Geochemistry, facies characteristics and palaeoenvironmental conditions of the storm-dominated phosphate-bearing deposits of eastern Tethyan Ocean; A case study from Zagros region, SW Iran

Armin Salsani; Abdolhossein Amini; Shahram shariati; Seyed Ali Aghanabati; Mohsen Aleali; Armin Salsani; Abdolhossein Amini; Shahram shariati; Seyed Ali Aghanabati; Mohsen Aleali

doi:10.3934/geosci.2020019

AIMS Geosciences

2020, Volume 6, Issue 3: 316-354. doi: 10.3934/geosci.2020019

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Research article

Geochemistry, facies characteristics and palaeoenvironmental conditions of the storm-dominated phosphate-bearing deposits of eastern Tethyan Ocean; A case study from Zagros region, SW Iran

1.
Department of Geology, Science and Research Branch, Islamic Azad University, Tehran, Iran
2.
School of Geology, College of Science, University of Tehran, Tehran, Iran
3.
Department of Geology, Sari branch, Islamic Azad University, Sari, Iran
4.
Earth Sciences Research Institute, Geological Survey of Iran, Tehran, Iran

Received: 25 June 2020 Accepted: 17 August 2020 Published: 08 September 2020

Phosphate deposits in south-western Iran are part of the South Tethyan Phosphogenic Province, a huge carbonate-dominated strata that extends to the Middle East. The Tethyan phosphorites of Iran are dated Eocene-Oligocene (Pabdeh Formation) and categorized as low-grade ore deposits on a global scale. Depositional conditions of the facies indicate that the Pabdeh Formation was deposited on a carbonate ramp setting as a distally steepened ramp. Under such an environment, turbidity currents transported phosphate particles from the back-shoal setting to the deeper middle and outer ramp of the ocean where they were suspended and deposited as shell-lag and phosphate lamination. Microfacies studies demonstrate that all the phosphatic ooids and phosphatized foraminifera, fish scales, bones and phosphatic intraclasts reworked from shallow parts of the Tethyan Ocean to deeper parts with the help of turbidity currents. Analysis and interpretation of the data reveal positive correlation between. REE+Y and P₂O₅ in all studied sections which attests to their strong coherence as a geochemical group. The shale normalized REE patterns of Mondun phosphorites are characterized by negative Ce anomalies. This anomaly indicate that the depositional environment was oxic and highly reworked, bioturbated with higher energy realm during phosphate deposition, conversely Nill section with Ce enrichment reflect conditions of relatively deeper water sedimentation. These geochemical findings are in accordance with microfacies studies which indicate shallow and high energy condition for Mondun section with negative cerium anomalies and a deep ramp setting for Nill and Siah sections which denote a positive cerium anomalies in REE patterns.

Keywords:

Citation: Armin Salsani, Abdolhossein Amini, Shahram shariati, Seyed Ali Aghanabati, Mohsen Aleali. Geochemistry, facies characteristics and palaeoenvironmental conditions of the storm-dominated phosphate-bearing deposits of eastern Tethyan Ocean; A case study from Zagros region, SW Iran[J]. AIMS Geosciences, 2020, 6(3): 316-354. doi: 10.3934/geosci.2020019

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Abstract

1. Introduction

In this paper, we consider the following unconstrained optimization problem:

$\begin{eqnarray} \min\{f(x)\mid x\in \mathbb{R}^n\}, \end{eqnarray}$

(1.1)

where $f:\mathbb{R}^n\to \mathbb{R}$ is a continuously differentiable function and its gradient is denoted by $g(x) = \nabla f(x)$ . Generally, (1.1) iterates along the following form:

$\begin{eqnarray} x_{k+1} = x_k+s_k, \text{ for all }\; k\geq0, \end{eqnarray}$

(1.2)

where $s_{k} = \alpha_{k}d_{k}$ , $d_{k}$ is the search direction, $\alpha_{k} > 0$ is a step length often obtained by a line search along $d_{k}$ .

There are many methods to solve unconstrained optimization problem (1.1). Among these methods, The Newton method has a second-order convergence rate when the Hessian matrix $\nabla^2f(x_k)$ is positive definite, but it cannot ensure that the direction chosen for the objective function at $x_k$ is a descent while solving unconstrained optimization problems. In addition, every iterations of the Newton method requires the second-order gradient of the objective function, namely the Hessian matrix, which is computationally complex for large-scale problems. To address large-scale problems more effectively, the quasi-Newton method was proposed. The quasi-Newton approach updates the search direction using an approximation Hessian matrix instead of the true Hessian matrix used in the Newton method. This approach can reduce the computation of second-order derivatives, lower the computational complexity, and handle non-differentiable functions in some special cases ^[1].

The quasi-Newton method is recognized as one of the excellent iterative methods for solving large-scale unconstrained optimization problems (for relevant research, see [22–25]). The fundamental concept of the quasi-Newton technique is to substitute an approximate matrix $B_{k}$ for the Hessian matrix $\nabla^2f(x_k)$ in the Newton method ^[1]. In order to satisfy the secant equation, i.e.,

$\begin{eqnarray} B_{k+1}s_k = y_k, \end{eqnarray}$

(1.3)

where $s_{k} = x_{k+1}-x_{k}$ , $y_k = g_{k+1}-g_k$ , the approximate matrix $B_{k}$ of the Hessian matrix is constructed using the quasi-Newton update formula. The direction of the quasi-Newton method can be calculated directly by the following formula:

$\begin{eqnarray} \begin{aligned} d_0 = -g_0, d_{k} = -H_{k}g_{k}, \text{ for all }\;k\geq0, \end{aligned} \end{eqnarray}$

(1.4)

where $H_{k}$ is the approximation of $\nabla^2f\left(x_{k}\right)^{-1}$ and satisfies secant equation, $\begin{aligned}H_{k+1}y_k = s_k \end{aligned}$ , for all $k\geq0.$

The numerical performance of the quasi-Newton method is directly impacted by the updating of $H_{k}$ . There are many updated formulas of $H_{k}$ such as the DFP formula, the BFGS formula, and the SR1 formula. These methods, which are composed of different updating formulas, are particularly effective in solving unconstrained optimization problems and they have promising computational performance in practical problems.

The BFGS method, independently proposed by Broyden, Fletcher, Goldfarb, and Shanno [9–13], is one of the most widely used and successful quasi-Newton methods. $B_{k}$ is always positive definite during the computation; hence, the BFGS method keeps the convergence and stability for optimization problems. Moreover, the BFGS method has emerged as the preferred option for many applications, including neural networks, image processing, and machine learning (see [26–28]). The BFGS formula is as follows:

$\begin{eqnarray} \begin{aligned} B_{k+1}^{BFGS} = B_{k}+\frac{y_{k}y_{k}^{T}}{s_{k}^{T}y_{k}}-\frac{B_{k}s_{k}s_{k}^{T}B_{k}}{s_{k}^{T}B_{k}s_{k}}, \end{aligned} \end{eqnarray}$

(1.5)

$\begin{eqnarray} \begin{aligned} H_{k+1}^{BFGS} = H_{k}-\frac{s_{k}y_{k}^{T}H_{k}+H_{k}y_{k}s_{k}^{T}}{s_{k}^{T}y_{k}}+\left( 1+\frac{y_{k}^{T}H_{k}y_{k}}{s_{k}^{T}y_{k}}\right) \frac{s_{k}s_{k}^{T}}{s_{k}^{T}y_{k}}. \end{aligned} \end{eqnarray}$

(1.6)

The BFGS formula satisfies the secant equation (1.3). Many researchers have improved the formula (1.3) to enhance numerical stability and the accuracy of the approximation Hessian matrix. Zhang et al. ^[2], Zhang and Xu ^[3] put forward a modified secant condition

$\begin{eqnarray} \begin{aligned} B_{k+1}s_k = \check{y}_k, \end{aligned} \end{eqnarray}$

(1.7)

$\begin{eqnarray} \begin{aligned} \check{y}_{k} = y_{k}+\tau\frac{\check{\eta}_{k}}{s_{k}^{T}w_{k}}w_{k}, \end{aligned} \end{eqnarray}$

(1.8)

$\begin{eqnarray} \begin{aligned} \check{\eta}_{k} = 2(f_{k}-f_{k+1})+s_{k}^{T}(g_{k}+g_{k+1}), \end{aligned} \end{eqnarray}$

(1.9)

where $\tau > 0$ and $w_{k}$ is a vector parameter satisfying ${s}_{k}^{T}w_{k}\neq0$ .

On the other hand, in order to reduce memory storage and improve the computational efficiency of the algorithm, a memoryless quasi-Newton method is proposed to solve large-scale unconstrained optimization problems, where the inner product of multiple vectors is employed to determine the search direction ^[4]. Memoryless technology has been employed in numerous works; for example, Babaie-Kafaki and Aminifard ^[5], Aminifard, Babaie-Kafaki and Ghafoori ^[6], Babaie-Kafaki, Aminifard and Ghafoori ^[7], Jourak, Nezhadhosein, and Rahpeymaii ^[8] have applied a memoryless technique to design and develop new quasi-Newton methods for solving large-scale unconstrained optimization problems, and Narushima, Nakayama, Takemura et al. ^[29] propose a memoryless quasi-Newton method in Riemannian manifolds.

Our goal in this research is to present an effective augmented memoryless BFGS method for solving unconstrained optimization problems. The major contributions of this paper have at least three aspects as follows:

(1) To establish an effective optimization algorithm for large-scale unconstrained optimization problems, a new augmented memoryless BFGS updating formula is provided that is based on a specific modified secant equation.

(2) To enhance the effectiveness of the experiment, the condition number may be minimized in order to obtain the scaling parameters.

(3) We prove the global convergence of the algorithm and numerical experiments that show the efficiency of the algorithm.

The organization of the article is as follows. In Section 2, we present an augmented memoryless BFGS technique along with the algorithm framework. The descent property and the global convergence of the proposed algorithm are demonstrated in Section 3. In Section 4, the numerical experiment demonstrates the effectiveness of our method in solving large-scale unconstrained optimization problems and nonlinear equations. A conclusion to this work is presented in Section 5.

2. An augmented memoryless BFGS method

Inspired by Zhang and Xu ^[3] and Aminifard, Babaie-Kafaki, and Ghafoori ^[6], to improve the precision of the solution, we select $w_{k} = y_{k}$ in Eqs (1.7)–(1.9), get a modified secant equation,

$\begin{eqnarray} \begin{aligned} B_{k+1}s_{k} = \bar{y}_{k}, \end{aligned} \end{eqnarray}$

(2.1)

where $\bar{y}_{k} = (1+\tau_{k})y_{k}$ , $\tau_{k} = \tau\frac{\eta_{k}}{s_{k}^{T}y_{k}}$ and $\eta_{k} = \max\left\{0, \check{\eta}_k\right\}$ .

We modify the BFGS iteration formula (1.5) by a rank-1 correction, which we refer to as the augmented BFGS (ABFGS) formula

$\begin{eqnarray} \begin{aligned} B_{k+1}^{ABFGS} = B_{k+1}^{BFGS}+\tau_{k}\frac{y_{k}s_{k}^{T}}{s_{k}^{T}s_{k}}. \end{aligned} \end{eqnarray}$

(2.2)

$B_{k+1}^{ABFGS}s_k = \bar{y}_k$ because $B_{k+1}^{BFGS}s_k = y_k$ . Therefore, (2.2) satisfies the modified secant equation (2.1).

As is known by the quasi-Newton property, when steplength $\alpha_{k}$ satifies Wolfe line search,

$\begin{eqnarray} \begin{aligned} f\left(x_k+\alpha_kd_k\right)-f\left(x_k\right)\leq\delta\alpha_kg_k^Td_k, \end{aligned} \end{eqnarray}$

(2.3)

$\begin{eqnarray} \begin{aligned} \nabla f\left(x_k+\alpha_kd_k\right)^Td_k\geq\rho g_k^Td_k, \end{aligned} \end{eqnarray}$

(2.4)

with $0 < \delta < \rho < 1$ , ${s}_k^{T}{y}_k > 0$ is established. By Theorem 5.2.2 of ^[1], we know (1.5) is positive definite. Therefore, $B_{k}^{ABFGS}$ is a positive definite matrix since $\tau_{k}\geq 0$ . In other words, the positive definiteness is passed down to the ABFGS update formula. As a result, the search directions of ABFGS are descent.

The scaled memoryless BFGS (SMBFGS) method is considered an effective tool for solving large-scale unconstrained optimization problems. In order to obtain SMBFGS formula, $B_{k}$ can be replaced by $\frac1{\vartheta_{k}}I$ in (1.5), namely,

$\begin{eqnarray} \begin{aligned} B_{k+1}^{SMBFGS} = \frac{1}{\vartheta_{k}}I+\frac{y_{k}y_{k}^{T}}{s_{k}^{T}y_{k}}-\frac{1}{\vartheta_{k}}\frac{s_{k}s_{k}^{T}}{s_{k}^{T}s_{k}}, \end{aligned} \end{eqnarray}$

(2.5)

where $\vartheta_{k} > 0$ is called the scaling parameter. Similarly, by replacing $H_{k}$ with $\vartheta_{k}I$ , we can get the SMBFGS iteration formula for the inverse of the Hessian matrix

$\begin{eqnarray} \begin{aligned} H_{k+1}^{SMBFGS} = \vartheta_{k} I-\vartheta_{k}\frac{s_{k}y_{k}^{T}+y_{k}s_{k}^{T}}{s_{k}^{T}y_{k}}+\left( 1+\vartheta_{k}\frac{y_{k}^{T}y_{k}}{s_{k}^{T}y_{k}}\right) \frac{s_{k}s_{k}^{T}}{s_{k}^{T}y_{k}}. \end{aligned} \end{eqnarray}$

(2.6)

In the formula (2.2), we replace $B_{k+1}^{BFGS}$ with $B_{k+1}^{SMBFGS}$ to make a rank-1 correction, and then, using the Sheran-Morrison formula ^[1], we get the augmented memoryless BFGS formula (AMBFGS), i.e.,

$\begin{eqnarray} \begin{aligned} B_{k+1}^{AMBFGS} = B_{k+1}^{SMBFGS}+\tau_{k}\frac{y_{k}s_{k}^{T}}{s_{k}^{T}s_{k}} = \frac{1}{\vartheta_{k}}\left( I-\frac{s_{k}s_{k}^{T}}{s_{k}^{T}s_{k}}\right) +\frac{y_{k}y_{k}^{T}}{s_{k}^{T}y_{k}}+\tau_{k}\frac{y_{k}s_{k}^{T}}{s_{k}^{T}s_{k}}, \end{aligned} \end{eqnarray}$

(2.7)

$\begin{eqnarray} \begin{aligned} H_{k+1}^{AMBFGS} = H_{k+1}^{SMBFGS}-\frac{\tau_{k}(s_{k}^{T}y_{k}s_{k}s_{k}^{T}-\vartheta_{k}s_{k}^{T}y_{k}s_{k}y_{k}^{T}+\vartheta_{k}y_{k}^{T}y_{k}s_{k}s_{k}^{T})}{(1+\tau_{k})s_{k}^{T}y_{k}s_{k}^{T}y_{k}}. \end{aligned} \end{eqnarray}$

(2.8)

For the selection of parameter $\vartheta_{k}$ , Babaie-Kafaki ^[4] came up with well-structured upper bounds for the condition numbers of the scaled memoryless quasi-Newton formulas based on eigenvalue analysis. It was then demonstrated that the scaling parameter suggested by Oren and Spedicato ^[14] is the only value that can be found as the lowest of the upper bound given for the condition number of the scaled memoryless BFGS update formula. On the other hand, according to Oren and Luenberger ^[15], the scaling parameter is the distinct lowest value of the provided upper bound on the condition number of the scaled memoryless DFP update algorithm.

According to ^[5], the choice of parameter $\vartheta_{k}$ is addressed by minimizing the given upper bound for the condition number of the formula (2.8). Since Wolfe line search (2.4) ensures $s_k^{T}y_k > 0$ , we have $s_k\neq 0$ and $y_k\neq 0$ . So a set of mutually orthogonal unit vectors ${q_{k}^{(i)}}_{i = 1}^{n-2}$ exists for which $s_{k}^{T}q_{k}^{(i)} = y_{k}^{T}q_{k}^{(i)} = 0$ , yielding $B_{k+1}^{AMBFGS}q_{k}^{(i)}$ , for $i = 1, 2, ..., n - 2.$ Therefore, $(n - 2)$ eigenvalues of the matrix $H_{k+1}^{AMBFGS} (B_{k+1}^{AMBFGS})$ are equivalent to $\vartheta_{k} (\frac{1}{\vartheta_{k}})$ .

Using the relationship between the determinants and traces of the matrices $H_{k+1}^{AMBFGS}$ and $B_{k+1}^{AMBFGS}$ , other eigenvalues of $B_{k+1}$ and $H_{k+1}$ can be obtained. The relationships are as follows:

$\begin{eqnarray} \begin{aligned} \rho_{k}^{+}+\rho_{k}^{-} = \frac{1}{\vartheta_{k}}+\frac{\|y_{k}\|^{2}}{s_{k}^{T}y_{k}}+\frac{\tau_{k}s_{k}^{T}y_{k}}{\|s_{k}\|^{2}}, \end{aligned} \end{eqnarray}$

(2.9)

$\begin{eqnarray} \begin{aligned} \rho_{k}^{+}\rho_{k}^{-} = \frac{\vartheta_{k}\|s_{k}\|^{2}}{(1+\tau_{k})s_{k}^{T}y_{k}}. \end{aligned} \end{eqnarray}$

(2.10)

Formulas of the two other eigenvalues of $B_{k+1}^{AMBFGS}$ , namely, $\rho_{k}^{+}$ and $\rho_{k}^{-}$ , can be obtained,

$\begin{eqnarray} \begin{aligned} \rho_{k}^{\pm} = \frac{1}{2}\left( \frac{1}{\vartheta_{k}}+\frac{\Vert y_{k}\Vert ^{2}}{s_{k}^{T}y_{k}}+\frac{\tau_{k}s_{k}^{T}y_{k}}{\Vert s_{k}\Vert ^{2}}\right) \pm \frac{1}{2}\sqrt{\left( \frac{1}{\vartheta_{k}}+\frac{\|y_{k}\|^{2}}{s_{k}^{T}y_{k}}+\frac{\tau_{k}s_{k}^{T}y_{k}}{\|s_{k}\|^{2}}\right) ^{2}-4\frac{\vartheta_{k}\|s_{k}\|^{2}}{(1+\tau_{k})s_{k}^{T}y_{k}}} \end{aligned} \end{eqnarray}$

(2.11)

for which $0 < \rho_{k}^{-} < \rho_{k}^{+}$ . Additionally, since $H_{k+1}^{AMBFGS}$ is positive definite, the search direction of AMBFGS is descent. Furthermore, we have $0 < \rho_{k}^{-} < \frac{1}{\vartheta_{k}} < \rho_{k}^{+}$ after performing a few algebraic operations. Therefore, $\|H_{k+1}^{AMBFGS}\| = \rho_{k}^{+}$ and $\| {H_{k+1}^{AMBFGS}}^{-1}\| = \rho_{k}^{-}$ . Now, from (2.9) and (2.10) we can write

$\begin{eqnarray} \begin{aligned} \kappa(H_{k+1}^{AMBFGS})& = \frac{\rho_{k}^{+}}{\rho_{k}^{-}} = \frac{\rho_{k}^{+2}}{\rho_{k}^{+}\rho_{k}^{-}}\leq \frac{(\rho_{k}^{+}+\rho_{k}^{-})^{2}}{\rho_{k}^{+}\rho_{k}^{-}}\\ &\leq \frac{(s_{k}^{T}y_{k}\|s_{k}\|^{2}+\vartheta_{k}\|s_{k}\|^{2}\|y_{k}\|^{2}+\tau_{k}\vartheta_{k}( s_{k}^{T}y_{k}) ^{2})^{2}}{(1+\tau_{k})\vartheta_{k}(s_{k}^{T}y_{k})^{3}\|s_{k}\|^{2}}, \end{aligned} \end{eqnarray}$

(2.12)

where $\kappa(\cdot)$ expresses the spectral condition number.

It is generally acknowledged that decreasing the condition number in matrix-based computing can enhance the stability of numerical calculations ^[16]. From (2.12), we derive the minimizer of the upper bound of $\kappa(H_{k+1}^{AMBFGS})$ as follows:

$\begin{eqnarray} \begin{aligned} \vartheta_{k} = \frac{s_{k}^{T}y_{k}\|s_{k}\|^{2}}{\tau_{k}(s_{k}^{T}y_{k})^{2}+\vartheta_{k}\|s_{k}\|^{2}\|y_{k}\|^{2}}. \end{aligned} \end{eqnarray}$

(2.13)

Now, we present a new augmented memoryless BFGS algorithm for solving unconstrained optimization problems.

Algorithm 2.1. The AMBFGS algorithm

Step 1. Choose an initial point $x_{0}$ , choose parameters $0 < \delta < \rho < 1$ , $\epsilon > 0$ , $\epsilon_{1} > 0$ . Set $H_{0} = I$ , $d_{0} = -H_{0}g_{0}$ and $k : = 0$ .

Step 2. If $\Vert g_{k}\Vert < \epsilon$ , stop; otherwise, go to Step 3.

Step 3. Compute the step size $\alpha_{k}$ , so that it satisfies Wolfe line search (2.3) and (2.4). Set $x_{k+1} = x_{k}+\alpha_{k}d_{k}$ .

Step 4. Compute $\vartheta_{k}$ , if $\vartheta_{k} < \epsilon_{1}$ , let $\vartheta_{k} = \frac{s_{k}^{T}y_{k}}{\|y_{k}\|^{2}}$ , otherwise, obtains $\vartheta_{k}$ by (2.13).

Step 5. Update $H_{k+1}$ by (2.8) and compute the search direction $d_{k}$ using (1.4).

Step 6. Set $k: = k+1$ and go to Step 2.

Remark 2.1. Step 4 is set this way in order to avoid $\vartheta_{k}$ falling into a dilemma during the calculation of the algorithm.

3. The descent property and global convergence

In this section, we demonstrate that the direction is sufficiently descent and the algorithm is globally convergent. Our analysis is based on the following assumptions.

Assumption 3.1. For arbitrary $x_{0}\in\mathbb {R}^{n}$ , $\mathcal{L} = \{x\mid f(x)\leq f(x_{0})\}$ is a bounded set and in some neighborhood $\mathcal{U}$ of $\mathcal{L}$ , $\nabla f(x)$ is Lipschitz continuous, that is, there exists a constant $L > 0$ such that

$\begin{eqnarray} \begin{aligned} \Vert\nabla f(x)-\nabla f(\check{x})\Vert \leq L\Vert x-\check{x}\Vert,\forall x,\check{x} \in \mathcal{U}. \end{aligned} \end{eqnarray}$

(3.1)

Based on Assumption 3.1, we know that there is a positive constant $\Phi$ exists such that

$\begin{eqnarray} \begin{aligned} \Vert\nabla f(x)\Vert \leq \Phi, \forall x \in \mathcal{L}. \end{aligned} \end{eqnarray}$

(3.2)

Since AMBFGS directions are descent, from (2.4), we have $\{x_{k}\}\subset\mathcal{L}$ .

The boundedness of the parameter $\vartheta_{k}$ in (2.13) is important. We will prove $\vartheta_{k}\in\left[m, M\right]$ in Lemma 3.1.

Lemma 3.1. Considering $f$ is a uniformly convex function on a neighborhood $\mathcal{U}$ of $\mathcal{L}$ , the scaling parameter $\vartheta_{k}$ of the AMBFGS algorithm in (2.13) is well defined and bounded.

Proof. Let $f$ be uniformly convex on $\mathcal{U}$ , then, by Theorem 1.3.16 of ^[1], for any $x, y\in\mathcal{L}$ , we have

$\begin{eqnarray} \begin{aligned} f(y)\geq f(x)+\left < \nabla f(x),y-x \right > +\frac{1}{2}v\Vert y-x\Vert ^{2}, \end{aligned} \end{eqnarray}$

(3.3)

and

$\begin{eqnarray} \begin{aligned} f(x)\geq f(y)+\left < \nabla f(y),x-y \right > +\frac{1}{2}v\Vert x-y\Vert ^{2}, \end{aligned} \end{eqnarray}$

(3.4)

where $v > 0$ is a constant. Let $y = x_{k+1}$ , $x = x_{k}$ , adding (3.3) and (3.4), we can obtain

$\begin{aligned} \left < \nabla f(x_{k+1})-\nabla f(x_{k}),x_{k+1}-x_{k} \right > \geq v\Vert x_{k+1}-x_{k}\Vert ^{2},\forall k\geq 0.\nonumber \end{aligned}$

In this paper, $s_k = x_{k+1}-x_{k}, y_{k} = \nabla f(x_{k+1})-\nabla f(x_{k})$ , and then,

$\begin{eqnarray} \begin{aligned} s_k^{T}y_{k}\geq v\Vert s_{k}\Vert^{2},\forall k\geq 0. \end{aligned} \end{eqnarray}$

(3.5)

Through (3.1) and (3.5), we can get

$\begin{eqnarray} \begin{aligned} \frac{v}{L^{2}} \leq \frac{s_{k}^{T}y_{k}}{\Vert y_{k}\Vert^{2}}\leq \frac{\Vert s_{k}\Vert^2}{s_k^{T}y_k}\leq \frac{1}{v}. \end{aligned} \end{eqnarray}$

(3.6)

By mean value theorem, (1.9) can be rewritten by

$\begin{eqnarray} \begin{aligned} \check{\eta}& = 2(f_{k}-f_{k+1})+(g_{k}+g_{k+1})^{T}s_{k}\\& = 2\nabla f(\theta_{k})^{T}(x_k-x_{k+1})+(g_k+g_{k+1})^{T}s_k\\& = -2\nabla f(\theta_{k})^{T}s_{k}+(g_k+g_{k+1})^{T}s_k\\& = \left( g_k-g(\theta_{k})+g_{k+1}-g(\theta_{k})\right) ^{T}s_k, \end{aligned} \end{eqnarray}$

(3.7)

where $\theta_{k} = hx_k+(1-h)x_{k+1}, h\in (0, 1)$ . Hence, by (3.1) we have that

$\begin{eqnarray} \begin{aligned} \vert\check{\eta}\vert&\leq(\Vert g_k-g(\theta_{k})\Vert+\Vert g_{k+1}-g(\theta_{k})\Vert)\Vert s_k\Vert\\&\leq (L\Vert x_k-\theta_{k}\Vert+L\Vert x_{k+1}-\theta_{k}\Vert)\Vert s_k\Vert\\& = L\Vert s_k\Vert^{2}. \end{aligned} \end{eqnarray}$

(3.8)

So, we can get $0 < \tau_{k} \leq \frac{\tau L}{v}$ . In this case, using (3.6) and the definition of $\vartheta_{k}$ in (2.13), we have

$\begin{eqnarray} \begin{aligned} \left| \frac{1}{\vartheta_{k}}\right| = \left| \frac{\tau_{k}\|s_{k}^{T}y_{k}\|^{2}+\vartheta_{k}\|s_{k}\|^{2}\|y_{k}\|^{2}}{s_{k}^{T}y_{k}\|s_{k}\|^{2}}\right| \leq \frac{\tau_{k}s_{k}^{T}y_{k}}{\|s_{k}\|^{2}}+\frac{\|y_{k}\|^{2}}{s_{k}^{T}y_{k}} \leq \tau L+\frac{L^{2}}{v} = \frac{1}{m}. \end{aligned} \end{eqnarray}$

(3.9)

Moreover,

$\begin{eqnarray} \begin{aligned} \left| \frac{1}{\vartheta_{k}}\right| = \left| \frac{\tau_{k}\|s_{k}^{T}y_{k}\|^{2}+\vartheta_{k}\|s_{k}\|^{2}\|y_{k}\|^{2}}{s_{k}^{T}y_{k}\|s_{k}\|^{2}}\right| \geq \frac{\tau_{k}s_{k}^{T}y_{k}}{\|s_{k}\|^{2}} \geq \tau_{k}{v} = \frac{1}{M}. \end{aligned} \end{eqnarray}$

(3.10)

From (3.9) and (3.10), we have

$\begin{eqnarray} \begin{aligned} \vartheta_{k}\in\left[m,M\right], \end{aligned} \end{eqnarray}$

(3.11)

which shows the boundedness of $\vartheta_{k}$ .

Remark 3.1. In Step 4 of the algorithm, when $\vartheta_{k} < \epsilon_{1}$ , we set $\vartheta_{k} = \frac{s_{k}^{T}y_{k}}{\|y_{k}\|^{2}}$ . Then, by applying Eq (3.5), it can be deduced that $\vartheta_{k}$ is bounded.

The next lemma states an effective property of the direction (1.4).

Lemma 3.2. Let $f$ be uniformly convex on the neighborhood $\mathcal{U}$ of $\mathcal{L}$ , then search direction $\left\lbrace d_{k}\right\rbrace$ produced by Algorithm 2.1 is sufficient descent, that is

$\begin{eqnarray} \begin{aligned} d_{k}^{T}g_{k}\leq-\zeta\Vert g_{k}\Vert^{2}, \forall k > 0. \end{aligned} \end{eqnarray}$

(3.12)

Proof. By carefully studying the proof of Lemma 3.6 of ^[17], we can show that $tr(B_{k+1}^{AMBFGS})$ is bounded.

By Lemma 3.1, we get $s_k^{T}y_{k}\geq v\Vert s_{k}\Vert^{2}, \forall k\geq 0$ and $\vert\check{\eta}\vert\leq L\Vert s_k\Vert^{2}$ . So, considering (2.7), (3.1), (3.2), (3.5) and (3.11), we have

$\begin{eqnarray} \begin{aligned} tr(B_{k+1}^{AMBFGS})& = tr\left( \frac{1}{\vartheta_{k}}I+\frac{y_{k}y_{k}^{T}}{s_{k}^{T}y_{k}}-\frac{1}{\vartheta_{k}}\frac{s_{k}s_{k}^{T}}{s_{k}^{T}s_{k}}+\tau_{k}\frac{y_{k}s_{k}^{T}}{s_{k}^{T}s_{k}}\right) \\& = \frac{n}{\vartheta_{k}}+\frac{y_k^{T}y_k}{s_k^{T}y_k}-\frac{1}{\vartheta_{k}}\frac{s_{k}^{T}s_{k}}{s_{k}^{T}s_{k}}+\tau_{k}\frac{s_{k}^{T}y_{k}}{s_{k}^{T}s_{k}}\\ &\leq \frac{n-1}{m}+\frac{L^{2}}{v}+\tau L\\& = \frac{(n-1)v+mL^{2}+\tau Lmv}{mv}. \end{aligned} \end{eqnarray}$

(3.13)

So, from (1.4) and (3.13), we have

$\begin{eqnarray} \begin{aligned} g_{0}^{T}d_{0} = -\Vert g_{0}\Vert^{2} \end{aligned} \end{eqnarray}$

(3.14)

and

$\begin{eqnarray} \begin{aligned} g_{k+1}^{T}d_{k+1} = -g_{k+1}^{T}H_{k+1}g_{k+1} \leq -\frac{1}{tr(B_{k+1}^{AMBFGS})}\Vert g_{k+1}\Vert^{2} \leq -\frac{mv}{(n-1)v+mL^{2}+\tau Lmv}\Vert g_{k+1}\Vert^{2}. \end{aligned} \end{eqnarray}$

(3.15)

Finally, according to (3.14) and (3.15), let

$\begin{eqnarray} \begin{aligned} \zeta = {\min}\left\lbrace 1,\frac{mv}{(n-1)v+mL^{2}+\tau Lmv}\right\rbrace, \end{aligned} \end{eqnarray}$

(3.16)

then (3.12) is established and the proof is complete.

We next consider the convergence of AMBFGS algorithm. For this purpose, we make the following additional lemma.

Lemma 3.3. Suppose that Assumption 3.1 holds. Consider iterative form $x_{k+1} = x_{k}+\alpha_{k}d_{k}$ , where $\alpha_{k}$ satisfies the Wolfe conditions (2.3) and (2.4) and $d_{k}$ satisfies the sufficient descent condition (3.12). If

$\begin{eqnarray} \begin{aligned} \sum\limits_{k = 0}^{\infty}\frac{1}{||d_{k}||^{2}} = \infty, \end{aligned} \end{eqnarray}$

(3.17)

then,

$\begin{eqnarray} \begin{aligned} \lim\limits_{k\to\infty}\inf\Vert g_{k}\Vert = 0. \end{aligned} \end{eqnarray}$

(3.18)

Proof. Since $d_{k}$ is sufficiently descent by (3.12) and $\alpha_{k}$ satisfies the Wolfe conditions (2.3) and (2.4), the Zoutendijk condition^[20]

$\begin{eqnarray} \begin{aligned} \sum\limits_{k = 0}^\infty\frac{(g_k^Td_k)^2}{\|d_k\|^2} < \infty \end{aligned} \end{eqnarray}$

(3.19)

holds (see Theorem 3.2 of ^[18]). To prove this lemma by contradiction, we suppose that there exists a positive constant $\chi$ such that

$\begin{eqnarray} \begin{aligned} \Vert g_{k}\Vert > \chi, \forall k > 0. \end{aligned} \end{eqnarray}$

(3.20)

Inequalities (3.12) and (3.20) yield $g_{k}^{T}d_{k}\leq -\zeta \Vert g_{k}\Vert^{2}\leq -\zeta \chi^{2}$ , which implies $\frac{\zeta^{2}\chi^{4}}{\chi^{4}}\leq \frac{(g_{k}^{T}d_{k})^{2}}{\Vert d_{k}\Vert^{2}}$ . It follows from the above inequality and (3.19) that

$\begin{eqnarray} \begin{aligned} \sum\limits_{k = 0}^\infty\frac{\zeta^2\chi^4}{\|d_k\|^2}\leq\sum\limits_{k = 0}^\infty\frac{(g_k^Td_k)^2}{\|d_k\|^2} = \infty. \end{aligned} \end{eqnarray}$

(3.21)

Since this contradicts the Zoutendijk condition (3.19), the proof is complete.

Theorem 3.1. Suppose $f$ is uniformly convex on the neighborhood $\mathcal{U}$ of $\mathcal{L}$ , then the Algorithm 2.1 converges in the sense that (3.18) holds.

Proof. Lemma 3.2 shows that $d_{k}\neq 0, \forall k > 0$ , therefore, considering Lemma 3.3, it suffices to prove that $\Vert d_{k+1}\Vert$ is bounded.

From (2.8), (3.1), (3.2), (3.5)–(3.7) and (3.11), we can get

$\begin{eqnarray} \begin{aligned} \Vert H_{k+1}^{AMBFGS}\Vert & = \left\| H_{k+1}^{SMBFGS}-\frac{\tau_{k}(s_{k}^{T}y_{k}s_{k}^{T}s_{k}s_{k}s_{k}^{T}-\vartheta_{k}s_{k}^{T}y_{k}s_{k}^{T}s_{k}s_{k}y_{k}^{T}+\vartheta_{k}s_{k}^{T}s_{k}y_{k}^{T}y_{k}s_{k}s_{k}^{T})}{(1+\tau_{k})s_{k}^{T}y_{k}s_{k}^{T}y_{k}s_{k}^{T}s_{k}}\right\| \\ &\leq v_{k}+2v_{k}\frac{\Vert s_{k}\Vert \Vert y_{k}\Vert}{s_{k}^{T}y_{k}}+(1+v_{k} \frac{\Vert y_{k}\Vert^{2}}{s_{k}^{T}y_{k}})\frac{\Vert s_{k}\Vert^{2}}{s_{k}^{T}y_{k}}+\frac{\tau_{k}\Vert s_{k}\Vert^{2}}{(1+\tau_{k})s_{k}^{T}y_{k}}+\frac{\vartheta_{k}\Vert s_{k}\Vert^{2}\Vert y_{k}\Vert^{2}}{(1+\tau_{k})\Vert s_{k}^{T}y_{k}\Vert^{2}}+\frac{\vartheta_{k}}{1+\tau_{k}}\\ &\leq 2M+2M\frac{L}{v}+\frac{2}{v}+2M\frac{L^{2}}{v^{2}}\\ & = \Lambda. \end{aligned} \end{eqnarray}$

(3.22)

Hence, from (1.4) and (3.2), we get

$\begin{eqnarray} \begin{aligned} ||d_{k+1}||\leq||H_{k+1}^\text{AMBFGS}{ | | }||g_{k+1}||\leq\Lambda\Phi. \end{aligned} \end{eqnarray}$

(3.23)

Inequality (3.23) suggests that $d_{k}$ is bounded. Thus, by Lemma 3.3, we can conclude that the Algorithm 2.1 is convergent.

4. Numerical experiments

In this section, we compare the computational efficiency of SMABFGS (provided by Aminifard et al. ^[6]), AMBFGS-OS (provided by Algorithm 2.1 and $\vartheta_{k}$ adopts the parameters in ^[14]) with AMBFGS (provided by Algorithm 2.1). All codes are written in Matlab 2017a and run on a Dell PC with 2.50 GHz CPU processor and 16 GB RAM memory as well as Windows 11 operation system.

We employ the effective Wolfe conditions with parameters $\rho = 0.99$ and $\delta = 10^{-4}$ in the implementations, as detailed in (2.3) and (2.4). When either $k > 10000$ or $\|g_{k}\| < 10^{-6}$ , all algorithms come to an end. The selection of $\tau = 1, \epsilon_{1} = 10^{-6}$ is made for the AMBFGS parameters, the selection of $\tau = 1$ and $\vartheta_{k} = \frac{s_k^{T}y_k}{\|y_k\|^{2}}$ is made for the AMBFGS-OS parameters. Additionally, for SMABFGS, we set $p = 1$ , $\tau = 1$ , and $C = 0.001$ , if $\|g_{k}\| \geq 1$ , otherwise, $p = 3$ .

4.1. Experiment I: test for unconstrained optimization problems

For experiment Ⅰ, the 71 unconstrained problems are tested and compared, in which the 1–32 problems are taken from the CUTE library ^[21], and the others come from the unconstrained problem collections ^[30,31]. The number of iterations (Itr), the total number of gradient evaluations (Ng), CPU time (Tcpu), and the gradient value $g_{k}$ at the end of iteration are also reported in Table 1. The performance of these algorithms is visually described in terms of Tcpu, Itr, and Ng in Figures 1–3, respectively, using the performance profiles suggested by Dolan and Moré ^[19] (see ^[19] for further information). In general, the top curve indicates that the applicable approach is the winner for the interpretation of the performance profiles.

Table 1. Numerical results.

		SMABFGS	AMBFGS-OS	AMBFGS
$Problems$	$n$	$Itr/Ng/Tcpu/\\|g_{k}\\|$	$Itr/Ng/Tcpu/\\|g_{k}\\|$	$Itr/Ng/Tcpu/\\|g_{k}\\|$
cosine	30	121/764/0.028/9.37e-07	1006/3482/0.095/9.79e-07	163/402/0.012/9.32e-07
dixmaana	6000	190/821/63.073/7.68e-07	213/1050/95.180/3.35e-07	262/1506/118.343/9.93e-08
dixmaanb	1500	211/1055/6.160/9.89e-07	127/736/4.796/8.66e-07	202/820/7.594/8.81e-07
dixmaanb	6000	148/592/51.969/2.63e-07	224/987/98.010/4.90e-07	226/1205/99.291/8.95e-07
dixmaanc	2700	192/804/15.977/6.93e-07	148/667/15.514/9.33e-07	131/580/13.602/9.91e-07
dixmaanc	5400	191/751/53.219/2.28e-07	124/537/43.716/8.44e-07	105/312/36.888/2.25e-07
dixmaand	3000	183/767/18.534/7.16e-07	184/570/22.726/9.81e-07	136/237/16.804/4.81e-07
dixmaane	2400	981/1133/61.848/9.53e-07	1050/1219/83.691/6.97e-07	792/966/64.245/8.84e-07
dixmaanf	6000	1385/1652/470.789/6.93e-07	1817/2280/841.260/9.32e-07	1580/1869/712.104/9.99e-07
dixmaang	900	480/608/6.013/8.85e-07	772/1106/11.630/9.59e-07	468/614/7.269/8.28e-07
dixmaanh	1500	912/1084/23.925/9.97e-07	577/704/19.216/8.73e-07	574/679/20.013/9.28e-07
dixmaani	360	4818/5451/7.748/8.01e-07	3702/4439/7.275/9.46e-07	3576/4178/6.944/9.88e-07
dixmaanj	600	3860/4548/25.622/9.64e-07	5469/6408/47.034/9.88e-07	3924/4550/34.264/8.72e-07
dixmaank	300	2704/3225/3.677/9.84e-07	2419/2914/3.924/5.35e-07	2340/2808/3.804/9.18e-07
dixmaanl	300	2787/3166/3.776/9.84e-07	3010/3559/4.868/9.42e-07	2382/2826/3.845/8.92e-07
dixon3dq	100	2240/2517/0.521/8.87e-07	1402/1691/0.360/7.03e-07	1850/2132/0.467/9.71e-07
dqrtic	4000	245/345/39.458/3.01e-08	173/274/35.476/9.00e-07	156/253/31.649/8.29e-07
edensch	60	276/1719/0.151/9.85e-07	316/1913/0.150/4.02e-07	53/117/0.014/5.83e-07
eg2	90	1612/6930/0.572/7.94e-07	4074/33581/2.179/4.20e-07	2164/14247/1.013/6.40e-07
fletchcr	100	1336/11340/0.715/9.99e-07	4005/37731/2.399/9.91e-07	175/354/0.050/9.86e-07
freuroth	4	NaN/NaN/NaN/NaN	NaN/NaN/NaN/NaN	372/664/0.014/5.55e-07
genrose	10000	345/540/307.401/8.20e-07	256/467/301.828/8.74e-07	302/541/352.489/4.67e-07
himmelbg	7000	10/15/3.993/9.62e-89	10/15/5.002/3.73e-98	10/15/5.049/2.62e-97
liarwhd	30	200/531/0.028/7.86e-07	423/738/0.027/5.67e-07	305/833/0.026/9.99e-07
liarwhd	100	609/947/0.158/8.84e-07	1310/10452/0.734/8.18e-07	1278/7729/0.606/8.00e-07
penalty1	400	5931/62812/52.135/6.91e-07	NaN/NaN/NaN/NaN	3578/35368/47.665/9.25e-07
quartc	4000	245/345/39.672/3.01e-08	173/274/35.207/9.00e-07	156/253/31.673/8.29e-07
tridia	300	1317/1587/1.517/8.34e-07	1430/1748/2.059/8.80e-07	1297/1607/1.801/9.88e-07
woods	1200	586/987/11.051/1.58e-07	553/824/12.976/9.07e-07	456/639/9.842/8.86e-07
VARDIM	160	NaN/NaN/NaN/NaN	NaN/NaN/NaN/NaN	NaN/NaN/NaN/NaN
himmelh	300	NaN/NaN/NaN/NaN	174/519/0.222/6.38e-07	130/371/0.166/1.14e-07
engval1	1000	NaN/NaN/NaN/NaN	NaN/NaN/NaN/NaN	2536/24677/44.083/9.98e-07
bdexp	5000	27/28/6.250/0.00e+00	27/28/7.874/0.00e+00	27/28/7.818/0.00e+00
exdenschnb	1200	166/655/3.118/7.91e-07	178/811/4.156/9.15e-07	64/205/1.459/7.75e-07
exdenschnb	3000	180/522/17.121/7.91e-07	122/426/14.506/6.98e-08	177/724/21.051/6.74e-07
exdenschnb	6000	119/447/40.595/1.37e-07	173/645/71.994/9.40e-07	118/377/48.837/8.24e-07
exdenschnf	1200	161/628/2.992/7.98e-07	135/413/3.098/1.36e-07	133/512/3.120/8.58e-07
exdenschnf	9000	192/515/139.351/1.37e-07	209/864/225.577/4.80e-07	129/313/143.359/9.11e-07
genquartic	1600	134/467/4.055/7.79e-07	156/413/5.926/3.11e-07	112/385/4.263/5.12e-07
genquartic	9000	224/516/184.527/9.43e-07	160/455/179.843/8.94e-07	167/459/192.917/7.90e-07
biggsb1	500	5191/5993/25.611/8.34e-07	4187/4921/26.456/7.89e-07	3962/4625/25.344/8.53e-07
biggsb1	1000	9326/10891/124.762/7.79e-07	7898/9011/133.143/9.26e-07	8819/10180/146.283/7.83e-07
sine	9	99/344/0.011/4.35e-07	NaN/NaN/NaN/NaN	100/364/0.006/7.25e-07
fletcbv3	120	964/1304/0.172/4.69e-07	885/1312/0.186/6.01e-07	563/854/0.127/6.22e-07
nonscomp	500	3022/3601/15.610/6.80e-07	3861/4582/26.791/5.43e-07	2304/2793/15.428/9.01e-07
nonscomp	5000	2102/2671/503.967/9.83e-07	1779/2285/534.087/9.32e-07	1948/2775/583.989/9.88e-07
power1	160	4649/5423/2.199/9.92e-07	5390/6328/3.090/7.32e-07	4178/5012/2.379/9.92e-07
raydan1	600	787/1141/5.836/9.90e-07	1103/2017/10.611/4.99e-07	759/1070/7.263/9.66e-07
raydan2	2000	NaN/NaN/NaN/NaN	NaN/NaN/NaN/NaN	360/2390/19.695/9.93e-07
diagonal1	100	1553/12157/0.423/9.98e-07	NaN/NaN/NaN/NaN	NaN/NaN/NaN/NaN
diagonal2	1000	665/846/9.306/9.46e-07	496/630/8.655/3.64e-07	562/677/9.774/9.94e-07
diagonal3	60	989/7265/0.197/9.99e-07	NaN/NaN/NaN/NaN	167/236/0.014/7.36e-07
diagonal8	100	142/649/0.045/6.44e-07	131/817/0.040/8.78e-07	168/1377/0.059/9.95e-07
bv	2000	129/246/8.215/9.78e-07	118/235/8.906/9.89e-07	133/250/9.927/9.98e-07
bv	20000	0/1/1.842/1.25e-08	0/1/0.672/1.25e-08	0/1/0.738/1.25e-08
ie	500	95/301/22.696/6.48e-07	105/518/39.082/7.29e-08	85/305/23.095/1.03e-07
ie	1500	125/473/317.448/5.36e-07	155/674/395.686/9.85e-07	88/293/143.732/7.95e-08
singx	1000	783/1242/14.044/1.92e-07	1367/2357/32.845/9.61e-07	827/1175/19.524/6.34e-07
singx	2000	1277/2319/84.054/8.45e-07	1056/1598/78.703/6.97e-07	939/1388/69.404/6.82e-07
lin	100	218/1368/1.121/8.25e-07	242/1720/1.344/2.49e-07	123/901/0.704/6.16e-07
lin	500	258/1515/8.617/6.69e-07	276/1830/10.716/8.19e-07	264/1609/9.544/7.68e-07
osb2	11	1164/1457/0.119/9.97e-07	1361/1701/0.095/9.49e-07	1210/1493/0.071/8.77e-07
pen1	200	NaN/NaN/NaN/NaN	NaN/NaN/NaN/NaN	7019/72994/15.319/8.18e-07
pen2	120	1231/4906/1.216/9.31e-07	NaN/NaN/NaN/NaN	NaN/NaN/NaN/NaN
rosex	300	997/1943/1.191/9.34e-07	1281/2176/1.770/7.14e-07	827/1963/1.245/8.88e-07
rosex	700	803/1530/12.360/4.96e-07	831/1253/13.887/1.00e-06	712/1620/13.958/9.98e-07
trid	900	151/297/2.826/8.64e-07	128/226/2.633/8.40e-07	137/247/2.882/7.66e-07
trid	9000	268/603/273.360/9.25e-07	200/489/255.239/8.67e-08	117/194/135.029/4.13e-07
ExFreudenstein	100	NaN/NaN/NaN/NaN	NaN/NaN/NaN/NaN	NaN/NaN/NaN/NaN
ExBeale	100	436/982/0.101/9.67e-07	231/532/0.058/8.63e-07	398/598/0.077/5.35e-07
hager	150	1225/10722/0.741/9.90e-07	290/2112/0.169/9.98e-07	134/292/0.055/8.64e-07

| Show Table

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Figure 1. Performance profiles based on CPU time.

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Figure 2. Performance profiles based on number of iterations.

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Figure 3. Performance profiles based on number of gradient evaluation.

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As can be seen from Table 1, the algorithm presented in the paper is clearly effective for solving most of the tested problems, and it is competitive with the other two algorithms in Itr, Ng, and Tcpu on the tested problems. Figures 1–3 also indicate that the numerical results of the AMBFGS algorithm are better than that of the SMABFGS algorithm and the AMBFGS-OS algorithm. Compared with the SMABFGS algorithm and AMBFGS-OS algorithm, the AMBFGS algorithm is generally in an advantageous position, has better numerical performance, and can solve large-scale unconstrained optimization problems quickly and effectively.

4.2. Experiment II: test for nonlinear equations

For experiment Ⅱ, we compare the performance of SMABFGS, AMBFGS-OS with AMBFGS in solving nonlinear equations, and the following mathematical model is considered:

$\begin{aligned} \min\limits_{x\in\mathbb{R}^n}f(x) = \frac12\|F(x)\|_2^2.\nonumber \end{aligned}$

Define $F(x) = (F_{1}(x), F_{2}(x), \ldots, F_{n}(x))^{T}, x\in \mathbb{R}^n$ and 7 problems are shown below.

Problem 1. ^[32] Set $F_{i}(x) = e^{x_{i}}-1$ , for $i = 1, 2, \ldots, n and x\in \mathbb{R}^n$ .

Problem 2. ^[32] Set

$F(x) = \begin{pmatrix}2.5&1&0&...&0\\1&2.5&1&...&0\\0&1&2.5&...&0\\\vdots&\vdots&\vdots&\ddots&\vdots\\0&0&0&1&2.5\end{pmatrix}x+(-1,\ldots,-1)^{T},$

and $x\in \mathbb{R}^n$ .

Problem 3. ^[32] Set

$F(x) = \begin{pmatrix}2&-1&0&...&0\\0&2&-1&...&0\\0&0&2&...&0\\\vdots&\vdots&\vdots&\ddots&\vdots\\0&0&0&0&2\end{pmatrix}x+(\sin x_{1}-1,\ldots,\sin x_{n}-1)^{T},$

and $x\in \mathbb{R}^n$ .

Problem 4. ^[32] Set $F_{i}(x) = (e^{x_{i}})^2+3\sin x_{i} \cos x_{i}-1$ , for $i = 1, 2, \ldots, n$ and $x\in \mathbb{R}^n$ .

Problem 5. ^[32] Set $F_{i}(x) = (x_{i}-1)^2-1.01$ , for $i = 1, 2, \ldots, n$ and $x\in \mathbb{R}^n$ .

Problem 6. ^[33] Set

$\begin{aligned} F_{1}(x)& = x_{1}(x_{1}^2+x_{2}^2)-1,\\ F_{i}(x)& = x_{i}(x_{i-1}^2+2x_{i}^2+x_{i+1}^2)-1, \text{for }i = 2,3, \ldots ,n-1,\\ F_{n}(x)& = x_{n}(x_{n-1}^2+x_{n}^2)-1, \end{aligned}$

$and x\in \mathbb{R}^n$ .

Problem 7. ^[34] Set

$\begin{aligned} &F_{1}(x) = \sum\limits_{j = 1}^{n}x_{j}^{2},\\ &F_{i}(x) = -2x_{1}x_{i}, \text{for }i = 2,3,\ldots,n,\nonumber \end{aligned}$

and $x\in \mathbb{R}^n$ .

The number of iterations (Itr), the total number of gradient evaluations (Ng), CPU time (Tcpu), and the value $F_{k}$ at the end of iteration are also reported in Tables 2–8. The performance of these algorithms is visually described in terms of Tcpu, Itr, and Ng in –, respectively, using the performance profiles suggested by Dolan and Moré ^[19]. In general, the top curve indicates that the applicable approach is the winner for the interpretation of the performance profiles. For each problem, we select 4 to 5 initial points from the following 7 points, that is, $x_{1} = (1, 1, \ldots, 1)^{T}$ , $x_{2} = (0.1, 0.1, \ldots, 0.1)^{T}$ , $x_{3} = (\frac{1}{2}, \frac{1}{2^{2}}, \ldots, \frac{1}{2^{n}})^{T}$ , $x_{4} = (0, \frac{1}{n}, \ldots, \frac{n-1}{n})^{T}$ , $x_{5} = (1, \frac{1}{2}, \ldots, \frac{1}{n})^{T}$ , $x_{6} = (\frac{1}{n}, \frac{2}{n}, \ldots, 1)^{T}$ , $x_{7} = (1-\frac{1}{n}, 1- \frac{2}{n}, \ldots, 0)^{T}$ .

Table 2. Numerical results (Problem 1).

		SMABFGS	AMBFGS-OS	AMBFGS
$x_{0}$	$n$	$Itr/Ng/Tcpu/\\|F_{k}\\|$	$Itr/Ng/Tcpu/\\|F_{k}\\|$	$Itr/Ng/Tcpu/\\|F_{k}\\|$
$x_{2}$	50	164/904/0.030/1.27e-06	131/807/0.021/5.69e-06	153/1002/0.022/1.71e-06
	100	1/2/0.000/1.05e+00	1/2/0.001/1.05e+00	1/2/0.000/1.05e+00
	500	165/933/0.733/1.29e-06	191/1127/1.188/1.22e-06	166/1095/1.135/5.70e-06
$x_{3}$	50	63/197/0.015/1.80e-06	68/289/0.011/1.69e-06	41/213/0.006/3.79e-06
	100	63/197/0.015/1.80e-06	68/289/0.020/1.69e-06	41/213/0.010/3.79e-06
	500	63/197/0.275/1.80e-06	68/289/0.504/1.69e-06	41/213/0.353/3.79e-06
$x_{4}$	50	86/342/0.017/1.17e-06	72/183/0.007/1.73e-06	74/410/0.012/1.59e-06
	100	93/446/0.024/2.26e-06	77/423/0.022/1.32e-06	43/177/0.011/2.72e-06
	500	132/426/0.824/1.12e-06	75/343/0.650/2.14e-06	133/533/1.130/1.44e-06
$x_{6}$	50	136/356/0.030/2.31e-06	117/376/0.030/1.36e-06	178/496/0.041/3.44e-06
	100	96/423/0.041/1.00e-06	114/573/0.053/1.01e-06	61/309/0.027/1.36e-06
	500	111/363/0.695/1.00e-06	127/354/1.084/1.65e-06	91/390/0.788/1.29e-06
$x_{7}$	50	81/335/0.021/2.06e-06	72/183/0.016/1.73e-06	74/410/0.024/1.50e-06
	100	89/423/0.036/1.28e-06	77/423/0.035/1.25e-06	43/177/0.018/2.77e-06
	500	147/440/0.938/1.13e-06	74/342/0.634/2.85e-06	134/544/1.142/3.52e-06

| Show Table

DownLoad: CSV

Table 3. Numerical results (Problem 2).

		SMABFGS	AMBFGS-OS	AMBFGS
$x_{0}$	$n$	$Itr/Ng/Tcpu/\\|F_{k}\\|$	$Itr/Ng/Tcpu/\\|F_{k}\\|$	$Itr/Ng/Tcpu/\\|F_{k}\\|$
$x_{2}$	50	200/319/0.047/2.50e-01	187/279/0.029/2.50e-01	180/274/0.022/2.50e-01
	200	169/286/0.148/2.50e-01	194/356/0.204/2.50e-01	181/350/0.201/2.50e-01
	600	209/342/1.659/2.50e-01	181/327/1.751/2.50e-01	213/373/2.077/2.50e-01
$x_{5}$	50	197/290/0.031/2.50e-01	215/322/0.033/2.50e-01	190/301/0.033/2.50e-01
	200	182/293/0.166/2.50e-01	199/310/0.212/2.50e-01	206/359/0.219/2.50e-01
	600	200/403/1.537/2.50e-01	220/368/2.177/2.50e-01	190/330/1.878/2.50e-01
$x_{6}$	50	214/447/0.036/2.50e-01	166/359/0.028/2.50e-01	173/341/0.028/2.50e-01
	200	284/601/0.244/2.50e-01	217/514/0.217/2.50e-01	241/524/0.248/2.50e-01
	600	203/375/1.599/2.50e-01	219/462/2.124/2.50e-01	239/480/2.349/2.50e-01
$x_{7}$	50	169/265/0.026/2.50e-01	201/322/0.032/2.50e-01	192/318/0.028/2.50e-01
	200	283/624/0.259/2.50e-01	225/540/0.246/2.50e-01	240/579/0.245/2.50e-01
	600	202/440/1.545/2.50e-01	223/442/2.222/2.50e-01	214/427/2.024/2.50e-01

| Show Table

DownLoad: CSV

Table 4. Numerical results (Problem 3).

		SMABFGS	AMBFGS-OS	AMBFGS
$x_{0}$	$n$	$Itr/Ng/Tcpu/\\|F_{k}\\|$	$Itr/Ng/Tcpu/\\|F_{k}\\|$	$Itr/Ng/Tcpu/\\|F_{k}\\|$
$x_{2}$	60	179/632/0.064/2.69e-07	133/296/0.031/6.83e-07	118/255/0.028/4.13e-07
	100	101/366/0.037/7.11e-07	101/258/0.033/6.26e-07	78/292/0.030/1.22e-06
	500	109/285/0.683/5.31e-07	101/431/0.879/4.88e-07	89/194/0.762/4.85e-07
$x_{3}$	60	199/596/0.049/2.77e-07	126/241/0.026/4.99e-07	128/192/0.025/3.69e-07
	100	95/254/0.030/3.18e-07	142/505/0.054/3.51e-07	70/244/0.026/1.19e-06
	500	145/485/0.920/3.90e-07	145/314/1.229/6.28e-07	136/426/1.167/4.31e-07
$x_{4}$	60	116/427/0.033/3.10e-07	124/416/0.034/3.34e-07	149/400/0.037/4.58e-07
	100	190/536/0.062/5.61e-07	55/176/0.020/5.43e-07	83/201/0.027/8.31e-07
	500	109/203/0.683/2.50e-06	97/369/0.832/5.04e-07	105/181/0.884/1.16e-06
$x_{5}$	60	149/230/0.029/2.74e-07	92/229/0.022/4.93e-07	110/315/0.028/3.83e-07
	100	108/287/0.033/3.72e-07	130/396/0.045/7.37e-07	94/212/0.030/3.90e-07
	500	125/328/0.774/5.23e-07	78/245/0.688/3.87e-06	86/262/0.748/6.28e-07
$x_{7}$	60	128/283/0.031/4.34e-07	111/287/0.029/4.78e-07	111/321/0.032/3.34e-07
	100	118/347/0.046/5.85e-07	62/162/0.025/8.92e-07	85/264/0.035/3.85e-07
	500	109/223/0.807/3.90e-07	152/430/1.342/4.11e-07	111/193/0.925/5.41e-07

| Show Table

DownLoad: CSV

Table 5. Numerical results (Problem 4).

		SMABFGS	AMBFGS-OS	AMBFGS
$x_{0}$	$n$	$Itr/Ng/Tcpu/\\|F_{k}\\|$	$Itr/Ng/Tcpu/\\|F_{k}\\|$	$Itr/Ng/Tcpu/\\|F_{k}\\|$
$x_{1}$	60	212/1199/0.094/2.11e-07	180/1058/0.070/2.77e-07	193/1016/0.070/2.01e-07
	200	219/1117/0.255/2.58e-07	144/1001/0.206/2.38e-07	135/773/0.180/9.55e-07
	500	191/1030/1.304/2.36e-07	208/1125/1.820/2.16e-07	180/1099/1.608/2.62e-07
$x_{2}$	60	144/777/0.054/7.60e-06	164/848/0.063/1.66e-06	148/873/0.062/7.31e-07
	200	188/1035/0.223/2.60e-07	145/753/0.191/4.09e-06	196/1038/0.251/2.34e-07
	500	202/1282/1.452/2.01e-07	147/953/1.373/9.18e-07	141/863/1.318/2.10e-07
$x_{5}$	60	146/516/0.039/2.99e-07	97/364/0.022/2.39e-07	153/570/0.034/6.92e-07
	200	89/263/0.071/5.99e-07	131/259/0.126/6.19e-07	131/336/0.129/4.29e-07
	500	82/253/0.557/2.89e-07	89/233/0.845/2.55e-07	56/127/0.549/4.42e-07
$x_{6}$	60	156/399/0.037/2.16e-07	109/495/0.024/2.90e-07	160/619/0.034/2.54e-07
	200	189/639/0.181/2.10e-07	165/509/0.189/4.42e-07	81/369/0.101/2.36e-07
	500	82/255/0.620/8.53e-07	137/467/1.379/2.24e-07	117/526/1.110/2.14e-07

| Show Table

DownLoad: CSV

Table 6. Numerical results (Problem 5).

		SMABFGS	AMBFGS-OS	AMBFGS
$x_{0}$	$n$	$Itr/Ng/Tcpu/\\|F_{k}\\|$	$Itr/Ng/Tcpu/\\|F_{k}\\|$	$Itr/Ng/Tcpu/\\|F_{k}\\|$
$x_{1}$	50	0/1/0.004/0.00e+00	0/1/0.000/0.00e+00	0/1/0.000/0.00e+00
	200	0/1/0.000/0.00e+00	0/1/0.001/0.00e+00	0/1/0.000/0.00e+00
	600	0/1/0.000/0.00e+00	0/1/0.000/0.00e+00	0/1/0.000/0.00e+00
$x_{2}$	50	219/1315/0.047/5.46e-07	113/788/0.022/1.34e-06	165/1170/0.031/5.26e-07
	200	131/972/0.122/6.45e-07	169/889/0.171/3.38e-06	134/838/0.149/3.20e-06
	600	128/869/1.278/5.24e-07	129/839/1.586/6.79e-06	175/1070/1.930/5.12e-07
$x_{3}$	50	84/262/0.025/6.97e-06	106/482/0.016/7.14e-07	44/167/0.009/7.72e-06
	200	105/522/0.095/5.16e-07	90/289/0.084/4.12e-06	62/197/0.052/1.26e-06
	600	59/287/0.501/5.99e-07	72/282/0.757/6.40e-07	28/84/0.300/5.70e-06
$x_{5}$	50	101/433/0.031/1.01e+00	100/471/0.015/1.01e+00	111/347/0.013/1.01e+00
	200	79/290/0.070/1.01e+00	135/426/0.133/1.01e+00	102/383/0.105/1.01e+00
	600	97/292/0.819/1.01e+00	92/329/0.967/1.01e+00	55/173/0.587/1.01e+00
$x_{6}$	50	47/113/0.008/8.20e-07	112/453/0.024/5.22e-08	84/373/0.019/8.33e-07
	200	173/509/0.157/4.90e-07	171/602/0.189/5.13e-07	106/325/0.114/9.78e-07
	600	107/321/0.813/9.48e-07	105/380/1.040/5.60e-07	125/532/1.247/6.48e-07

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DownLoad: CSV

Table 7. Numerical results (Problem 6).

		SMABFGS	AMBFGS-OS	AMBFGS
$x_{0}$	$n$	$Itr/Ng/Tcpu/\\|F_{k}\\|$	$Itr/Ng/Tcpu/\\|F_{k}\\|$	$Itr/Ng/Tcpu/\\|F_{k}\\|$
$x_{1}$	50	215/347/0.074/3.84e-07	156/294/0.028/8.83e-07	179/349/0.022/7.99e-07
	200	141/236/0.139/1.31e-06	208/309/0.212/5.07e-07	216/445/0.256/3.49e-07
	500	236/366/2.194/6.31e-07	175/302/2.305/1.45e-06	172/402/2.117/5.08e-07
$x_{2}$	50	148/293/0.026/3.86e-07	197/339/0.034/4.32e-07	159/274/0.047/4.71e-07
	200	228/427/0.214/1.15e-06	117/282/0.137/1.44e-06	172/343/0.242/4.65e-07
	500	198/424/1.949/1.20e-06	166/253/2.328/1.16e-06	171/279/2.677/5.59e-07
$x_{3}$	50	693/761/0.163/3.10e-07	708/783/0.062/8.10e-07	412/496/0.038/7.75e-07
	200	2595/2733/3.698/6.50e-07	2645/2857/4.039/2.76e-07	1466/1539/2.077/1.89e-06
	500	6512/6606/55.638/3.15e-07	6425/6485/54.137/7.41e-07	3566/3639/29.952/1.89e-06
$x_{4}$	50	193/260/0.016/1.62e-06	195/331/0.015/1.30e-06	215/369/0.016/6.72e-07
	200	279/556/0.212/2.24e-07	270/371/0.208/1.31e-06	230/307/0.187/9.36e-07
	500	256/347/1.606/9.32e-07	310/402/2.603/7.21e-07	246/344/2.026/1.45e-06
$x_{7}$	50	132/247/0.014/5.59e-07	136/206/0.011/4.30e-07	156/363/0.014/6.96e-07
	200	159/301/0.123/4.65e-07	193/363/0.149/2.78e-07	156/246/0.118/5.53e-07
	500	171/261/1.060/5.29e-07	225/350/1.871/4.36e-07	170/268/1.386/2.69e-07

| Show Table

DownLoad: CSV

Table 8. Numerical results (Problem 7).

		SMABFGS	AMBFGS-OS	AMBFGS
$x_{0}$	$n$	$Itr/Ng/Tcpu/\\|F_{k}\\|$	$Itr/Ng/Tcpu/\\|F_{k}\\|$	$Itr/Ng/Tcpu/\\|F_{k}\\|$
$x_{2}$	50	81/176/0.029/6.38e-05	113/457/0.029/3.63e-05	103/450/0.029/6.93e-05
	200	31/76/0.031/6.62e-05	34/145/0.041/8.57e-05	20/37/0.021/8.73e-03
	500	65/147/0.392/5.95e-05	119/530/1.090/2.02e-04	20/40/0.165/1.19e-02
$x_{4}$	50	69/596/0.029/9.63e-05	74/596/0.028/2.45e-04	74/643/0.031/8.24e-05
	200	91/703/0.107/6.80e-05	171/1075/0.223/6.46e-05	114/796/0.152/8.24e-05
	500	NaN/NaN/NaN/NaN	NaN/NaN/NaN/NaN	NaN/NaN/NaN/NaN
$x_{6}$	50	135/253/0.024/4.99e-05	180/853/0.058/5.59e-05	132/456/0.035/7.84e-03
	200	81/214/0.082/1.26e-04	59/197/0.074/6.21e-05	98/520/0.133/6.58e-05
	500	93/221/0.656/3.83e-03	40/201/0.371/1.09e-04	83/219/0.769/1.29e-02
$x_{7}$	50	24/91/0.006/3.18e-04	74/470/0.028/6.85e-05	21/84/0.006/3.37e-04
	200	97/194/0.089/1.59e-04	269/893/0.321/1.38e-04	88/337/0.107/5.85e-05
	500	73/212/0.503/3.32e-05	148/840/1.311/4.55e-05	35/126/0.295/1.98e-04

| Show Table

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Figure 4. Performance profiles based on CPU time.

DownLoad: Full-Size Img PowerPoint

Figure 5. Performance profiles based on number of iterations.

DownLoad: Full-Size Img PowerPoint

Figure 6. Performance profiles based on number of gradient evaluation.

DownLoad: Full-Size Img PowerPoint

As can be seen from Tables 2–8, the algorithm presented in the paper is clearly effective for solving most of the tested problems and is competitive with the other two algorithms in Itr, Ng, and Tcpu on the tested problems. Figures 4–6 also indicate that the AMBFGS algorithm, when compared with the SMABFGS and AMBFGS-OS algorithms, generally occupies an advantageous position. It exhibits better numerical performance and can solve nonlinear equations quickly and effectively.

5. Conclusions

In this research, we presented an augmented memoryless BFGS algorithm based on a modified secant condition, which ensures a descent search direction. We determined the scaling parameter by reducing the upper bound of the condition number using an eigenvalue analysis. Global convergence of our approach has been demonstrated under appropriate assumptions. Finally, numerical results obtained by applying the AMBFGS method to solve large-scale unconstrained optimization problems and nonlinear equations demonstrate its encouraging efficiency, even when compared to the SMABFGS method and AMBFGS-OS method.

Author contributions

Yulin Cheng and Jing Gao: Methodology, Software, Visualization, Writing-original draft. All authors of this article have been contributed equally. All authors have read and approved the final version of the manuscript for publication.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work is supported by the Scientific and Technological Developing Scheme of Jilin Province (YDZJ202101ZYTS167, YDZJ202201ZYTS303, 20230508184RC); the project of education department of Jilin Province (JJKH20210030KJ, JJKH20230054KJ); the doctoral research project start-up fund of Beihua University; the graduate innovation project of Beihua University (2023037).

Conflict of interest

All authors declare no conflicts of interest in this paper.

References

[1]	Versfelt Jr PL (2001) Major Hydrocarbon potential in Iran, In: Downey MW, Threet JC, Morgan WA, Petroleum provinces of the twenty first century, American Association of Petroleum Geologists Memorial, 74: 417-427.
[2]	Alavi M (2004) Regional stratigraphy of the Zagros fold-thrust belt of Iran and its proforeland evolution. Am J Sci 304: 1-20. doi: 10.2475/ajs.304.1.1
[3]	James GA, Wynd JG (1965) Stratigraphic Nomenclature of Iranian Oil Consortium Agreement Area. AAPG Bull 49: 2182-2245.
[4]	Ala MA, Kinghorn RRF, Rahman M (1980) Organic geochemistry and source rock characteristics of the Zagros petroleum province: South West Iran. J Pet Geol 3: 6l-89.
[5]	Bordenave ML, Bunwood R (1990) Source rock distribution and maturation in the Zagros orogenic belt-provenance of the Asmari and Bangestan reservoir oil accumulations. Org Geochem 16: 369-378. doi: 10.1016/0146-6380(90)90055-5
[6]	Bordenave ML, Huc AY (1995) The Cretaceous source rocks in the Zagros Foothills of Iran. Rev Inst Fr Pet 50: 721-752.
[7]	Daraei M, Amini A, Ansari M (2015) Facies analysis and depositional environment study of the mixed carbonate-evaporite Asmari Formation (Oligo-Miocene) in the sequence stratigraphic framework, NW Zagros, Iran. Carbonates Evaporites 30: 253-272. doi: 10.1007/s13146-014-0207-4
[8]	Berberian M, King GCP (1981) Towards a paleogeography and tectonic evolution of Iran. Can J Earth Sci 18: 210-265. doi: 10.1139/e81-019
[9]	Lees GM, Falcon NL (1952) The geographical history of the Mesopotamian plains. Geogr J 118: 24-39. doi: 10.2307/1791234
[10]	Purser BH (1973) Sedimentation around bathymetric highs in the southern Persian Gulf, In: Purser BH, The Persian Gulf: Holocene carbonate sedimentation and diagenesis in a shallow epicontinental sea, Berlin and New York: Springer-Verlag, 157-177.
[11]	Kassler P (1973) The structural and geomorphic evolution of the Persian Gulf, In: PurserBH, The Persian Gulf: Holocene carbonate sedimentation and diagenesis in a shallow epicontinental sea, Berlin and New York: Springer-Verlag, 11-32.
[12]	Baltzer F, Purser BH (1990) Modern alluvial fan and deltaic sedimentation in a foreland Tectonic setting: the lower Mesopotamian plain and the Arabian Gulf. Sediment Geol 67: 175-195. doi: 10.1016/0037-0738(90)90034-Q
[13]	Stocklin JA (1974) Northern Iran: Alborz Mountains, Mesozoic-Cenozoic orogenic Belt, data for orogenic studies. Geological Society of London Special Publication, 4: 213-234. doi: 10.1144/GSL.SP.2005.004.01.12
[14]	Takin M (1972) Iranian geology and continental drift in the Middle East. Nature 235: 147-150. doi: 10.1038/235147a0
[15]	Alavi M (2007) Structures of the Zagros fold-thrust belt in Iran. Am J Sci 307: 1064-1095. doi: 10.2475/09.2007.02
[16]	Sharland PR, Archer DM, Casey RB, et al. (2001) Arabian plate sequence stratigraphy. Gulf Petro link: Manama, Bahrain, 371-384.
[17]	Mohseni H, Al-Aasm IS (2004) Tempestite deposits on the storm influenced carbonate ramp: An example from the Pabdeh Formation (Paleogene), Zagros basin, SW Iran. J Pet Geol 27: 163-178. doi: 10.1111/j.1747-5457.2004.tb00051.x
[18]	Mohseni H, Behbahani R, Khodabakhsh S, et al. (2011) Depositional environments and trace fossil assemblages in the Pabdeh Formation (Paleogene). Zagros Basin, Iran. Neues Jahrb Geol Paläontol 262: 59-77.
[19]	Bolourchifard F, Fayazi F, Mehrabi B, et al. (2019) Evidence of high-energy storm and shallow water facies in Pabdeh sedimentary phosphate deposit, Kuhe-Lar-anticline, SW Iran. Carbonate Evaporite 34: 1703-1721. doi: 10.1007/s13146-019-00520-4
[20]	Bahrami M (2009) Microfacies and Sedimentary Environments of Gurpi and Pabdeh Formations in Southwest of Iran. Am J Appl Sci 6: 1295-1300. doi: 10.3844/ajassp.2009.1295.1300
[21]	Rezaee P, Nejad SAA (2014) Depositional evolution and sediment facies pattern of the tertiary basin in southern Zagros, South Iran. Asian J Earth Sci 7: 27-39. doi: 10.3923/ajes.2014.27.39
[22]	Wilson JL (1975) Carbonate Facies in Geologic History. Berlin, Germany: Springer-Verlag, 471.
[23]	Carozzi AV, Gerber MS (1978) Sedimentary Chert Breccia: A Mississippian Tempestite. J Sediment Pet 48: 705-708.
[24]	Bourgeois JA (1980) Transgressive Shelf Sequence Exhibiting Hummocky Stratification: The Cape Sebastian Sandstone (Upper Cretaceous), Southwestern Oregon. J Sediment Pet 50: 681-702. doi: 10.1306/212F7AC2-2B24-11D7-8648000102C1865D
[25]	Kreisa RD (1981) Storm-Generated Sedimentary Structures in Subtidal Marine with Examples from the Middle and Upper Ordovician of Southwestern Virginia. J Sediment Pet 51: 823-848.
[26]	Walker RG (1984) General Introduction: facies, facies sequences and facies models, In: Walker RG, Facies Models, Canada: Geoscience, 1-9.
[27]	Vera JA, Molina JM (1998) Shallowing-upward cycles in pelagic troughs. (Upper Jurassic, Subbetic, Southern Spain). Sediment Geol 119: 103-121.
[28]	Cheel RJ, Leckie DA (1993) Coarse grained storm beds of the upper Cretaceous Chungo Member (Wapiabi Formation), Southern Alberta, Canadian. J Sediment Pet 62: 933-945.
[29]	Molina JM, Ruiz-Ortiz PA, Vera JA (1997) Calcareous tempestites in pelagic facies (Jurassic, Betic Cordilleras, Southern Spain). J Sediment Pet 109: 95-109.
[30]	Hips K (1998) Lower Triassic storm-dominated ramp sequence in northern Hungary: an example of evolution from homoclinal through distally steepened ramp to Middle Triassic flat-toped platform, In: Wright V, Burchette TP, Carbonate Ramps, London: Geological Society of London, Special Publication, 149: 315-338.
[31]	Abed AM (2013) The eastern Mediterranean phosphorite giants: An interplay between tectonics and upwelling. GeoArabia 18: 67-94.
[32]	Jasinski SM (2003) Phosphate rock. United States Geological Survey Minerals Yearbook, 56.1-56.5.
[33]	Jasinski SM (2011) Phosphate rock. Mineral Commodity Summaries 2011, United States Geological Survey, United States Government Printing Office, Washington, D.C.
[34]	Van Kauwenbergh SJ (2010) World phosphate rock reserves and resources. International Fertilizer Development Center (IFDC), Technical Bulletin no. 75, Muscle Shoals, Alabama, USA, 58.
[35]	Lucas J, Prévôt L (1975) Les marges continentales, pièges géochimiques, l'example de la marge atlantique de l'Afrique a la limite Crétacé-Tertiaire. Bull Soc Géol Fr 7: 496-501. doi: 10.2113/gssgfbull.S7-XVII.4.496
[36]	Notholt AJG (1980) Economic phosphatic sediments-mode of occurrence and stratigraphical distribution. J Geol Soc Lond 137: 793-805. doi: 10.1144/gsjgs.137.6.0793
[37]	Lucas J, Prévôt L (1995) Tethyan phosphates and bioproductites, In: Nairn AEM, Stehli FG, The Ocean Basins and Margins-The Tethys Ocean, Plenum Press, 8: 367-391.
[38]	Follmi KB (1996) The phosphorus cycle, phosphogenesis and marine phosphate-rich deposits. Earth-Sci Rev 40: 55-124. doi: 10.1016/0012-8252(95)00049-6
[39]	Soudry D, Glenn CR, Nathan Y, et al. (2006) Evolution of the Tethyan phosphogenesis along the northern edges of the Arabian-African shield during the Cretaceous-Eocene as deduced from temporal variations in Ca and Nd isotopes and rates of P accumulation. Earth-Sci Rev 78: 27-57. doi: 10.1016/j.earscirev.2006.03.005
[40]	Al-Bassam KS (1989) The Akashat phosphate deposits, Iraq. In: NothoIt AJG, Sheldon RP, Davison DF, Phosphate Deposit of the World, Phosphate Rock Resources, Cambridge Univ. Press, 2: 316-322.
[41]	Jones RW, Racey A (1994) Cenozoic stratigraphy of the Arabian Peninsula and Gulf. In: Simmons MD, Micropalaeontology and Hydrocarbon Exploration in the Middle East. London, Chapman and Hall, 273-307.
[42]	Bramkamp RA (1941) Unpublished report, In: Cavelier C, Geological description of the Qatar Peninsula, Qatar, Bureau de recherches géologiques et minières, 39.
[43]	Soudry D, Nathan Y, Ehrlich S (2013) Geochemical diagenetic trends during phosphorite formation-economic implications: The case of the Negev Campanian phosphorites, Southern. Sedimentology 60: 800-819. doi: 10.1111/j.1365-3091.2012.01361.x
[44]	Follmi KB, Garrison RE (1991) Phosphatic sediments, ordinary or extraordinary deposits? The example of the Miocene Monterey Formation (California), In: Muller DW, McKenzie JA, Weissert H, Geologic Events and Non-Uniform Sedimentation, London, Academic Press, 55-84.
[45]	Zaïer A, Beji-Sassi A, Sassi S, et al. (1998) Basin evolution and deposition during the Early Paleocene in Tunisia. In: Macgregor DS, Moody RTJ, ClarkLowes DD, Petroleum Geology of North Africa, London, Geological Society of London Special Publication, 132: 375-393.
[46]	Burollet PF (1956) Contribution à l'étude stratigraphique de la Tunisie Centrale. Ann Mines Géol 18: 352.
[47]	Fournier D (1978) Nomenclature lithostratigraphique des série du Crétacé supérieur au Tertiaire de Tunisie. Bull Cent Rech Explor Prod Elf-Aquitaine 2: 97-148.
[48]	Sassi S (1974) La sédimentation phosphatée au Paléocène dans le Sud et le Centre Ouest de la Tunisie. Thesis Doctorat. d'Etat ès-Sciences, Université de Paris-Orsay, France, 300.
[49]	Chaabani F (1995) Dynamique de la partie orientale du bassin de Gafsa au Crétacé ET au Paléogène: Etude minéralogique ET géochimique de la série phosphate Eocène, Tunisie méridionale. Thèse Doc. Etat, Univ. Tunis II. Tunisie.
[50]	Henchiri M, Slim-S'Himi N (2006) Silicification of sulfate evaporites and their carbonate replacement in Eocene marine sediments, Tunisia, two diagenetic trends. Sedimentology 53: 1135-1159. doi: 10.1111/j.1365-3091.2006.00806.x
[51]	Chaabani F, Ounis A (2008) Sequence stratigraphy and depositional environment of phosphorite deposits evolution: case of the Gafsa basin, Tunisia. Conference abstract at the Intern. Geol. Cong. Oslo.
[52]	Galfati I, Sassi AB, Zaier A, et al. (2010) Geochemistry and mineralogy of Paleocene-Eocene Oum El Khecheb phosphorites (Gafsa-Metlaoui Basin) Tunisia. Geochem J 44: 189-210. doi: 10.2343/geochemj.1.0062
[53]	Kocsis L, Gheerbrant E, Mouflih M, et al. (2014) Comprehensive stable isotope investigation of marine biogenic apatite from the late Cretaceous-early Eocene phosphate series of Morocco. Palaeogeogr Palaeoclimatol Palaeoecol 394: 74-88. doi: 10.1016/j.palaeo.2013.11.002
[54]	Messadi AM, Mardassi B, Ouali JA, et al. (2016) Sedimentology, diagenesis, clay mineralogy and sequential analysis model of Upper Paleocene evaporitecarbonate ramp succession from Tamerza area (Gafsa Basin: Southern Tunisia). J Afr Earth Sci 118: 205-230. doi: 10.1016/j.jafrearsci.2016.02.020
[55]	El-Naggar ZR, Saif SI, Abdennabi A (1982) Stratigraphical analysis of the phosphate deposits in Northwestern Saudi Arabia. Progress report 1-4 submitted to SANCST, Riyadh. Saudi Arabia.
[56]	Baioumy H, Tada R (2005) Origin of Late Cretaceous phosphorites in Egypt. Cretaceous Res 26: 261-275. doi: 10.1016/j.cretres.2004.12.004
[57]	Scotese CR (2014) The PALEOMAP Project Paleo Atlas for ArcGIS, version 2, Volume 1, Cenozoic Plate Tectonic, Paleogeographic, and Paleoclimatic Reconstructions, PALEOMAP Project, Evanston, IL, Maps 1-15.
[58]	Wade BS, Pearson PN, Berggren, WA, et al. (2011) Review and revision of Cenozoic tropical planktonic foraminiferal biostratigraphy and calibration to the geomagnetic polarity and astronomical time scale. Earth-Sci Rev 104: 111-142. doi: 10.1016/j.earscirev.2010.09.003
[59]	Dunham RJ (1962) Classifications of carbonate rocks according to depositional texture. In: Ham WE, Classification of Carbonate Rocks-A Symposium, AAPG Special Publications, 108-121.
[60]	Flugel E (2010) Microfacies of Carbonate Rocks, Analysis, Interpretation and Application, 2nd Edition. Berlin: Springer-Verlag, 984.
[61]	Omidvar M, Safari A, Vaziri-Moghaddam H, et al. (2016) Facies analysis and paleoenvironmental reconstruction of upper cretaceous sequences in the eastern Para-Tethys basin, NW Iran. Geol Acta 14: 363-384.
[62]	Baioumy H, Omran M, Fabritius T (2017) Mineralogy, geochemistry and the origin of high-phosphorus oolitic iron ores of Aswan, Egypt. Ore Geol Rev 80: 185-199. doi: 10.1016/j.oregeorev.2016.06.030
[63]	Donnelly TH, Shergold JH, Southgate PN, et al. (1990) Events leading to global phosphogenesis around the Proterozoic-Cambrian transition, In: Notholt AJG, Jarvis I, Phosphorite research and development, London: Journal Geology Society London. Special Publication, 52: 273-287. doi: 10.1144/GSL.SP.1990.052.01.20
[64]	Trappe J (1998) Phanerozoic phosphorite depositional systems: A dynamic model for a sedimentary resource system, Lecture Notes in Earth Science 76: 1-316.
[65]	Piper DZ (1991) Geochemistry of a Tertiary sedimentary phosphate deposit: Baja California Sur, Mexico. Chem Geol 92: 283-316. doi: 10.1016/0009-2541(91)90075-3
[66]	Soudry D, Ehrlich S, Yoffe O, et al. (2002) Uranium oxidation state and related variations in geochemistry of phosphorites from the Negev (southern). Chem Geol 189: 213-230. doi: 10.1016/S0009-2541(02)00144-4
[67]	Jarvis I, Burnett WC, Nathan Y, et al. (1994) Phosphorite geochemistry: state-of-the-art and environmental concerns. Eclogae Geol Helv 87: 643-700.
[68]	Garnit H, Bouhlel S, Jarvis I (2017) Geochemistry and depositional environments of Paleocene-Eocene phosphorites: Metlaoui Group, Tunisia. J Afr Earth Sci 134: 704-736. doi: 10.1016/j.jafrearsci.2017.07.021
[69]	McArthur JM (1985) Francolite geochemistry-compositional controls during formation, diagenesis, metamorphism and weathering. Geochim Cosmochim Acta 49: 23-35. doi: 10.1016/0016-7037(85)90188-7
[70]	McArthur JM (1978) Systematic variations in the contents of Na, Sr, CO₂ and SO₄ in marine carbonate fluorapatite and their relation to weathering. Chem Geol 21: 41-52. doi: 10.1016/0009-2541(78)90008-6
[71]	Altschuler ZS (1980) The geochemistry of trace elements in marine phosphorites. Part I: characteristic abundances and enrichment, In: Bentor YK, Marine Phosphorites, London: SEPM Special Publication, 29: 19-30.
[72]	Prévôt L (1990) Geochemistry, Petrography, Genesis of Cretaceous-Eocene Phosphorites. Soc Geol Fr Mem 158: 1-232.
[73]	Jiang SY, Zhao HX, Chen YQ, et al. (2007) Trace and rare earth element geochemistry of phosphate nodules from the lower Cambrian black shale sequence in the Mufu Mountain of Nanjing, Jiangsu province, China. Chem Geol 244: 584-604. doi: 10.1016/j.chemgeo.2007.07.010
[74]	Prevot L, Lucas J (1980) Behaviour of some trace elements in phosphatic sedimentary formations. Society of Economic Paleontologists and Mineralogists, Special Publication 29: 31-40.
[75]	Gulbrandsen RA (1966) Chemical composition of the phosphorites of the Phosphoria Formation. Geochim cosmochim Acta. 30: 769-778. doi: 10.1016/0016-7037(66)90131-1
[76]	Slansky M (1986) Geology of sedimentary phosphate (Studies in Geology), 1st English edition, North Oxford Academic Publishers Limited, 59-159.
[77]	Iylin AV, Volkov RI (1994) Uranium Geochemistry in Vendian-Cambrian Phosphorites. Geokhimiya 32: 1042-1051.
[78]	Sokolov AS (1996) Evolution of Uranium Mineralization in Phosphorites. Geokluntiva 11:1117-1119.
[79]	Zanin YN, Zanrirailova AG, Gilinskaya LG, et al. (2000) Uranium in the Sedimentary Apatite during Catagenesis. Geokhimiya 5: 502-509.
[80]	Baturin GN, Kochenov AV (2001) Uranium in phosphorites. Lithol Miner Resour 36: 303-321. doi: 10.1023/A:1010406103447
[81]	Smirnov KM, Men'shikov YA, Krylova OK, et al. (2015) Form of Uranium Found in Phosphate Ores in Northern Kazakhstan. At Energy 118: 337-340. doi: 10.1007/s10512-015-0003-9
[82]	Cunha CSM, da Silva YJAB, Escobar MEO, et al. (2018) Spatial variability and geochemistry of rare earth elements in soils from the largest uranium-phosphate deposit of Brazil. Environ Geochem Health 40: 1629-1643. doi: 10.1007/s10653-018-0077-0
[83]	Lucas J, Prevot J, Larnboy M (1978) Les phosphorites de in marge norde de I' Espagne, Cl imie mineralogy, Genese. Oceanol Acta 1: 55-72.
[84]	Saigal N, Banerjee DM (1987) Proterozoic phosphorites of India: updated information. Part 1: Petrography, In: Kale VS, Phansalkar VG, Purana Basins of Peninsular India, India: Geological Society of India Memoir, 6: 471-486.
[85]	Dooley JH (2001) Baseline Concentrations of Arsenic, Beryllium and Associated Elements in Glauconite and Glauconitic Soils in the New Jersey Coastal Plain, The New Jersey Geological Survey, Investigation Report, Trenton NJ, 238.
[86]	Barringer JL, Reilly PA, Eberl DD, et al. (2011) Arsenic in sediments, groundwater, and stream water of a glauconitic Coastal Plain terrain, New Jersey, USA. Appl Geochem 26: 763-776. doi: 10.1016/j.apgeochem.2011.01.034
[87]	Liu YG, Mia MRU, Schmitt RA (1988) Cerium: A chemical tracer for paleo oceanic redox conditions. Geochim Cosmochim Acta 52: 2362-2371.
[88]	German CR, Elderfield H (1990) Application of the Ce anomaly as a paleoredox indicator: the ground rules. Paleoceanography 5: 823-833. doi: 10.1029/PA005i005p00823
[89]	Murray RW, Brink MRB, Ten Gerlach DC, et al. (1991) Rare earth, major, and trace elements in chert from the Franciscan Complex and Monterey Group, California: Assessing REE sources to fine-grained marine sediments. Geochim Cosmochim Acta 55: 1875-1895. doi: 10.1016/0016-7037(91)90030-9
[90]	Nath BN, Roelandts I, Sudhakar M, et al. (1992) Rare Earth Element patterns of the Central Indian Basin sediments related to their lithology. Geophys Res Lett 19: 1197-1200. doi: 10.1029/92GL01243
[91]	Madhavaraju J, Ramasamy S (1999) Rare earth elements in Limestones of Kallankurichchi Formation of Ariyalur Group, Tiruchirapalli Cretaceous, Tamil Nadu. J Geol Soc India 54: 291-301.
[92]	Armstrong-Altrin JS, Verma SP, Madhavaraju J, et al. (2003) Geochemistry of Late Miocene Kudankulam Limestones, South India. Int Geol Rev 45: 16-26. doi: 10.2747/0020-6814.45.1.16
[93]	Madhavaraju J, González-León CM, Lee YI, et al. (2010) Geochemistry of the Mural Formation (Aptian-Albian) of the Bisbee Group, Northern Sonora, Mexico. Cretac Res 31: 400-414. doi: 10.1016/j.cretres.2010.05.006
[94]	Wang YL, Liu YG, Schmitt RA (1986) Rare earth element geochemistry of South Atlantic deep sea sediments: Ce anomaly change at ~54 My. Geochim Cosmochim Acta 50: 1337-1355. doi: 10.1016/0016-7037(86)90310-8
[95]	Kato Y, Nakao K, Isozaki Y (2002) Geochemistry of Late Permian to Early Triassic pelagic cherts from southwest Japan: implications for an oceanic redox change. Chem Geol 182: 15-34. doi: 10.1016/S0009-2541(01)00273-X
[96]	Kamber BS, Webb GE (2001) The geochemistry of late Archaean microbial carbonate: implications for ocean chemistry and continental erosion history. Geochim Cosmochim Acta 65: 2509-2525. doi: 10.1016/S0016-7037(01)00613-5
[97]	Kemp RA, Trueman CN (2003) Rare earth elements in Solnhofen biogenic apatite: geochemical clues to the palaeoenvironment. Sediment Geol 155: 109-127. doi: 10.1016/S0037-0738(02)00163-X
[98]	Bakkiaraj D, Nagendra R, Nagarajan R, et al. (2010) Geochemistry of Siliciclastic rocks of Sillakkudi Formation, Cauvery Basin, Southern India; Implications for Provenance. J Geol Soc India 76: 453-467. doi: 10.1007/s12594-010-0128-3
[99]	McArthur JM, Walsh JN (1984) Rare-earth geochemistry of phosphorites. Chem Geol 47: 191-220. doi: 10.1016/0009-2541(84)90126-8
[100]	Kidder DL, Krishnaswamy R, Mapes RH (2003) Elemental mobility in phosphatic shales during concretion growth and implications for provenance analysis. Chem Geol 198: 335-353. doi: 10.1016/S0009-2541(03)00036-6
[101]	Cossa M (1878) Sur la diffusion de cerium, du lanthane et du didyme, extract of a letter from Cossa to Sella, presented by Freny. C R Ac Sci 87: 378-388.
[102]	McKelvey VE (1967) Rare Earths in Western Phosphate Rocks. US Geological Survey Professional paper, 1-17.
[103]	Kocsis L, Gheerbrant E, Mouflih M, et al. (2016) Gradual changes in upwelled seawater conditions (redox, pH) from the late Cretaceous through early Paleogene at the northwest coast of Africa: negative Ce anomaly trend recorded in fossil bio-apatite. Chem Geol 421: 44-54. doi: 10.1016/j.chemgeo.2015.12.001
[104]	Altschuler ZS, Berman S, Cuttiti F (1967) Rare earths in phosphorites-geochemistry and potential recovery, U. S. Geological Survey Professional paper, 575: 1-9.
[105]	Kon Y, Hoshino M, Sanematsu K, et al. (2014) Geochemical characteristics of apatite in heavy REE-rich deep-sea mud from Minami-Torishima area, southeastern Japan. Resour Geol 64: 47-57. doi: 10.1111/rge.12026
[106]	Taylor SR (1985) An examination of the geochemical record preserved in sedimentary rocks. In: Taylor SR, McLennan SM, The Continental Crust; Its composition and evolution, Blackwell, Oxford, 312.
[107]	Shields G, Stille P (2001) Diagenetic constraints on the use of cerium anomalies as palaeoseawater redox proxies: an isotopic and REE study of Cambrian phosphorites. Chem Geo 175: 29-48. doi: 10.1016/S0009-2541(00)00362-4
[108]	Elderfield H, Greaves MJ (1982) The rare earth elements in seawater. Nature 296: 214-219. doi: 10.1038/296214a0
[109]	De Baar HJW, Bacon MP, Brewer PG (1985) Rare earth elements in the Pacific and Atlantic Oceans. Geochim Cosmochim Acta 49: 1943-1959. doi: 10.1016/0016-7037(85)90089-4
[110]	Pattan JN, Pearce NJG, Mislankar PG (2005) Constraints in using Cerium-anomaly of bulk sediments as an indicator of paleo bottom water redox environment: A case study from the Central Indian Ocean Basin. Chem Geol 221: 260-278. doi: 10.1016/j.chemgeo.2005.06.009
[111]	Daesslé LW, Carriquiry JD (2008) Rare Earth and Metal Geochemistry of Land and SubmarinePhosphorites in the Baja California Peninsula, Mexico. Mar Georesour Geotechnol 26: 340-349. doi: 10.1080/10641190802382633
[112]	Stoneley R (1990) The Arabian continental margin in Iran during the late Cretaceous, In: Roberston AHF, Searl MP, Ries A, The geology and tectonics of the Oman region, London: Geological Society of London Special Publication, 49: 787-795.
[113]	Bolourchifard F, Fayazi F, Mehrabi B, et al. (2019) Evidence of high-energy storm and shallow water facies in Pabdeh sedimentary phosphate deposit, Kuhe-Lar-anticline, SW Iran. Carbonate Evaporite 34: 1703-1721. doi: 10.1007/s13146-019-00520-4
[114]	Daneshian J, Shariati S, Salsani A (2015) Biostratigraphy and planktonic foraminiferal abundance in the phosphate bearing pabdeh Formation of the lar mountains (SW Iran). Neues Jahrb Geol Paläontol 278: 175-189.
[115]	Rees A, Thomas A, Lewis M, et al. (2014) Lithostratigraphy and palaeoenvironments of the Cambrian in SW Wales. Geol Soc London 42: 33-100. doi: 10.1144/M42.2
[116]	Vaziri-Moghaddam H, Kimiagari M, Taheri A (2006) Depositional environment and sequence stratigraphy of the Oligo-MioceneAsmari Formationin SW Iran. Facies 52: 41-51. doi: 10.1007/s10347-005-0018-0
[117]	Murris RJ (1980) Middle East: stratigraphic evolution and oil habitat. Am Assoc Pet Geol Bull 64: 597-618.
[118]	Price NB, Calvert SE (1978) The geochemistry of phosphorites from the Namibian shelf. Chem Geol 23: 151-170. doi: 10.1016/0009-2541(78)90072-4
[119]	Froelich PN, Arthur MA, Burnett WC, et al. (1988) Early diagenesis in organic matter in Peru continental margin sediments: Phosphorite precipitation. Mar Geol 80: 309-343. doi: 10.1016/0025-3227(88)90095-3
[120]	Tribovillard N, Algeo TJ, Lyons T, et al. (2006) Trace metals as paleoredox and paleoproductivity proxies: an update. Chem Geol 232: 12-32. doi: 10.1016/j.chemgeo.2006.02.012
[121]	MacLeod KG, Irving AJ (1996) Correlation of cerium anomalies with indicators of paleoenvironment. J Sediment Res 66: 948-988.
[122]	Shields G, Stille P, Brasier MD, et al. (1997) Stratified oceans and oxygenation of the late Proterozoic environment: a post glacial geochemical record from the Neoproterozoic of W Mongolia. Terra Nov 9: 218-222. doi: 10.1111/j.1365-3121.1997.tb00016.x

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