begin{array}{l}(\sigma -1){\text{h}}_k^T{\text{z}}_k\le {({\text{h}}_{k+1}-{\text{h}}_k)}^T{\text{z}}_k={\text{q}}_k^T{\text{z}}_k\le ||{\text{q}}_k||||{\text{z}}_k||,\end{array}$

combining with Lipschitz continuity, we obtain

$\begin{array}{l}(\sigma -1){\text{h}}_k^T{\text{z}}_k\le {\alpha }_{k}L||{\text{z}}_k|{|}^2.\end{array}$

Since z_k is a descent direction and σ < 1, formula (14 holds immediately.

Zoutendijk condition [43] is often used to prove the global convergence of the CG method. The following lemma shows that the Zoutendijk condition holds for the proposed method under the weak Wolfe line search conditions formulas (4) and (5).

Lemma 2.4. Suppose Assumption 2.2 holds and consider any iterative expression formula (2, where z_k is generated by formula formula (9). If α_k is calculated by weak Wolfe line search formulas (4) and (5), then the following so-called Zoutendijk condition holds:

$\begin{array}{ll}{\displaystyle \sum _{k=1}^{\infty }}\frac{{({\text{h}}_k^T{\text{z}}_k)}^2}{||{\text{z}}_k|{|}^2}<+\infty .& (15)\end{array}$

Proof: From weak Wolfe condition (4), we have

$\begin{array}{l}f({\text{r}}_k)-f({\text{r}}_k+{\alpha }_k{\text{z}}_k)\ge -\delta {\alpha }_k{\text{h}}_k^T{\text{z}}_k,\end{array}$

combining with formula (14), we get

$\begin{array}{ll}f({\text{r}}_k)-f({\text{r}}_k+{\alpha }_k{\text{z}}_k)\ge \frac{\delta (1-\sigma ){({\text{h}}_k^T{\text{z}}_k)}^2}{L||{\text{z}}_k|{|}^2}.& (16)\end{array}$

Summing up both sides of formula (16), and applying the condition (A1) in Assumption 2.2, zoutendijk condition (15) holds.

Lemma 2.5. Suppose that Assumption 2.2 holds and consider the sequences {h_k} and {z_k} are generated by Algorithm 1, where α_k is calculated by weak Wolfe line search (4–(5, then

$\begin{array}{ll}\frac{||{\text{z}}_k|{|}^2}{{({\text{h}}_k^T{\text{z}}_k)}^2}\le {\displaystyle \sum _{i=1}^k}\frac{1}{||{\text{h}}_i|{|}^2}.& (17)\end{array}$

Proof: From formula (9), we have

$\begin{array}{ll}{\text{z}}_k+{\theta }_k^{FMSD}{\text{h}}_k={\beta }_k^{FMSD}{\text{z}}_{k-1}.& (18)\end{array}$

Squaring up both sides of formula (18) and using the first condition in Lemma 2.1, we obtain

$\begin{array}{l}||{\text{z}}_k|{|}^2={({\beta }_k^{FMSD})}^2||{\text{z}}_{k-1}|{|}^2-2{\theta }_k^{FMSD}{\text{h}}_k^T{\text{z}}_k-{({\theta }_k^{FMSD})}^2||{\text{h}}_k|{|}^2\\ \le {\left(\frac{{\text{h}}_k^T{\text{z}}_k}{{\text{h}}_{k-1}^T{\text{z}}_{k-1}}\right)}^2||{\text{z}}_{k-1}|{|}^2-2{\theta }_k^{FMSD}{\text{h}}_k^T{\text{z}}_k\\ -{({\theta }_k^{FMSD})}^2||{\text{h}}_k|{|}^2,\end{array}$

multiplying up both sides by $\frac{1}{{({\text{h}}_k^T{\text{z}}_k)}^2}$, we get

$\begin{array}{l}\frac{||{\text{z}}_k|{|}^2}{{({\text{h}}_k^T{\text{z}}_k)}^2}\le {\left(\frac{||{\text{z}}_{k-1}||}{{\text{h}}_{k-1}^T{\text{z}}_{k-1}}\right)}^2-\frac{2{\theta }_k^{FMSD}}{{\text{h}}_k^T{\text{z}}_k}-\frac{{({\theta }_k^{FMSD})}^2||{\text{h}}_k|{|}^2}{{\left({\text{h}}_k^T{\text{z}}_k\right)}^2}\\ ={\left(\frac{||{\text{z}}_{k-1}||}{{\text{h}}_{k-1}^T{\text{z}}_{k-1}}\right)}^2-{\left(\frac{1}{||{\text{h}}_k||}+\frac{{\theta }_k^{FMSD}||{\text{h}}_k||}{{\text{h}}_k^T{\text{z}}_k}\right)}^2+\frac{1}{||{\text{h}}_k|{|}^2}\\ \le {\left(\frac{||{\text{z}}_{k-1}||}{{\text{h}}_{k-1}^T{\text{z}}_{k-1}}\right)}^2+\frac{1}{||{\text{h}}_k|{|}^2}.\end{array}$

Since z₁ = −h₁ holds, we obtain

$\begin{array}{l}\frac{||{\text{z}}_k|{|}^2}{{({\text{h}}_k^T{\text{z}}_k)}^2}\le {\left(\frac{||{\text{z}}_{k-1}||}{{\text{h}}_{k-1}^T{\text{z}}_{k-1}}\right)}^2+\frac{1}{||{\text{h}}_k|{|}^2}\\ \le {\left(\frac{||{\text{z}}_{k-2}||}{{\text{h}}_{k-2}^T{\text{z}}_{k-2}}\right)}^2+\frac{1}{||{\text{h}}_{k-1}|{|}^2}+\frac{1}{||{\text{h}}_k|{|}^2}\\ \le {\left(\frac{||{\text{z}}_{k-3}||}{{\text{h}}_{k-3}^T{\text{z}}_{k-3}}\right)}^2+\frac{1}{||{\text{h}}_{k-2}|{|}^2}+\frac{1}{||{\text{h}}_{k-1}|{|}^2}+\frac{1}{||{\text{h}}_k|{|}^2}\\ \le ...\le {\displaystyle \sum _{i=1}^k}\frac{1}{||{\text{h}}_i|{|}^2}.\end{array}$

The proof is finished.

Based on Lemmas 2.1, 2.4, and 2.5, we can establish the theorem of global convergence of the FMSD method.

Theorem 2.6. Suppose that Assumption 2.2 is satisfied. Consider {r_k} is generated by Algorithm 1, where α_k is calculated by weak Wolfe line search (4–5), then

$\begin{array}{ll}{\displaystyle \underset{k\to \infty }{lim inf}}||{\text{h}}_k||=0.& (19)\end{array}$

Proof: We prove this theorem by contradiction. Suppose that formula (19) is not true, then there exists a positive constant a > 0 such that

$\begin{array}{l}||{\text{h}}_k||\ge a,\forall k\ge 1.\end{array}$

Using the above relation and formula (17), we obtain

$\begin{array}{l}\frac{||{\text{z}}_k|{|}^2}{{({\text{h}}_k^T{\text{z}}_k)}^2}\le {\displaystyle \sum _{i=1}^k}\frac{1}{||{\text{h}}_i|{|}^2}\le \frac{k}{a^2}.\end{array}$

It implies

$\begin{array}{l}{\displaystyle \sum _{k=1}^{\infty }}\frac{{({\text{h}}_k^T{\text{z}}_k)}^2}{||{\text{z}}_k|{|}^2}\ge {\displaystyle \sum _{k=1}^{\infty }}\frac{a^2}{k}=+\infty ,\end{array}$

which contradicts with the Zoutendijk condition in formula (15). Hence, formula (19) is true. The proof is finished.

3. Numerical experiments

In this part, we report the numerical experiments of the FMSD method and compare the computational performance with the JYJLL method proposed by Jian et al. [7]. Both the methods were coded in MATLAB 2019a and ran using a personal computer with an Intel Core i7 processor, 16 GB RAM, 64 bit Windows 10 Pro operating system. The comparisons are made under the weak Wolfe line search (4–5) with σ = 0.2 and δ = 0.02 for the FMSD method and σ = 0.1 and δ = 0.01 for the JYJLL method. We tested 132 unconstrained problems in the CUTEr library suggested by Andrei [6, 44] and Moré et al. [45] with dimensions from 2 to 1,000,000. Mostly, we used two different dimensions for the problem and the iteration stopped using the $\left|\right|{\text{h}}_k|{|}_{\infty }\le 10^{-6}$ criteria. The initial point used for all problems can be seen in Jiang et al. [25]. Table 1 details the test function and dimensions of the test problems.

TABLE 1

Table 1. The problems and their dimensions.

Detailed numerical results are provided in Table 2 which include the number of iterations (NOI), the total number of function evaluations (NOF), and the CPU time in seconds (CPU). In Table 2, “-” indicates that the methods failed to solve the corresponding problems within 2000 iterations.

TABLE 2

Table 2. Numerical results.

To clearly determine a method that has good computational performance, here we use the performance profiles suggested by Dolan and Moré [46] to show the performance under NOI, NOF, and CPU time, respectively. Comparison results are obtained by running a solver on a set P of problems and recording relevant information such as NOI, NOF, and CPU time. Suppose that S is the set of solvers under consideration and assume S is made up of n_s solvers and P is made up of n_p problems. For each problem p ∈ P and solver s ∈ S, we denote t_p,s as the CPU time (or NOI or NOF, etc.) required to solve problem p ∈ P by solver s ∈ S. The comparison between different solvers is based on the performance ratio described by

$\begin{array}{l}{r}_{p,s}=\frac{t_{p,s}}{min\left\{t_{p,s}:s\in S\right\}}.\end{array}$

Let ρ_s(τ) be the probability for solver s ∈ S that a performance ratio r_p,s is within a factor τ ∈ ℝⁿ. For example, the value of ρ_s(1) is the probability that the solver will win over the rest of the solvers. The formula of ρ_s(τ) is defined as follows:

$\begin{array}{l}{\rho }_s(\tau )=\frac{1}{n_p}size\left\{p\in P:log{r}_{p,s}\le \tau \right\}.\end{array}$

According to the rule of the performance profile above, we can describe the performance curves based on Table 2 as in Figures 1–3. Based on the three figures, we can see that the FMSD method is superior to the JYJLL method under the unconstrained problems in Table 1.

FIGURE 1

Figure 1. Performance profiles on NOI.

FIGURE 2

Figure 2. Performance profiles on NOF.

FIGURE 3

Figure 3. Performance profiles on CPU Time.

4. Application to regression models of COVID-19

SARS-CoV-2 virus popularly known as the COVID-19 infection was first reported in the Asian continent from Wuhan province, Hubei city of China toward the end of 2019. As of 20 June 2022, almost all the countries in Asia except Turkmenistan have reported at least one case of the infection [47]. However, countries that include India, South Korea, Vietnam, Japan, and Iran recorded the highest rates of confirmed cases of the infection [48]. The first positive COVID-19 case in Indonesia was recorded on March 2, 2020, but within the first 6 weeks, the presence of the virus has been confirmed in almost all the provinces of the country [49]. Despite the early wide-scale response from the government, the country has recorded a high number of deaths from the positive cases of the infection [50]. According to the WHO, Indonesia has so far recorded a total of 156,695 deaths from a total of 6,069,255 confirmed cases of the infection as of 20 June 2022 [51] of which more than 750 deaths are front-line health workers. Based on recent figures, we can say that Indonesia has been able to contain the disease outbreak. This can be attributed to the admirable resilience of the country's front-line health workers, strict health protocols, and successful vaccination programs. Data from the WHO shows that the total of people that have been administered the vaccine doses as of 15 June 2020 stands at 417,522,347 [51].

In recent times, several works of literature have employed different mathematical and numerical approaches for modeling the COVID-19 outbreak [see [5, 32, 52]]. This paper aims to study the performance of the proposed method on a parameterized COVID-19 regression model. For deriving the COVID-19 regression model, the study will consider the total Indonesian monthly positive confirmed cases of the infection from March 2020 (the month of the first recorded case) until May 2022. The obtained data would be transformed into an unconstrained optimization model which would later be solved using the proposed method.

A regression analysis function of the form:

$\begin{array}{ll}y=h(x_1,x_2,...,x_p+\epsilon ),& (20)\end{array}$

has the response variable denoted by y, ε represents the error, and the predictor is given as x_i, i = 1, 2, …, p, p > 0. The type of function plays an important role in the statistical modeling of problems in applied sciences, physical sciences, management sciences, and more. Based on the above description, we can describe regression analysis as a statistical procedure employed to estimate the relationships between a dependent and one or several independent variables. For any given regression analysis-related problem, the linear regression function can be derived by computing y such that

$\begin{array}{ll}y=a_0+a_1{x}_1+a_2{x}_2+\dots +a_p{x}_p+\epsilon & (21)\end{array}$

with a₀, …, a_p representing the regression parameters. These parameters a₀, a₁, …, a_p are estimated to minimize the error ε value. Based on several works of literature, the linear regression process rarely occurs in situations because most problems are often nonlinear in nature. Based on the non-linearity of the problems, studies usually consider the nonlinear regression process [5]. This and other considerations motivated the idea of using the nonlinear regression procedure in this study.

To construct the parameterized regression model, we considered the death cases recorded from those infected by the COVID-19 virus from the first month Indonesia confirmed the first case; March 2020 until May 2022, totaling 27 months. The data were obtained from the Indonesia COVID Coronavirus Statistics Worldometer [53] and the detailed description of the model formulation process was presented as follows. Note: it may be confirmed that the statistics of recorded cases are less than the actual number, this might be a result of limited testing. From the data presented in Table 3, the x-variable would represent the months considered while the y-variable represents the confirmed death cases for that month. Also, only data of 26 months (March 2020 to April 2022) would be considered for data fitting because data for May 2022 would be reserved for error analysis.

TABLE 3

Table 3. Statistics of confirmed positive cases and death recorded from COVID-19 infection in Indonesia from March 2020 to May, 2022.

Based on the data of x and y given in Table 3, the approximate function for the nonlinear least square method was obtained as follows:

$\begin{array}{ll}f(x)=-842.24+35865.66x-909.17x^2.& (22)\end{array}$

The above function (22) will be utilized when approximating the y data values based on x data values. Since this study considered the monthly confirmed cases, the x_j would be used to denote the months while y_j will present the confirmed cases for that month. Based on this information, the least squares method defined by function (22) would be transformed into an unconstrained minimization problem of the form:

$\begin{array}{ll}{\displaystyle \underset{x\in {ℝ}^n}{min}}f(x)={\displaystyle \sum _{j=1}^n}{\left(\left(u_0+u_1{x}_j+u_2{x}_j^2\right)-y_j\right)}^2.& (23)\end{array}$

The data for the first 26 months from Table 3 will be used to derive the nonlinear quadratic function for the least square method. The derived function would be extended to construct the unconstrained optimization function. Based on the above discussion, it is obvious that there exist some parabolic relations between the regression parameters u₀, u₁, u₂, the regression function (20) with the data x_j and the value of y_j.

$\begin{array}{ll}{\displaystyle \underset{x\in {ℝ}^2}{min}}{\displaystyle \sum _{j=1}^n}{E}_j^2={\displaystyle \sum _{j=1}^n}{\left(\left(u_0+u_1{x}_j+u_2{x}_j^2\right)-y_j\right)}^2.& (24)\end{array}$

To define the nonlinear quadratic unconstrained minimization model, Equation (24 would be transformed using data from Table 3 as follows:

$\begin{array}{l}26u_1^2+702u_1{u}_2+12402u_1{u}_3-209806038u_1+2610621u_2^2\\ +246402u_2{u}_3-17172778u_2+15333u_3^2\\ -4006838782u_3+4152673772991.& (25)\end{array}$

The above nonlinear quadratic model was constructed using data from the first month until the 26th month because the data for the 27th month was reserved for relative error analysis of the predicted data. Now, we can apply the proposed method to solve the model (25). The results presented in Table 4 illustrate the performance of the proposed FMSD algorithm for problem (25) under the weak Wolfe line search conditions (4–5).

TABLE 4

Table 4. Performance results of FMSD method for optimization of the quadratic model Equation (25).

The proposed method was employed as an alternative method to compute the values of u₀, u₁, u₂ because of the difficulty faced when using matrix inverse. For the proposed method, different initial points were considered for the model. The iteration was terminated if the iterations exceeded 1,000 or the method was unable to solve the problem.

4.1. Trend line method

In finance and related areas, one of the easiest processes to boost the likelihood of making a successful trade is to understand the direction of an underlying trend because it assures that the overall market dynamics are in your favor. Trend lines are bounding lines that traders use to connect a sequence of prices of security on charts. It is created when three or more price pivot points or more can be connected diagonally. In this section, the proposed FMSD and existing least squares methods were employed to estimate data from Table 3. Microsoft Excel software was used to plot the trend line for data for the first 26 months. The graph demonstrated in Figure 4 was obtained by plotting the real data from Table 3 with x and y denoting the x-axis and y-axis respectively.

FIGURE 4

Figure 4. Nonlinear quadratic trend line for indonesia COVID-19 cases.

The efficiency of the proposed method is further demonstrated by comparing the approximation functions of FMSD with those of the trend line and least squares methods. Table 5 presents the estimation Point and relative Errors for the three methods based on the reserved data for the 27th month.

TABLE 5

Table 5. Relative error analysis using the data of the 27th month.

From Table 5, we can see that the error ε has been minimized which agrees with the main purpose of regression analysis. This shows that the proposed FMSD method is efficient and promising, and thus, can find a wider range of other real-life applications.

5. Conclusions

In this paper, we have presented a spectral conjugate gradient method for solving unconstrained optimization problems by modifying the spectral parameter of the JYJLL method in Jian et al. [7]. Based on some conditions, the global convergence properties were established under a weak Wolfe line search. A numerical comparison of the proposed method with the JYJLL method shows that the proposed method is efficient, fast, and robust. Moreover, our proposed method can solve the COVID-19 case model in Indonesia.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

MM and FN: conceptualization. MM and IS: methodology, numerical experiments, and writing—original draft preparation. MM and DA: formal analysis. IS: application. All authors have read and agreed to the published version of the manuscript.

Funding

This research is funded by Hibah Riset Penugasan FMIPA UI (Grant No. 002/UN2.F3.D/PPM.00.02 /2022).

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher's note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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