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Endoplasmic reticulum, oxidative stress and their complex crosstalk in neurodegeneration: proteostasis, signaling pathways and molecular chaperones

  • Cellular stress caused by protein misfolding, aggregation and redox imbalance is typical of neurodegenerative disorders such as Parkinson’s disease (PD) and Amyotrophic Lateral Sclerosis (ALS). Activation of quality control systems, including endoplasmic reticulum (ER)-mediated degradation, and reactive oxygen species (ROS) production are initially aimed at restoring homeostasis and preserving cell viability. However, persistent damage to macromolecules causes chronic cellular stress which triggers more extreme responses such as the unfolded protein response (UPR) and non-reversible oxidation of cellular components, eventually leading to inflammation and apoptosis. Cell fate depends on the intensity and duration of stress responses converging on the activation of transcription factors involved in the expression of antioxidant, autophagic and lysosome-related genes, such as erythroid-derived 2-related factor 2 (Nrf2) and transcription factor EB respectively. In addition, downstream signaling pathways controlling metabolism, cell survival and inflammatory processes, like mitogen activated protein kinase and nuclear factor-kB, have a key impact on the overall outcome.
    Molecular chaperones and ER stress modulators play a critical role in protein folding, in the attenuation of UPR and preservation of mitochondrial and lysosomal activity. Therefore, the use of chaperone molecules is an attractive field of investigation for the development of novel therapeutic strategies and disease-modifying drugs in the context of neurodegenerative diseases such as PD and ALS.

    Citation: Giulia Ambrosi, Pamela Milani. Endoplasmic reticulum, oxidative stress and their complex crosstalk in neurodegeneration: proteostasis, signaling pathways and molecular chaperones[J]. AIMS Molecular Science, 2017, 4(4): 424-444. doi: 10.3934/molsci.2017.4.424

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  • Cellular stress caused by protein misfolding, aggregation and redox imbalance is typical of neurodegenerative disorders such as Parkinson’s disease (PD) and Amyotrophic Lateral Sclerosis (ALS). Activation of quality control systems, including endoplasmic reticulum (ER)-mediated degradation, and reactive oxygen species (ROS) production are initially aimed at restoring homeostasis and preserving cell viability. However, persistent damage to macromolecules causes chronic cellular stress which triggers more extreme responses such as the unfolded protein response (UPR) and non-reversible oxidation of cellular components, eventually leading to inflammation and apoptosis. Cell fate depends on the intensity and duration of stress responses converging on the activation of transcription factors involved in the expression of antioxidant, autophagic and lysosome-related genes, such as erythroid-derived 2-related factor 2 (Nrf2) and transcription factor EB respectively. In addition, downstream signaling pathways controlling metabolism, cell survival and inflammatory processes, like mitogen activated protein kinase and nuclear factor-kB, have a key impact on the overall outcome.
    Molecular chaperones and ER stress modulators play a critical role in protein folding, in the attenuation of UPR and preservation of mitochondrial and lysosomal activity. Therefore, the use of chaperone molecules is an attractive field of investigation for the development of novel therapeutic strategies and disease-modifying drugs in the context of neurodegenerative diseases such as PD and ALS.


    Systems of nonlinear equations are fundamental to a diverse range of applications, including power flow analysis [1], economic equilibrium modeling [2], the development of generalized Bregman distance proximal point methods [3], and traffic assignment [4]. Meanwhile, these systems also involve monotone variational inequalities [5,6] and compression sensing problems [7,8]. Given the ubiquity and significance of such problems across these varied domains, the study and development of numerical methods to efficiently solve systems of nonlinear equations are of considerable practical importance. In this paper, we focus on a specific class of systems of nonlinear equations subject to convex constraints, which can be formulated as follows:

    θ(a)=0,aΘ, (1.1)

    where ΘRn is a non-empty, and closed convex set. The function θ:RnRn is assumed to possess monotonicity and continuous differentiability, satisfying the following condition:

    θ(a)θ(b),ab0,a,bRn.

    Generally, the gradient-type method generates a sequence {ak}, defined as follows:

    ak+1=ak+tkdk,k0,

    where tk is the step length, and dk denotes the search direction. The choice of the search direction dk gives rise to various gradient-type methods, such as the steepest descent methods, Newton's methods, and quasi-Newton methods [9,10]. Newton's and quasi-Newton methods, along with their numerous variants, have been extensively studied due to their strong local linear convergence properties. For instance, Mahdavi et al. [11] proposed and analyzed a nonmonotone quasi-Newton algorithm for strongly convex multiobjective optimization, demonstrating its global convergence and local superlinear convergence rate under certain conditions. Sihwail et al. [12] proposed a novel hybrid method, Newton-Harris hawks optimization, which combines Newton's methods and Harris hawks optimization to effectively solve systems of nonlinear equations. Moreover, Krutikov et al. [13] demonstrated that quasi-Newton methods, when applied to strongly convex functions with a Lipschitz gradient, achieve geometric convergence without relying on local quadratic approximations. However, despite these advantages, Newton's and quasi-Newton methods involve the computation of the Hessian matrix or its approximation value at each iteration, which significantly increases computational complexity. This requirement can be a limiting factor, particularly for large-scale problems where the Hessian matrix is difficult to compute and store efficiently.

    The conjugate gradient method [14,15] is one of the most effective approaches in the field of the gradient-type methods. It is highly recognized for its efficiency, simplicity, lower storage requirements, and reliable convergence properties. These characteristics make it particularly well-suited for solving large-scale systems of nonlinear equations [16,17]. The method's search direction is typically defined as follows:

    dk={θk,k=0,θk+βkdk1,k1,

    where θkθ(ak) and βk is known as the conjugate parameter. The choice of βk differentiates various conjugate gradient methods. Several advancements have been made in the development of conjugate gradient methods with different conjugate parameters [18]. For instance, Ma et al. [19] proposed a modified inertial three-term conjugate gradient method for solving nonlinear monotone equations with convex constraints. This method is notable for its global and Q-linear convergence properties, and has demonstrated superior numerical performance in applications such as sparse signal recovery and image restoration in compressed sensing. Furthermore, Liu et al. [20] introduced a spectral conjugate gradient method with an inertial factor for solving nonlinear pseudo-monotone equations over a convex set. Additionally, Sabiu et al. [21] developed an optimal scaled Perry conjugate gradient method for solving large-scale systems of monotone nonlinear equations. This method ensures global convergence under the conditions of monotonicity and Lipschitz continuity.

    Inspired by the classical Liu-Story (LS) [22] and Rivai-Mohamad-Ismail-Leong (RMIL) [23] conjugate parameters, as well as incorporating the hybrid technique (e.g, [24,25]) and the projection approach, we develop a modified LS-RMIL-type conjugate gradient projection algorithm. The proposed algorithm is specifically designed for solving systems of nonlinear equations with convex constraints. In this paper, and \langle \cdot, \cdot \rangle represent the Euclidean norm and the inner product of vectors, respectively.

    In this section, we refine and enhance the search direction employed in the optimization processes of the LS and RMIL methods. Specifically, Liu et al. [22] and Rivai et al. [23] introduced conjugate parameters, defined respectively as:

    \begin{equation*} \beta_k^{LS} = \frac{\langle \theta_k, y_{k-1} \rangle}{- \langle \theta_{k-1}, d_{k-1} \rangle}, \quad \beta_k^{RMIL} = \frac{\langle \theta_k, y_{k-1} \rangle}{\|d_{k-1}\|^2}. \end{equation*}

    Based on the insights derived from these parameters, we adopt the hybrid technique (e.g., [24,25]) that combines their key features. This leads to the formulation of a new conjugate parameter, which is subsequently incorporated into the framework of a three-term search direction. Our primary objective is to construct a novel search direction that ensures both the sufficient descent and trust-region properties, which are critical for the robustness and efficiency of the optimization process. To accomplish this, we carefully design a novel search direction tailored to meet these requirements. The designed search direction d_k is defined as follows:

    \begin{equation} d_k = \left\{ \begin{array}{ll} -\theta_k, & k = 0, \\ -\theta_k + \beta_k d_{k-1} + \varpi_k y_{k-1}, & k \geq 1, \end{array} \right. \end{equation} (2.1)

    where the conjugate parameter \beta_k and the scalar parameter \varpi_k are given by

    \begin{equation} \beta_k = \frac{\langle \theta_k, y_{k-1} \rangle}{c_k} - \frac{\|y_{k-1}\|^2 \langle \theta_{k}, d_{k-1} \rangle}{c_k^2} \quad \text{and} \quad \varpi_k = \frac{\nu_k \langle \theta_k, d_{k-1} \rangle}{c_k}, \end{equation} (2.2)

    with y_{k-1} = \theta_k - \theta_{k-1} . One scalar parameter c_k is crucial for maintaining stability in the iterative process, and is defined by

    c_k = \max \left\{ \mu \|d_{k-1}\| \|y_{k-1}\|, - \langle \theta_{k-1}, d_{k-1} \rangle, \| d_{k-1} \|^2 \right\},

    where \mu is a positive constant. Another scalar parameter \nu_k is introduced to fine-tune the adjustment of the search direction. It is defined as \nu_k = \min\{\tilde{\nu}, \max\{\bar{\nu}_k, 0\}\} with 0 < \tilde{\nu} < 1 , and \bar{\nu}_k = \frac{\langle \theta_k, y_{k-1} - s_{k-1} \rangle}{\|\theta_k\|^2}, where s_{k-1} = a_k - a_{k-1} represents the difference between the iterative points a_k and a_{k-1} of the optimization variable.

    Before delving into the sufficient descent and trust-region properties of the designed search direction (2.1), we can deduce some important bounds from the definition of \beta_k and \varpi_k . We consider the bound for \beta_k :

    |\beta_k| \leq \frac{\|\theta_k\| \|y_{k-1}\|}{c_k} + \frac{\|y_{k-1}\|^2 \|\theta_k\| \|d_{k-1}\|}{c_k^2},

    which can be further bounded by

    |\beta_k| \leq \frac{\|\theta_k\| \|y_{k-1}\|}{\mu \|d_{k-1}\| \|y_{k-1}\|} + \frac{\|y_{k-1}\|^2 \|\theta_k\| \|d_{k-1}\|}{\mu^2 \|d_{k-1}\|^2 \|y_{k-1}\|^2}.

    Simplifying this, we obtain

    \begin{equation} |\beta_k| \leq \left( \frac{1}{\mu}+\frac{1}{\mu^2} \right) \frac{\|\theta_k\|}{\|d_{k-1}\|}. \end{equation} (2.3)

    Next, we consider the bound for \varpi_k :

    \begin{equation} |\varpi_k| \leq \frac{\nu_k \|\theta_k\| \|d_{k-1}\|}{c_k} \leq \frac{\tilde{\nu}_k \|\theta_k\| \|d_{k-1}\|}{\mu \|d_{k-1}\| \|y_{k-1}\|} = \frac{\tilde{\nu}}{\mu} \frac{\|\theta_k\|}{\|y_{k-1}\|}. \end{equation} (2.4)

    Lemma 1. The search direction d_k generated by (2.1) satisfies the sufficient descent property:

    \langle \theta_k, d_k \rangle \leq -M \|\theta_k\|^2,

    where M = 1 - \frac{1}{4} (1 + \tilde{\nu})^2 .

    Proof. For k = 0 , the conclusion is straightforward, which implies \langle \theta_0, d_0 \rangle = - \|\theta_0\|^2 \leq -M \|\theta_0\|^2 . For k \geq 1 , together with the search direction generated by (2.1), we have:

    \begin{equation} \begin{array}{lll} \langle \theta_k, d_k \rangle & = & \langle \theta_k, -\theta_k + \beta_k d_{k-1} + \varpi_k y_{k-1} \rangle \\ & = & -\|\theta_k\|^2 + \frac{\langle \theta_k, y_{k-1} \rangle \langle \theta_{k}, d_{k-1} \rangle}{c_k} - \frac{\|y_{k-1}\|^2 \langle \theta_{k}, d_{k-1} \rangle^2}{c_k^2} + \frac{\nu_k \langle \theta_k, y_{k-1} \rangle \langle \theta_{k}, d_{k-1} \rangle}{c_k} \\ & = & -\|\theta_k\|^2 + (1+\nu_k) \frac{\langle \theta_k, y_{k-1} \rangle \langle \theta_{k}, d_{k-1} \rangle}{c_k} - \frac{\|y_{k-1}\|^2 \langle \theta_{k}, d_{k-1} \rangle^2}{c_k^2} . \end{array} \end{equation} (2.5)

    By applying the inequality \langle e_k, g_k \rangle \leq \frac{1}{2} (\|e_k\|^2+\|g_k\|^2) , where e_k = (1 + \nu_k) \theta_k / \sqrt{2} and g_k = \sqrt{2} \langle \theta_k, d_{k-1} \rangle y_{k-1} / c_k , we obtain the following result:

    \begin{equation} \frac{(1+\nu_k)\langle \theta_k, y_{k-1} \rangle \langle \theta_{k}, d_{k-1} \rangle}{c_k} \leq \frac{1}{4} (1+\nu_k)^2 \|\theta_k\|^2 + \frac{\langle \theta_k, d_{k-1} \rangle^2 \|y_{k-1}\|^2 }{c_k^2}. \end{equation} (2.6)

    Substituting (2.6) into (2.5), we obtain

    \langle \theta_k, d_k \rangle \leq -\|\theta_k\|^2 + \frac{1}{4} (1 + \tilde{\nu}_k)^2 \|\theta_k\|^2 \leq -\left(1-\frac{1}{4}(1+\tilde{\nu})^2\right) \|\theta_k\|^2.

    Thus, the result holds.

    Lemma 2. The search direction d_k generated by (2.1) satisfies the trust-region property:

    M \|\theta_k\| \leq \|d_k\| \leq N \|\theta_k\|,

    where N = 1 + \frac{1}{\mu} + \frac{1}{\mu^2} + \frac{\tilde{\nu}}{\mu} .

    Proof. From Lemma 1, we have -\|\theta_k\| \|d_k\| \leq \langle \theta_k, d_k \rangle \leq -M \|\theta_k\|^2 , which implies:

    \|d_k\| \geq M \|\theta_k\|.

    Additionally, together with (2.1), we obtain:

    \|d_k\| = \|- \theta_k + \beta_k d_{k-1} + \varpi_k y_{k-1}\| \leq \|\theta_k\| + |\beta_k| \|d_{k-1}\| + |\varpi_k| \|y_{k-1}\|.

    Substituting the inequalities |\beta_k| and |\varpi_k| defined in (2.3) and (2.4), respectively, into the above equality, we have

    \|d_k\| \leq \left( 1 + \frac{1}{\mu} + \frac{1}{\mu^2} + \frac{\tilde{\nu}}{\mu} \right) \|\theta_k\|.

    Thus, the result holds.

    Before delving into the specifics of our proposed algorithm, it is essential to first clarify the line search approach, the projection operator, and the iterative update rule employed in the proposed algorithm. These foundational components play a crucial role in the overall efficacy of the proposed algorithm.

    First, in the proposed algorithm, the line search approach is used to determine an appropriate step length t_k = \eta \rho^{i_k} . Specifically, this step length is computed based on the following procedure, where i_k = \{i: i = 0, 1, \ldots\} is the smallest non-negative integer i that satisfies the following inequality:

    \begin{equation} -\langle \theta(a_k + \eta \rho^i d_k), d_k \rangle \geq \sigma \eta \rho^i \| \theta(a_k + \eta \rho^i d_k) \| \|d_k\|^2, \end{equation} (2.7)

    where \eta > 0 , \rho \in (0, 1) , and \sigma > 0 are algorithmic parameters.

    Furthermore, the projection operator P_{\Theta}[\cdot] is a critical component that ensures the iterative points remain within the feasible region \Theta . Specifically, the projection of a point a \in \mathbb{R}^n onto the set \Theta is defined as

    P_{\Theta}[a] = \arg\min \{\|a - b\| : b \in \Theta\}, \quad a \in \mathbb{R}^n.

    This operator identifies the point in \Theta closest to a in the Euclidean norm. Moreover, the projection operator is non-expansive, meaning it satisfies the property:

    \begin{equation} \| P_{\Theta}[a]-P_{\Theta}[b] \| \leq \| a-b \|. \end{equation} (2.8)

    Finally, the iterative update rule forms the core of the proposed algorithm, indicating how the next iterative point a_{k+1} is computed from the current iterative point a_k . Specifically, the update is performed by using the following formula:

    \begin{equation} a_{k+1} = P_{\Theta}\left[ a_k - \gamma w_k \theta(z_k) \right], \quad w_k = \frac{\langle \theta(z_k), a_k-z_k\rangle}{||\theta(z_k)||^2}, \end{equation} (2.9)

    where z_k = a_k + t_k d_k and \gamma \in (0, 2) . This projection-based update ensures that the new iterative point remains feasible and moves towards reducing the objective function.

    With the foundational components described above, we now present the detailed steps of an improved LS-RMIL-type conjugate gradient projection algorithm (Abbr. ILR algorithm), which is described as Algorithm 1.

    Algorithm 1 An improved LS-RMIL-type conjugate gradient projection algorithm
      1: Initialization: a_0\in \mathbb{R}^n , \mu > 0 , \tilde{\nu}\in(0, 1) , \eta, \sigma > 0 , \rho\in(0, 1) , \tau > 0 , and set k: = 0 .
      2: while \|\theta_k\| > \tau do
      3:        Evaluate two parameters \beta_k and \varpi_k from (2.2) and search direction d_k from (2.1).
      4:        Evaluate the step length t_k from (2.7) and set the trial point z_k = a_k+t_kd_k .
      5:        if z_k\in\Theta and \|\theta(z_k)\| < \tau then
      6:                break.
      7:        else
      8:                Evaluate the next iterative point a_{k+1} from (2.9).
      9:        end if
    10:        Set k: = k+1 .
    11: end while

    In this section, we provide a comprehensive analysis of the global convergence properties of the ILR algorithm. To facilitate this analysis, we introduce the following key assumptions.

    Assumption B:

    (B1) The solution set \Theta_* of problem (1.1) is non-empty.

    (B2) The function \theta(a) exhibits a monotonicity property, i.e.,

    \langle \theta(a) - \theta(b), a - b \rangle \geq 0, \quad \forall a, b \in \mathbb{R}^n.

    These assumptions are fundamental in establishing the convergence behavior of the ILR algorithm as they ensure that the iterative process converges to a solution within the feasible region of problem (1.1).

    The following lemma demonstrates that the line search approach defined in (2.7) of the ILR algorithm is indeed well-defined and can be successfully applied in the iterative process.

    Lemma 3. Consider the sequence \{t_k\} generated by the ILR algorithm. Then, there exists a step length t_k at each iteration that satisfies the line search approach defined in (2.7).

    Proof. We proceed by contradiction and assume that inequality (2.7) does not hold. Specifically, suppose there exists a positive index k_0 such that, for all i \in \{0\} \cup \mathbb{N} , the following inequality is satisfied:

    \begin{equation*} - \langle \theta(a_{k_0} + \eta \rho^i d_{k_0}), d_{k_0} \rangle < \sigma \eta \rho^i \| \theta(a_{k_0} + \eta \rho^i d_{k_0}) \| \|d_{k_0}\|^2. \end{equation*}

    By utilizing the continuity of \theta and taking the limit as i \to \infty , the above inequality yields:

    \begin{equation} - \langle \theta(a_{k_0}), d_{k_0} \rangle \leq 0. \end{equation} (3.1)

    On the other hand, invoking Lemma 1 and again taking the limit as i \to \infty , we obtain:

    \begin{equation*} -\langle \theta(a_{k_0}), d_{k_0} \rangle \geq M \| \theta(a_{k_0}) \|^2 > 0, \end{equation*}

    which clearly contradicts inequality (3.1). This contradiction implies that the initial assumption must be false, and therefore inequality (2.7) must hold.

    The following lemma establishes that the sequence \{a_k\} generated by the ILR algorithm exhibits monotonic behavior with respect to the solutions set \Theta_* of problem (1.1).

    Lemma 4. Consider the sequences \{a_k\} and \{z_k\} generated by the ILR algorithm. Then, the following properties hold:

    (i) The sequence \{a_k\} is bounded, meaning that there exists a constant D > 0 such that \|a_k\|\leq D for all k \geq 0 .

    (ii) The sequence \{z_k\} converges to the sequence \{a_k\} , i.e., \lim\limits_{k \to \infty} \|z_k-a_k\| = 0 .

    Proof. From the definition of the projection operator P_{\Theta}[\cdot] and the non-expensive property defined in (2.8), we can derive the following inequality:

    \begin{equation} \begin{array}{lll} \|a_{k+1} - a_\ast \|^2 & = & \| P_\Theta\left[ a_k - \gamma w_k \theta(z_k) \right] - P_\Theta[a_\ast]\|^2 \\ & \leq & \|a_k - \gamma w_k \theta(z_k) - a_\ast \|^2 \\ & = & \|a_k - a_\ast \|^2 - 2 \gamma w_k \langle \theta(z_k), a_k - a_\ast \rangle + \gamma^2 w_k^2 \|\theta(z_k)\|^2, \end{array} \end{equation} (3.2)

    where a_* denotes a solution of problem (1.1). Next, starting from Assumption B2, the definition of z_k , and the line search approach (2.7), we can further establish the following inequality:

    \begin{equation} \begin{array}{lll} \langle \theta(z_k), a_k - a_\ast \rangle & = & \langle \theta(z_k), a_k - z_k \rangle + \langle \theta(z_k), z_k - a_\ast \rangle - \langle \theta(a_*), z_k-a_* \rangle \\ & \geq & \langle \theta(z_k), a_k - z_k \rangle \\ & \geq & \sigma t_k^2 \|\theta(z_k)\| \|d_k\|^2. \end{array} \end{equation} (3.3)

    Combining with (3.2), (3.3), and the definition of w_k , we can derive

    \begin{equation} \begin{array}{lll} \|a_{k+1} - a_\ast \|^2 & \leq & \|a_k - a_\ast \|^2 - 2 \gamma w_k \langle \theta(z_k), a_k - z_k \rangle + \gamma^2 w_k^2 \|\theta(z_k)\|^2, \\ & = & \|a_k - a_\ast \|^2 - 2 \gamma w_k^2 \|\theta(z_k)\|^2 + \gamma^2 w_k^2 \|\theta(z_k)\|^2, \\ & = & \|a_k - a_\ast \|^2 - (2\gamma-\gamma^2) w_k^2 \|\theta(z_k)\|^2, \end{array} \end{equation} (3.4)

    Given the definition of w_k and (3.3), we have

    \|\theta(z_k)\|^2 w_k = \langle \theta(z_k), a_k - z_k \rangle \geq \sigma t_k^2 \|\theta(z_k)\| \|d_k\|^2,

    which implies that \|\theta(z_k)\| w_k \geq \sigma t_k^2 \|d_k\|^2 . Substituting this into (3.4), we obtain

    \begin{equation} \begin{array}{lll} \|a_{k+1} - a_\ast \|^2 & \leq & \|a_k - a_\ast \|^2 - (2\gamma-\gamma^2) (\sigma t_k^2 \|d_k\|^2)^2, \\ & = & \|a_k - a_\ast \|^2 - (2\gamma-\gamma^2) \sigma^2 t_k^4 \|d_k\|^4, \\ & = & \|a_k - a_\ast \|^2 - (2\gamma-\gamma^2) \sigma^2 \|a_k-z_k\|^4. \end{array} \end{equation} (3.5)

    This result indicates that the sequence \{\|a_k-a_*\|\} is monotonically decreasing, meaning that it consistently reduces as k increases. Hence, the sequence \{a_k\} is bounded.

    By reorganizing the formula defined in (3.5), we obtain

    \begin{equation} \begin{array}{lll} (2\gamma-\gamma^2) \sigma^2 \sum\limits_{k = 0}^{\infty} \|a_k-z_k\|^4 & \leq & \sum\limits_{k = 0}^{\infty} \left(\|a_k - a_\ast \|^2 - \|a_{k+1} - a_\ast \|^2\right) \\ & \leq & \|a_0 - a_* \|. \end{array} \end{equation} (3.6)

    This implies that \lim\limits_{k \to \infty} \|z_k-a_k\| = 0.

    Theorem 1. Consider the sequence \{\theta_k\} generated by the ILR algorithm. Then, the following conclusion is satisfied:

    \begin{equation} \lim\limits_{k \to \infty} \| \theta_k \| = 0. \end{equation} (3.7)

    Proof. To demonstrate the desired result, we begin by assuming the contrary. Suppose that there exists a constant A_1 > 0 such that \|\theta_k\| > A_1 for all k\geq0 . This assumption, combined with Lemma 2, gives us the following relation \|d_k\| \geq M \|\theta_k\| > M \ A_1 for all k\geq0 . Given the continuity of the function \theta(a) and the boundedness of the sequence \{a_k\} , it follows that the sequence \{\theta_k\} is also bounded. In other words, there exists a non-negative constant A_2 such that \|\theta_k\| \leq A_2 for all k \geq 0. By incorporating this bound with Lemma 2, we obtain \|d_k\| \leq N\|\theta_k\| \leq N A_2 for all k\geq0 . The two inequalities derived above imply that the sequence \{d_k\} is bounded. Together with Lemma 4(ii) and the definition of z_k , we have \lim_{k \to \infty} \|z_k-a_k\| = \lim_{k \to \infty} \|a_k + t_k d_k-a_k\| = \lim_{k \to \infty} t_k \| d_k\| = 0 , which leads to the conclusion that \lim_{k \to \infty} t_k = 0 with the boundedness of the sequence \{d_k\} .

    Since the sequences \{a_k\} and \{d_k\} are both bounded, we can extract two convergent subsequences, \{a_{k_i}\} and \{d_{k_i}\} , such that \lim_{i \to \infty, i \in \mathcal{K}} a_{k_i} = \bar{a} and \lim_{i \to \infty, i \in \mathcal{K}} d_{k_i} = \bar{d} , where \mathcal{K} denotes an infinite index set. Utilizing Lemma 1, we have -\langle \theta_{k_i}, d_{k_i} \rangle \geq M \|\theta_{k_i}\|^2. Taking the limit as i \to \infty in the above inequality and invoking the continuity of \theta(a) , we obtain

    -\langle \theta(\bar{a}), \bar{d} \rangle \geq M\|\theta(\bar{a})\|^2 \geq M A_1^2 > 0.

    Furthermore, we adopt the line search approach defined in (2.7), which implies the following inequality holds: -\langle \theta(a_{k_i} + (\eta\rho)^{-1}t_{k_i}d_{k_i}), d_{k_i} \rangle < \sigma \eta (\eta\rho)^{-1} t_{k_i}\|\theta(a_{k_i} + (\eta\rho)^{-1} t_{k_i}d_{k_i})\|\|d_{k_i}\|^2 . Taking the limit as i \to \infty in the above inequality, and using the continuity of \theta(a) , we conclude

    -\langle \theta(\bar{a}), \bar{d} \rangle \leq 0.

    These two results directly contradict each other. Therefore, the assumption that \|\theta_k\| > A_1 for all k \geq 0 must be false, and the desired result follows.

    In this section, we evaluate the effectiveness of the proposed ILR algorithm through a comprehensive set of numerical experiments. These experiments are designed to solve large-scale systems of nonlinear equations with convex constraints, thereby assessing the algorithm's computational efficiency. For benchmarking purposes, we compare the ILR algorithm with two established methods (e.g., VRMILP and DFPRPMHS) across various test problems, initial points, and dimensional settings.

    In this section, we utilize the ILR algorithm to address large-scale systems of nonlinear equations with convex constraints. We then compare it with two existing algorithms: the VRMILP algorithm [26] and the DFPRPMHS algorithm [27]. All experimental codes are executed on a 64-bit Ubuntu 20.04.2 LTS operating system with an Intel(R) Xeon(R) Gold 5115 2.40GHz CPU. The parameters for the ILR algorithm are set as follows:

    \mu = 0.02, \quad \tilde{\nu} = 0.105, \quad \eta = 1, \quad \sigma = 10^{-4}, \quad \rho = 0.74, \quad \tau = 10^{-5}.

    For the VRMILP and DFPRPMHS algorithms, we adhere to the parameter settings provided in their respective original works. Seven test problems are selected for evaluation, with problem dimensions set at {5,000 10,000 50,000 100,000 150,000}. Each test problem is initialized by the following points: a_1 = \left(\frac{1}{2}, \frac{1}{2^2}, \ldots, \frac{1}{2^n}\right) , a_2 = \left(0, \frac{1}{n}, \frac{2}{n}, \ldots, \frac{n-1}{n}\right) , a_3 = (1, \frac{1}{2}, \ldots, \frac{1}{n}) , a_4 = \left(\frac{1}{n}, \frac{2}{n}, \ldots, \frac{n}{n}\right) , a_5 = \left(\frac{1}{3}, \frac{1}{3^2}, \ldots, \frac{1}{3^n}\right) , a_6 = (2, 2, \ldots, 2) a_7 = \left(1-\frac{1}{n}, 1-\frac{2}{n}, \ldots, 1-\frac{n}{n}\right) , a_8 \in [0, 1]^n . The stopping criteria for all algorithms is set to either \theta_k \leq \tau or a maximum of 3000 iterations. Here, \theta(a) = (\theta_1(a), \theta_2(a), \ldots, \theta_n(a))^\text{T} with a = (a_1, a_2, \ldots, a_n)^\text{T} . The seven test problems are described as follows:

    Problem 1 [7]:

    \theta_i(a) = e^{a_i}-1,\; \; \; \text{for}\; \; i = 1,2,\ldots,n,

    with the constraint set \Theta = \mathbb{R}^n_+ . The unique solution is a_* = (0, 0, \ldots, 0)^\text{T} .

    Problem 2 [7]:

    \theta_i(a) = \frac{i}{n}e^{a_i}-1,\; \; \; \text{for}\; \; i = 1,2,\ldots,n,

    with the constraint set \Theta = \mathbb{R}^n_+ .

    Problem 3 [5]:

    \theta_i(a) = \log(a_i+1)-\frac{a_i}{n},\; \; \; \text{for}\; i = 1,2,\ldots,n,

    with the constraint set \Theta = [-1, +\infty) .

    Problem 4 [5]:

    \theta_i(a) = (e^{a_i})^2+3\sin(a_i)\cos(a_i)-1,\; \; \; \text{for}\; i = 1,2,\ldots,n,

    with the constraint set \Theta = \mathbb{R}^n_+ .

    Problem 5 [5]:

    \begin{eqnarray*} \theta_1(a) & = & 2a_1+\sin(a_1)-1, \\ \theta_i(a) & = & 2a_{i-1}+2a_i+\sin(a_i)-1,\; \; \; \text{for}\; \; i = 2,3,\ldots,n-1, \\ \theta_n(a) & = & 2a_n+\sin(a_n)-1, \end{eqnarray*}

    with the constraint set \Theta = \mathbb{R}^n_+ .

    Problem 6 [7]:

    \theta_i(a) = \frac{1}{n}e^{a_i}-1,\; \; \; \text{for}\; \; i = 1,2,\ldots,n,

    with the constraint set \Theta = \mathbb{R}^n_+ .

    Problem 7 [5]:

    \theta_i(a) = a_i-2\sin(|a_i-1|),\; \; \; \text{for}\; \; i = 1,2,\ldots,n,

    with the constraint set \Theta = \mathbb{R}^n_+ .

    The performance of the ILR, VRMILP, and DFPRPMHS algorithms are systematically evaluated through a series of test problems, with the numerical results presented in Tables 17. In these tables, "Init" refers to the initial point used in each test problem, " n " refers to the problem dimension multiplied by 1000, "CPUT" refers to the CPU time in seconds, "Nfunc" refers to the number of function evaluations, and "Niter" refers to the number of iterations. A notable observation from the numerical results is that all three algorithms successfully solve the test problem across various initial points and problem dimensions. To be specific, the ILR algorithm demonstrates superior performance in most cases compared to the other two algorithms.

    Table 1.  Numerical results for Problem 1.
    Inti( n ) ILR VRMILP DFPRPMHS
    CPUT/Nfunc/Niter CPUT/Nfunc/Niter CPUT/Nfunc/Niter
    a_1 (5) 1.93\times10^{-3} / \textbf{4} / \textbf{1} \mathbf{4.45\times10^{-4}} /4/1 1.57\times10^{-2} /332/7
    a_2 (5) \mathbf{1.84\times10^{-3}} / \textbf{23} / \textbf{7} 3.47\times10^{-3} /60/15 2.15\times10^{-2} /487/24
    a_3 (5) \mathbf{2.11\times10^{-3}} / \textbf{29} /9 2.16\times10^{-3} /37/ \textbf{8} 1.75\times10^{-2} /447/10
    a_4 (5) \mathbf{1.55\times10^{-3}} / \textbf{23} / \textbf{7} 2.13\times10^{-3} /39/9 1.86\times10^{-2} /405/31
    a_5 (5) \mathbf{1.57\times10^{-3}} / \textbf{23} / \textbf{7} 3.33\times10^{-3} /60/15 2.04\times10^{-2} /487/24
    a_6 (5) 5.08\times10^{-4} / \textbf{7} / \textbf{1} \mathbf{4.16\times10^{-4}} /7/1 2.00\times10^{-3} /9/1
    a_7 (5) \mathbf{1.92\times10^{-3}} / \textbf{23} / \textbf{7} 3.46\times10^{-3} /60/15 2.07\times10^{-2} /487/24
    a_8 (5) 2.21\times10^{-3} / \textbf{30} / \textbf{9} \mathbf{2.16\times10^{-3}} /39/9 2.30\times10^{-2} /563/14
    a_1 (10) 6.36\times10^{-4} / \textbf{4} / \textbf{1} \mathbf{4.22\times10^{-4}} /4/1 2.50\times10^{-2} /332/7
    a_2 (10) \mathbf{3.69\times10^{-3}} / \textbf{23} / \textbf{7} 6.19\times10^{-3} /57/14 4.11\times10^{-2} /487/24
    a_3 (10) 4.66\times10^{-3} / \textbf{29} /9 \mathbf{3.96\times10^{-3}} /37/ \textbf{8} 3.44\times10^{-2} /447/10
    a_4 (10) \mathbf{3.48\times10^{-3}} / \textbf{23} / \textbf{7} 4.12\times10^{-3} /39/9 3.59\times10^{-2} /408/32
    a_5 (10) \mathbf{3.45\times10^{-3}} / \textbf{23} / \textbf{7} 6.12\times10^{-3} /57/14 3.98\times10^{-2} /487/24
    a_6 (10) 8.35\times10^{-4} / \textbf{7} / \textbf{1} \mathbf{7.34\times10^{-4}} /7/1 1.99\times10^{-3} /9/1
    a_7 (10) \mathbf{3.56\times10^{-3}} / \textbf{23} / \textbf{7} 6.10\times10^{-3} /57/14 3.88\times10^{-2} /487/24
    a_8 (10) 6.37\times10^{-3} / \textbf{41} /13 \mathbf{5.28\times10^{-3}} /48/ \textbf{12} 4.40\times10^{-2} /563/14
    a_1 (50) 3.75\times10^{-3} / \textbf{4} / \textbf{1} \mathbf{1.64\times10^{-3}} /4/1 9.06\times10^{-2} /332/7
    a_2 (50) \mathbf{1.35\times10^{-2}} / \textbf{23} / \textbf{7} 2.48\times10^{-2} /60/15 1.51\times10^{-1} /596/26
    a_3 (50) \mathbf{1.27\times10^{-2}} / \textbf{29} /9 1.72\times10^{-2} /37/ \textbf{8} 1.02\times10^{-1} /447/10
    a_4 (50) \mathbf{9.17\times10^{-3}} / \textbf{23} / \textbf{7} 1.11\times10^{-2} /39/9 1.39\times10^{-1} /602/28
    a_5 (50) \mathbf{1.39\times10^{-2}} / \textbf{23} / \textbf{7} 2.47\times10^{-2} /60/15 1.49\times10^{-1} /596/26
    a_6 (50) \mathbf{2.67\times10^{-3}} / \textbf{7} / \textbf{1} 4.30\times10^{-3} /7/1 5.98\times10^{-3} /9/1
    a_7 (50) \mathbf{1.13\times10^{-2}} / \textbf{23} / \textbf{7} 1.95\times10^{-2} /60/15 1.32\times10^{-1} /596/26
    a_8 (50) \mathbf{1.06\times10^{-2}} / \textbf{23} / \textbf{7} 1.86\times10^{-2} /56/14 1.30\times10^{-1} /576/16
    a_1 (100) 7.76\times10^{-3} / \textbf{4} / \textbf{1} \mathbf{2.44\times10^{-3}} /4/1 1.21\times10^{-1} /332/7
    a_2 (100) \mathbf{1.49\times10^{-2}} / \textbf{23} / \textbf{7} 3.51\times10^{-2} /66/17 1.44\times10^{-1} /487/24
    a_3 (100) 1.69\times10^{-2} / \textbf{29} /9 \mathbf{1.64\times10^{-2}} /37/ \textbf{8} 1.24\times10^{-1} /447/10
    a_4 (100) \mathbf{1.49\times10^{-2}} / \textbf{23} / \textbf{7} 1.56\times10^{-2} /39/9 1.84\times10^{-1} /602/28
    a_5 (100) \mathbf{1.24\times10^{-2}} / \textbf{23} / \textbf{7} 2.76\times10^{-2} /66/17 1.38\times10^{-1} /487/24
    a_6 (100) 2.91\times10^{-3} / \textbf{7} / \textbf{1} \mathbf{2.55\times10^{-3}} /7/1 5.45\times10^{-3} /9/1
    a_7 (100) \mathbf{1.22\times10^{-2}} / \textbf{23} / \textbf{7} 2.67\times10^{-2} /66/17 1.50\times10^{-1} /487/24
    a_8 (100) \mathbf{1.32\times10^{-2}} / \textbf{23} / \textbf{7} 3.29\times10^{-2} /60/15 1.92\times10^{-1} /677/18
    a_1 (150) 3.49\times10^{-3} / \textbf{4} / \textbf{1} \mathbf{2.63\times10^{-3}} /4/1 1.13\times10^{-1} /332/7
    a_2 (150) \mathbf{1.88\times10^{-2}} / \textbf{23} / \textbf{7} 3.88\times10^{-2} /75/19 2.84\times10^{-1} /602/28
    a_3 (150) 1.95\times10^{-2} / \textbf{29} /9 \mathbf{1.88\times10^{-2}} /37/ \textbf{8} 1.71\times10^{-1} /447/10
    a_4 (150) \mathbf{1.65\times10^{-2}} / \textbf{23} / \textbf{7} 2.08\times10^{-2} /39/9 2.36\times10^{-1} /602/28
    a_5 (150) \mathbf{1.65\times10^{-2}} / \textbf{23} / \textbf{7} 3.80\times10^{-2} /75/19 2.40\times10^{-1} /602/28
    a_6 (150) 5.11\times10^{-3} / \textbf{7} / \textbf{1} \mathbf{4.06\times10^{-3}} /7/1 1.05\times10^{-2} /9/1
    a_7 (150) \mathbf{1.75\times10^{-2}} / \textbf{23} / \textbf{7} 4.80\times10^{-2} /75/19 2.46\times10^{-1} /602/28
    a_8 (150) \mathbf{1.95\times10^{-2}} / \textbf{23} / \textbf{7} 2.97\times10^{-2} /45/11 2.34\times10^{-1} /575/18

     | Show Table
    DownLoad: CSV
    Table 2.  Numerical results for Problem 2.
    Inti( n ) ILR VRMILP DFPRPMHS
    CPUT/Nfunc/Niter CPUT/Nfunc/Niter CPUT/Nfunc/Niter
    a_1 (5) 8.86\times10^{-3} / \textbf{29} / \textbf{11} \mathbf{8.84\times10^{-3}} /96/25 3.91\times10^{-2} /576/23
    a_2 (5) 2.64\times10^{-2} /400/52 \mathbf{1.73\times10^{-2}} / \textbf{281} / \textbf{31} 5.06\times10^{-2} /893/43
    a_3 (5) \mathbf{2.85\times10^{-3}} / \textbf{30} / \textbf{11} 6.36\times10^{-3} /88/23 2.04\times10^{-2} /300/34
    a_4 (5) 2.69\times10^{-2} /414/53 \mathbf{1.86\times10^{-2}} / \textbf{310} / \textbf{30} 4.14\times10^{-2} /692/44
    a_5 (5) \mathbf{4.95\times10^{-3}} / \textbf{54} / \textbf{20} 7.76\times10^{-3} /107/28 2.56\times10^{-2} /406/35
    a_6 (5) 1.57\times10^{-2} /232/31 \mathbf{7.92\times10^{-3}} / \textbf{114} / \textbf{25} 5.95\times10^{-2} /1056/34
    a_7 (5) \mathbf{5.33\times10^{-3}} / \textbf{54} / \textbf{20} 7.78\times10^{-3} /107/28 2.59\times10^{-2} /406/35
    a_8 (5) 4.65\times10^{-2} /727/76 \mathbf{1.90\times10^{-2}} / \textbf{305} / \textbf{36} 4.12\times10^{-2} /688/41
    a_1 (10) \mathbf{1.22\times10^{-2}} / \textbf{56} / \textbf{20} 1.26\times10^{-2} /84/22 5.03\times10^{-2} /411/37
    a_2 (10) 5.78\times10^{-2} /435/47 \mathbf{4.50\times10^{-2}} / \textbf{398} /34 7.79\times10^{-2} /703/ \textbf{22}
    a_3 (10) \mathbf{6.49\times10^{-3}} / \textbf{29} / \textbf{11} 1.56\times10^{-2} /82/22 5.64\times10^{-2} /353/55
    a_4 (10) 8.59\times10^{-2} /582/57 \mathbf{4.34\times10^{-2}} / \textbf{387} /36 8.26\times10^{-2} /721/ \textbf{25}
    a_5 (10) \mathbf{1.05\times10^{-2}} / \textbf{50} / \textbf{18} 1.92\times10^{-2} /128/32 4.28\times10^{-2} /300/37
    a_6 (10) 3.89\times10^{-2} /276/35 \mathbf{1.51\times10^{-2}} / \textbf{108} / \textbf{24} 9.90\times10^{-2} /851/40
    a_7 (10) \mathbf{1.07\times10^{-2}} / \textbf{50} / \textbf{18} 1.87\times10^{-2} /128/32 4.04\times10^{-2} /300/37
    a_8 (10) 8.75\times10^{-2} /673/60 \mathbf{6.39\times10^{-2}} / \textbf{541} /52 1.09\times10^{-1} /966/ \textbf{39}
    a_1 (50) 5.65\times10^{-2} / \textbf{57} / \textbf{20} \mathbf{5.36\times10^{-2}} /90/24 1.69\times10^{-1} /320/41
    a_2 (50) 4.89\times10^{-1} /997/80 4.34\times10^{-1} /951/57 \mathbf{3.28\times10^{-1}} / \textbf{731} / \textbf{24}
    a_3 (50) \mathbf{2.44\times10^{-2}} / \textbf{35} / \textbf{13} 5.55\times10^{-2} /99/26 1.42\times10^{-1} /327/35
    a_4 (50) 4.42\times10^{-1} /926/83 3.80\times10^{-1} /957/55 \mathbf{3.09\times10^{-1}} / \textbf{731} / \textbf{24}
    a_5 (50) \mathbf{4.08\times10^{-2}} / \textbf{55} / \textbf{19} 6.92\times10^{-2} /124/31 1.93\times10^{-1} /409/36
    a_6 (50) 2.48\times10^{-1} /510/51 \mathbf{5.96\times10^{-2}} / \textbf{114} / \textbf{25} 4.73\times10^{-1} /1119/36
    a_7 (50) \mathbf{3.77\times10^{-2}} / \textbf{55} / \textbf{19} 6.56\times10^{-2} /124/31 1.99\times10^{-1} /409/36
    a_8 (50) 4.88\times10^{-1} /1011/86 3.19\times10^{-1} /699/50 \mathbf{3.04\times10^{-1}} / \textbf{686} / \textbf{20}
    a_1 (100) 7.76\times10^{-2} / \textbf{75} /27 \mathbf{7.69\times10^{-2}} /92/ \textbf{24} 4.11\times10^{-1} /613/31
    a_2 (100) 8.14\times10^{-1} /1187/92 8.34\times10^{-1} /1259/65 \mathbf{4.43\times10^{-1}} / \textbf{727} / \textbf{24}
    a_3 (100) 9.47\times10^{-2} /96/33 \mathbf{7.82\times10^{-2}} / \textbf{94} / \textbf{25} 2.83\times10^{-1} /413/39
    a_4 (100) 7.69\times10^{-1} /1073/90 8.34\times10^{-1} /1193/66 \mathbf{5.16\times10^{-1}} / \textbf{728} / \textbf{24}
    a_5 (100) \mathbf{9.05\times10^{-2}} / \textbf{76} / \textbf{26} 1.01\times10^{-1} /119/31 3.28\times10^{-1} /493/29
    a_6 (100) 3.62\times10^{-1} /505/46 \mathbf{9.80\times10^{-2}} / \textbf{122} / \textbf{27} 6.84\times10^{-1} /1006/46
    a_7 (100) \mathbf{8.67\times10^{-2}} / \textbf{76} / \textbf{26} 9.83\times10^{-2} /119/31 3.27\times10^{-1} /493/29
    a_8 (100) 1.18\times 10^0 /1610/106 5.32\times10^{-1} /800/52 \mathbf{4.04\times10^{-1}} / \textbf{608} / \textbf{25}
    a_1 (150) \mathbf{8.00\times10^{-2}} / \textbf{54} / \textbf{20} 1.15\times10^{-1} /99/26 3.48\times10^{-1} /330/40
    a_2 (150) 1.15\times 10^0 /1130/87 1.34\times 10^0 /1408/76 \mathbf{7.09\times10^{-1}} / \textbf{726} / \textbf{24}
    a_3 (150) \mathbf{8.44\times10^{-2}} / \textbf{57} / \textbf{18} 1.25\times10^{-1} /102/26 4.08\times10^{-1} /400/35
    a_4 (150) 1.17\times 10^0 /1165/95 1.26\times 10^0 /1365/69 \mathbf{6.80\times10^{-1}} / \textbf{726} / \textbf{24}
    a_5 (150) \mathbf{9.90\times10^{-2}} / \textbf{69} / \textbf{25} 1.44\times10^{-1} /124/32 4.11\times10^{-1} /407/38
    a_6 (150) 7.33\times10^{-1} /714/63 \mathbf{1.32\times10^{-1}} / \textbf{117} / \textbf{26} 8.80\times 10^0 /8853/275
    a_7 (150) \mathbf{1.24\times10^{-1}} / \textbf{69} / \textbf{25} 1.73\times10^{-1} /124/32 4.47\times10^{-1} /407/38
    a_8 (150) 1.33\times 10^0 /1224/91 1.27\times 10^0 /1255/71 \mathbf{7.01\times10^{-1}} / \textbf{657} / \textbf{29}

     | Show Table
    DownLoad: CSV
    Table 3.  Numerical results for Problem 3.
    Inti( n ) ILR VRMILP DFPRPMHS
    CPUT/Nfunc/Niter CPUT/Nfunc/Niter CPUT/Nfunc/Niter
    a_1 (5) 8.33\times10^{-3} /23/ \textbf{8} \mathbf{2.50\times10^{-3}} / \textbf{21} /9 3.90\times10^{-2} /544/11
    a_2 (5) 8.23\times10^{-3} / \textbf{61} /21 \mathbf{8.17\times10^{-3}} /82/22 6.76\times10^{-2} /977/ \textbf{19}
    a_3 (5) 5.10\times10^{-3} / \textbf{45} / \textbf{16} \mathbf{5.06\times10^{-3}} /60/16 3.11\times10^{-2} /492/30
    a_4 (5) 7.16\times10^{-3} / \textbf{66} /23 \mathbf{6.87\times10^{-3}} /82/22 5.68\times10^{-2} /977/ \textbf{19}
    a_5 (5) \mathbf{6.90\times10^{-3}} / \textbf{61} /21 7.31\times10^{-3} /82/22 5.94\times10^{-2} /977/ \textbf{19}
    a_6 (5) \mathbf{2.28\times10^{-3}} / \textbf{19} / \textbf{7} 3.53\times10^{-3} /33/14 6.56\times10^{-2} /1084/21
    a_7 (5) 7.05\times10^{-3} / \textbf{61} /21 \mathbf{6.94\times10^{-3}} /82/22 5.87\times10^{-2} /977/ \textbf{19}
    a_8 (5) \mathbf{4.85\times10^{-3}} / \textbf{44} / \textbf{15} 8.76\times10^{-3} /105/28 5.83\times10^{-2} /977/19
    a_1 (10) 6.23\times10^{-3} /23/ \textbf{8} \mathbf{3.88\times10^{-3}} / \textbf{21} /9 5.47\times10^{-2} /544/11
    a_2 (10) \mathbf{9.75\times10^{-3}} / \textbf{49} / \textbf{17} 1.26\times10^{-2} /85/23 1.05\times10^{-1} /986/21
    a_3 (10) 1.12\times10^{-2} / \textbf{54} /19 \mathbf{8.97\times10^{-3}} /60/ \textbf{16} 5.92\times10^{-2} /499/31
    a_4 (10) 1.67\times10^{-2} / \textbf{78} /28 \mathbf{1.27\times10^{-2}} /85/23 1.07\times10^{-1} /986/ \textbf{21}
    a_5 (10) \mathbf{1.04\times10^{-2}} / \textbf{49} / \textbf{17} 1.24\times10^{-2} /85/23 1.09\times10^{-1} /986/21
    a_6 (10) \mathbf{4.08\times10^{-3}} / \textbf{19} / \textbf{7} 5.89\times10^{-3} /33/14 1.06\times10^{-1} /977/19
    a_7 (10) \mathbf{1.10\times10^{-2}} / \textbf{49} / \textbf{17} 1.44\times10^{-2} /85/23 1.11\times10^{-1} /986/21
    a_8 (10) \mathbf{8.57\times10^{-3}} / \textbf{44} / \textbf{15} 1.47\times10^{-2} /98/27 9.76\times10^{-2} /890/24
    a_1 (50) 2.84\times10^{-2} /23/ \textbf{8} \mathbf{1.39\times10^{-2}} / \textbf{21} /9 1.18\times10^{-1} /437/9
    a_2 (50) 4.56\times10^{-2} / \textbf{74} /26 \mathbf{3.72\times10^{-2}} /94/25 3.08\times10^{-1} /978/ \textbf{19}
    a_3 (50) \mathbf{2.33\times10^{-2}} / \textbf{44} / \textbf{15} 2.67\times10^{-2} /60/16 1.04\times10^{-1} /279/31
    a_4 (50) \mathbf{4.24\times10^{-2}} / \textbf{61} /21 4.43\times10^{-2} /93/25 2.94\times10^{-1} /978/ \textbf{19}
    a_5 (50) 5.26\times10^{-2} / \textbf{74} /26 \mathbf{4.02\times10^{-2}} /94/25 2.85\times10^{-1} /978/ \textbf{19}
    a_6 (50) \mathbf{1.01\times10^{-2}} / \textbf{19} / \textbf{7} 1.48\times10^{-2} /35/15 2.93\times10^{-1} /977/19
    a_7 (50) 4.55\times10^{-2} / \textbf{74} /26 \mathbf{4.41\times10^{-2}} /94/25 2.97\times10^{-1} /978/ \textbf{19}
    a_8 (50) \mathbf{4.27\times10^{-2}} / \textbf{63} /22 5.60\times10^{-2} /102/27 2.92\times10^{-1} /978/ \textbf{19}
    a_1 (100) 1.51\times10^{-2} /20/ \textbf{7} \mathbf{1.14\times10^{-2}} / \textbf{18} /7 1.52\times10^{-1} /437/9
    a_2 (100) \mathbf{5.81\times10^{-2}} / \textbf{72} /25 6.66\times10^{-2} /104/28 3.86\times10^{-1} /978/ \textbf{19}
    a_3 (100) 4.40\times10^{-2} / \textbf{58} /20 \mathbf{3.25\times10^{-2}} /60/ \textbf{16} 1.36\times10^{-1} /364/19
    a_4 (100) \mathbf{4.69\times10^{-2}} / \textbf{61} /21 5.44\times10^{-2} /104/28 4.02\times10^{-1} /978/ \textbf{19}
    a_5 (100) \mathbf{5.39\times10^{-2}} / \textbf{72} /25 5.64\times10^{-2} /104/28 3.98\times10^{-1} /978/ \textbf{19}
    a_6 (100) \mathbf{1.69\times10^{-2}} / \textbf{19} / \textbf{7} 2.70\times10^{-2} /35/15 4.70\times10^{-1} /1085/21
    a_7 (100) \mathbf{5.06\times10^{-2}} / \textbf{72} /25 5.60\times10^{-2} /104/28 4.14\times10^{-1} /978/ \textbf{19}
    a_8 (100) 6.54\times10^{-2} / \textbf{76} /26 \mathbf{5.91\times10^{-2}} /109/29 3.98\times10^{-1} /978/ \textbf{19}
    a_1 (150) 1.99\times10^{-2} /20/ \textbf{7} \mathbf{1.50\times10^{-2}} / \textbf{18} /7 2.04\times10^{-1} /437/9
    a_2 (150) \mathbf{7.71\times10^{-2}} / \textbf{71} /25 8.53\times10^{-2} /99/27 5.09\times10^{-1} /978/ \textbf{19}
    a_3 (150) 5.08\times10^{-2} / \textbf{51} /17 \mathbf{4.93\times10^{-2}} /60/ \textbf{16} 1.65\times10^{-1} /274/19
    a_4 (150) \mathbf{8.24\times10^{-2}} / \textbf{74} /25 8.47\times10^{-2} /99/27 5.37\times10^{-1} /978/ \textbf{19}
    a_5 (150) 7.76\times10^{-2} / \textbf{71} /25 \mathbf{6.93\times10^{-2}} /99/27 5.39\times10^{-1} /978/ \textbf{19}
    a_6 (150) \mathbf{2.37\times10^{-2}} / \textbf{22} / \textbf{8} 3.91\times10^{-2} /35/15 5.64\times10^{-1} /978/19
    a_7 (150) \mathbf{8.44\times10^{-2}} / \textbf{71} /25 8.50\times10^{-2} /99/27 5.38\times10^{-1} /978/ \textbf{19}
    a_8 (150) \mathbf{6.24\times10^{-2}} / \textbf{62} /22 7.60\times10^{-2} /105/28 5.27\times10^{-1} /978/ \textbf{19}

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    Table 4.  Numerical results for Problem 4.
    Inti( n ) ILR VRMILP DFPRPMHS
    CPUT/Nfunc/Niter CPUT/Nfunc/Niter CPUT/Nfunc/Niter
    a_1 (5) 1.58\times10^{-1} / \textbf{3} / \textbf{1} \mathbf{4.47\times10^{-3}} /3/1 2.10\times10^{-2} /3/1
    a_2 (5) \mathbf{1.70\times10^{-2}} / \textbf{44} / \textbf{5} 1.91\times10^{-2} /149/16 4.55\times10^{-2} /651/16
    a_3 (5) \mathbf{9.99\times10^{-3}} / \textbf{63} /8 1.86\times10^{-2} /247/26 2.06\times10^{-2} /259/ \textbf{7}
    a_4 (5) \mathbf{3.70\times10^{-3}} / \textbf{44} / \textbf{5} 1.33\times10^{-2} /225/24 2.86\times10^{-2} /509/12
    a_5 (5) \mathbf{5.19\times10^{-3}} / \textbf{44} / \textbf{5} 1.36\times10^{-2} /149/16 3.72\times10^{-2} /651/16
    a_6 (5) 8.76\times10^{-4} / \textbf{4} / \textbf{1} \mathbf{5.51\times10^{-4}} /4/1 2.28\times10^{-3} /4/1
    a_7 (5) \mathbf{3.48\times10^{-3}} / \textbf{44} / \textbf{5} 9.75\times10^{-3} /149/16 3.18\times10^{-2} /651/16
    a_8 (5) \mathbf{3.25\times10^{-3}} / \textbf{44} / \textbf{5} 1.24\times10^{-2} /200/21 3.11\times10^{-2} /626/14
    a_1 (10) 1.03\times10^{-3} / \textbf{3} / \textbf{1} \mathbf{5.12\times10^{-4}} /3/1 3.02\times10^{-3} /3/1
    a_2 (10) \mathbf{5.99\times10^{-3}} / \textbf{44} / \textbf{5} 2.83\times10^{-2} /243/26 6.79\times10^{-2} /660/16
    a_3 (10) \mathbf{8.56\times10^{-3}} / \textbf{63} /8 2.97\times10^{-2} /246/26 2.56\times10^{-2} /259/ \textbf{7}
    a_4 (10) \mathbf{5.47\times10^{-3}} / \textbf{44} / \textbf{5} 1.59\times10^{-2} /141/15 5.05\times10^{-2} /509/12
    a_5 (10) \mathbf{5.66\times10^{-3}} / \textbf{44} / \textbf{5} 2.84\times10^{-2} /243/26 6.01\times10^{-2} /660/16
    a_6 (10) 1.04\times10^{-3} / \textbf{4} / \textbf{1} \mathbf{6.07\times10^{-4}} /4/1 1.69\times10^{-3} /4/1
    a_7 (10) \mathbf{5.08\times10^{-3}} / \textbf{44} / \textbf{5} 2.48\times10^{-2} /243/26 5.86\times10^{-2} /660/16
    a_8 (10) \mathbf{5.28\times10^{-3}} / \textbf{44} / \textbf{5} 2.45\times10^{-2} /229/24 3.55\times10^{-2} /364/17
    a_1 (50) 8.54\times10^{-3} / \textbf{3} / \textbf{1} \mathbf{1.09\times10^{-3}} /3/1 3.14\times10^{-3} /3/1
    a_2 (50) \mathbf{2.88\times10^{-2}} / \textbf{44} / \textbf{5} 6.72\times10^{-2} /159/17 1.44\times10^{-1} /509/12
    a_3 (50) \mathbf{2.57\times10^{-2}} / \textbf{63} /8 7.24\times10^{-2} /237/25 6.79\times10^{-2} /259/ \textbf{7}
    a_4 (50) \mathbf{1.77\times10^{-2}} / \textbf{44} / \textbf{5} 9.94\times10^{-2} /245/26 1.31\times10^{-1} /509/12
    a_5 (50) \mathbf{1.74\times10^{-2}} / \textbf{44} / \textbf{5} 6.06\times10^{-2} /159/17 1.66\times10^{-1} /509/12
    a_6 (50) \mathbf{2.83\times10^{-3}} / \textbf{4} / \textbf{1} 3.46\times10^{-3} /4/1 4.31\times10^{-3} /4/1
    a_7 (50) \mathbf{1.88\times10^{-2}} / \textbf{44} / \textbf{5} 5.47\times10^{-2} /159/17 1.40\times10^{-1} /509/12
    a_8 (50) \mathbf{1.87\times10^{-2}} / \textbf{44} / \textbf{5} 8.57\times10^{-2} /244/26 1.81\times10^{-1} /725/17
    a_1 (100) 3.06\times10^{-3} / \textbf{3} / \textbf{1} \mathbf{2.70\times10^{-3}} /3/1 4.55\times10^{-3} /3/1
    a_2 (100) \mathbf{2.59\times10^{-2}} / \textbf{44} / \textbf{5} 1.15\times10^{-1} /250/27 1.79\times10^{-1} /509/12
    a_3 (100) \mathbf{3.02\times10^{-2}} / \textbf{63} /8 8.74\times10^{-2} /197/21 8.90\times10^{-2} /259/ \textbf{7}
    a_4 (100) \mathbf{2.46\times10^{-2}} / \textbf{44} / \textbf{5} 8.31\times10^{-2} /176/19 1.76\times10^{-1} /509/12
    a_5 (100) \mathbf{2.40\times10^{-2}} / \textbf{44} / \textbf{5} 1.17\times10^{-1} /250/27 1.83\times10^{-1} /509/12
    a_6 (100) 1.20\times10^{-2} /11/ \textbf{1} 1.08\times10^{-2} /11/1 \mathbf{5.98\times10^{-3}} / \textbf{4} /1
    a_7 (100) \mathbf{2.57\times10^{-2}} / \textbf{44} / \textbf{5} 1.21\times10^{-1} /250/27 1.97\times10^{-1} /509/12
    a_8 (100) \mathbf{2.37\times10^{-2}} / \textbf{44} / \textbf{5} 9.33\times10^{-2} /187/20 1.82\times10^{-1} /509/12
    a_1 (150) 3.87\times10^{-3} / \textbf{3} / \textbf{1} \mathbf{1.92\times10^{-3}} /3/1 6.30\times10^{-3} /3/1
    a_2 (150) \mathbf{3.76\times10^{-2}} / \textbf{44} / \textbf{5} 1.33\times10^{-1} /201/22 2.57\times10^{-1} /509/12
    a_3 (150) \mathbf{4.78\times10^{-2}} / \textbf{63} /8 1.15\times10^{-1} /179/19 1.22\times10^{-1} /259/ \textbf{7}
    a_4 (150) \mathbf{3.00\times10^{-2}} / \textbf{44} / \textbf{5} 8.73\times10^{-2} /132/14 2.34\times10^{-1} /509/12
    a_5 (150) \mathbf{3.16\times10^{-2}} / \textbf{44} / \textbf{5} 1.34\times10^{-1} /201/22 2.51\times10^{-1} /509/12
    a_6 (150) 1.60\times10^{-2} /11/ \textbf{1} 1.42\times10^{-2} /11/1 \mathbf{1.33\times10^{-2}} / \textbf{8} /1
    a_7 (150) \mathbf{3.09\times10^{-2}} / \textbf{44} / \textbf{5} 1.32\times10^{-1} /201/22 2.46\times10^{-1} /509/12
    a_8 (150) \mathbf{3.17\times10^{-2}} / \textbf{44} / \textbf{5} 2.09\times10^{-1} /301/32 3.02\times10^{-1} /642/24

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    Table 5.  Numerical results for Problem 5.
    Inti( n ) ILR VRMILP DFPRPMHS
    CPUT/Nfunc/Niter CPUT/Nfunc/Niter CPUT/Nfunc/Niter
    a_1 (5) 5.45\times10^{-2} / \textbf{283} /36 \mathbf{3.93\times10^{-2}} /309/ \textbf{34} 6.74\times10^{-2} /694/43
    a_2 (5) \mathbf{2.50\times10^{-2}} / \textbf{235} / \textbf{30} 3.07\times10^{-2} /330/36 9.82\times10^{-2} /1113/46
    a_3 (5) \mathbf{2.09\times10^{-2}} / \textbf{191} / \textbf{24} 2.60\times10^{-2} /294/32 5.82\times10^{-2} /663/50
    a_4 (5) \mathbf{2.41\times10^{-2}} / \textbf{235} / \textbf{30} 3.07\times10^{-2} /330/36 8.84\times10^{-2} /1125/46
    a_5 (5) \mathbf{2.04\times10^{-2}} / \textbf{212} / \textbf{27} 3.62\times10^{-2} /289/31 1.26\times10^{-1} /1142/56
    a_6 (5) 2.66\times10^{-2} / \textbf{253} / \textbf{32} \mathbf{2.25\times10^{-2}} /286/32 1.41\times10^{-1} /1265/113
    a_7 (5) \mathbf{2.84\times10^{-2}} / \textbf{212} / \textbf{27} 3.94\times10^{-2} /289/31 1.22\times10^{-1} /1142/56
    a_8 (5) \mathbf{3.33\times10^{-2}} / \textbf{247} / \textbf{32} 3.92\times10^{-2} /331/35 2.61\times10^{-1} /2555/55
    a_1 (10) \mathbf{6.23\times10^{-2}} / \textbf{254} / \textbf{32} 9.28\times10^{-2} /356/39 1.72\times10^{-1} /757/57
    a_2 (10) \mathbf{5.84\times10^{-2}} / \textbf{239} / \textbf{31} 6.88\times10^{-2} /308/33 1.68\times10^{-1} /800/48
    a_3 (10) \mathbf{5.94\times10^{-2}} / \textbf{258} / \textbf{33} 6.19\times10^{-2} /296/33 1.20\times10^{-1} /565/46
    a_4 (10) \mathbf{5.24\times10^{-2}} / \textbf{232} / \textbf{30} 6.53\times10^{-2} /308/33 1.60\times10^{-1} /800/48
    a_5 (10) \mathbf{5.56\times10^{-2}} / \textbf{239} / \textbf{31} 6.79\times10^{-2} /325/35 1.65\times10^{-1} /789/62
    a_6 (10) \mathbf{4.87\times10^{-2}} / \textbf{205} / \textbf{26} 6.13\times10^{-2} /282/32 1.81\times10^{-1} /887/53
    a_7 (10) \mathbf{5.46\times10^{-2}} / \textbf{239} / \textbf{31} 6.68\times10^{-2} /325/35 1.63\times10^{-1} /789/62
    a_8 (10) \mathbf{6.69\times10^{-2}} / \textbf{275} / \textbf{36} 7.43\times10^{-2} /350/37 2.56\times10^{-1} /1297/54
    a_1 (50) \mathbf{2.56\times10^{-1}} / \textbf{211} / \textbf{27} 3.13\times10^{-1} /308/33 5.78\times10^{-1} /657/46
    a_2 (50) \mathbf{2.42\times10^{-1}} / \textbf{229} / \textbf{29} 2.56\times10^{-1} /279/31 1.24\times 10^0 /1319/111
    a_3 (50) \mathbf{2.35\times10^{-1}} / \textbf{250} / \textbf{32} 2.68\times10^{-1} /308/33 6.76\times10^{-1} /682/59
    a_4 (50) \mathbf{2.38\times10^{-1}} / \textbf{229} / \textbf{29} 2.83\times10^{-1} /279/31 1.17\times 10^0 /1276/105
    a_5 (50) \mathbf{2.33\times10^{-1}} / \textbf{228} / \textbf{29} 3.54\times10^{-1} /340/38 9.50\times10^{-1} /983/83
    a_6 (50) \mathbf{2.17\times10^{-1}} / \textbf{219} / \textbf{28} 2.92\times10^{-1} /324/35 8.49\times10^{-1} /937/52
    a_7 (50) \mathbf{2.46\times10^{-1}} / \textbf{228} / \textbf{29} 3.07\times10^{-1} /340/38 9.22\times10^{-1} /983/83
    a_8 (50) \mathbf{2.32\times10^{-1}} / \textbf{241} / \textbf{31} 3.16\times10^{-1} /364/39 1.09\times 10^0 /1238/56
    a_1 (100) \mathbf{3.79\times10^{-1}} / \textbf{220} / \textbf{28} 4.93\times10^{-1} /325/35 1.12\times 10^0 /753/44
    a_2 (100) \mathbf{4.04\times10^{-1}} / \textbf{236} / \textbf{30} 4.62\times10^{-1} /283/32 1.24\times 10^0 /768/61
    a_3 (100) \mathbf{4.28\times10^{-1}} / \textbf{250} / \textbf{32} 4.92\times10^{-1} /317/34 1.87\times 10^0 /1229/56
    a_4 (100) \mathbf{3.82\times10^{-1}} / \textbf{236} / \textbf{30} 4.27\times10^{-1} /273/31 1.24\times 10^0 /768/61
    a_5 (100) \mathbf{4.77\times10^{-1}} / \textbf{262} / \textbf{33} 5.63\times10^{-1} /331/36 1.40\times 10^0 /892/65
    a_6 (100) \mathbf{4.27\times10^{-1}} / \textbf{254} / \textbf{33} 5.55\times10^{-1} /314/34 1.66\times 10^0 /1020/50
    a_7 (100) \mathbf{4.73\times10^{-1}} / \textbf{262} / \textbf{33} 5.60\times10^{-1} /331/36 1.30\times 10^0 /892/65
    a_8 (100) \mathbf{4.79\times10^{-1}} / \textbf{291} / \textbf{38} 5.78\times10^{-1} /380/41 1.96\times 10^0 /1246/58
    a_1 (150) \mathbf{5.67\times10^{-1}} / \textbf{221} / \textbf{28} 6.78\times10^{-1} /287/31 2.40\times 10^0 /1010/50
    a_2 (150) \mathbf{5.10\times10^{-1}} / \textbf{206} / \textbf{26} 8.62\times10^{-1} /375/41 2.98\times 10^0 /1327/123
    a_3 (150) \mathbf{5.57\times10^{-1}} / \textbf{228} / \textbf{29} 6.93\times10^{-1} /325/35 1.95\times 10^0 /832/52
    a_4 (150) \mathbf{5.48\times10^{-1}} / \textbf{206} / \textbf{26} 8.80\times10^{-1} /375/41 3.41\times 10^0 /1363/126
    a_5 (150) \mathbf{5.66\times10^{-1}} / \textbf{214} / \textbf{27} 8.50\times10^{-1} /358/40 2.31\times 10^0 /1031/78
    a_6 (150) \mathbf{4.97\times10^{-1}} / \textbf{240} / \textbf{31} 7.32\times10^{-1} /313/34 1.83\times 10^0 /875/46
    a_7 (150) \mathbf{5.95\times10^{-1}} / \textbf{214} / \textbf{27} 9.02\times10^{-1} /358/40 2.42\times 10^0 /1031/78
    a_8 (150) \mathbf{6.91\times10^{-1}} / \textbf{291} / \textbf{38} 1.13\times 10^0 /456/49 2.85\times 10^0 /1242/58

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    Table 6.  Numerical results for Problem 6.
    Inti( n ) ILR VRMILP DFPRPMHS
    CPUT/Nfunc/Niter CPUT/Nfunc/Niter CPUT/Nfunc/Niter
    a_1 (5) 7.08\times10^{-3} / \textbf{34} / \textbf{14} \mathbf{6.92\times10^{-3}} /84/22 4.32\times10^{-2} /672/21
    a_2 (5) 7.84\times10^{-3} / \textbf{69} /27 \mathbf{5.48\times10^{-3}} /71/ \textbf{22} 2.96\times10^{-2} /513/40
    a_3 (5) \mathbf{4.14\times10^{-3}} / \textbf{34} / \textbf{14} 7.29\times10^{-3} /111/27 2.58\times10^{-2} /504/34
    a_4 (5) 7.08\times10^{-3} /74/28 \mathbf{4.61\times10^{-3}} / \textbf{71} / \textbf{22} 2.23\times10^{-2} /414/41
    a_5 (5) 5.92\times10^{-3} / \textbf{69} /27 \mathbf{4.87\times10^{-3}} /71/ \textbf{22} 2.85\times10^{-2} /513/40
    a_6 (5) 3.27\times10^{-3} /32/13 \mathbf{2.77\times10^{-3}} / \textbf{28} / \textbf{11} 3.18\times10^{-2} /670/20
    a_7 (5) 6.28\times10^{-3} / \textbf{69} /27 \mathbf{4.86\times10^{-3}} /71/ \textbf{22} 2.74\times10^{-2} /513/40
    a_8 (5) 6.68\times10^{-3} / \textbf{79} /30 \mathbf{5.76\times10^{-3}} /84/ \textbf{25} 2.66\times10^{-2} /446/40
    a_1 (10) \mathbf{6.94\times10^{-3}} / \textbf{31} / \textbf{13} 8.81\times10^{-3} /67/20 4.26\times10^{-2} /424/29
    a_2 (10) \mathbf{6.88\times10^{-3}} / \textbf{34} / \textbf{14} 1.25\times10^{-2} /95/27 6.07\times10^{-2} /672/21
    a_3 (10) \mathbf{6.52\times10^{-3}} / \textbf{31} / \textbf{13} 1.68\times10^{-2} /142/31 3.96\times10^{-2} /384/30
    a_4 (10) \mathbf{6.86\times10^{-3}} / \textbf{34} / \textbf{14} 1.36\times10^{-2} /90/26 6.88\times10^{-2} /672/21
    a_5 (10) \mathbf{7.18\times10^{-3}} / \textbf{34} / \textbf{14} 1.23\times10^{-2} /95/27 6.25\times10^{-2} /672/21
    a_6 (10) 7.48\times10^{-3} /35/14 \mathbf{4.64\times10^{-3}} / \textbf{33} / \textbf{13} 5.18\times10^{-2} /561/18
    a_7 (10) \mathbf{6.86\times10^{-3}} / \textbf{34} / \textbf{14} 1.20\times10^{-2} /95/27 6.47\times10^{-2} /672/21
    a_8 (10) \mathbf{6.92\times10^{-3}} / \textbf{34} / \textbf{14} 1.10\times10^{-2} /86/25 6.35\times10^{-2} /672/21
    a_1 (50) \mathbf{3.20\times10^{-2}} / \textbf{41} / \textbf{17} 3.59\times10^{-1} /1099/94 2.04\times10^{-1} /698/31
    a_2 (50) \mathbf{2.57\times10^{-2}} / \textbf{36} / \textbf{15} 4.34\times10^{-2} /105/29 1.82\times10^{-1} /572/23
    a_3 (50) \mathbf{2.34\times10^{-2}} / \textbf{38} / \textbf{16} 2.38\times10^{-1} /769/56 1.70\times10^{-1} /511/38
    a_4 (50) \mathbf{2.23\times10^{-2}} / \textbf{36} / \textbf{15} 3.88\times10^{-2} /101/28 1.87\times10^{-1} /572/23
    a_5 (50) \mathbf{2.68\times10^{-2}} / \textbf{36} / \textbf{15} 4.40\times10^{-2} /105/29 1.84\times10^{-1} /572/23
    a_6 (50) 2.29\times10^{-2} /37/15 \mathbf{1.08\times10^{-2}} / \textbf{25} / \textbf{10} 2.13\times10^{-1} /672/21
    a_7 (50) \mathbf{2.70\times10^{-2}} / \textbf{36} / \textbf{15} 5.36\times10^{-2} /105/29 1.88\times10^{-1} /572/23
    a_8 (50) \mathbf{2.61\times10^{-2}} / \textbf{36} / \textbf{15} 4.65\times10^{-2} /102/28 1.88\times10^{-1} /572/23
    a_1 (100) \mathbf{3.96\times10^{-2}} / \textbf{41} / \textbf{17} 5.61\times10^{-2} /84/25 2.28\times10^{-1} /569/22
    a_2 (100) \mathbf{5.49\times10^{-2}} / \textbf{57} /24 6.17\times10^{-2} /100/28 2.95\times10^{-1} /676/ \textbf{23}
    a_3 (100) \mathbf{3.64\times10^{-2}} / \textbf{41} / \textbf{17} 4.73\times10^{-2} /84/23 2.08\times10^{-1} /411/43
    a_4 (100) \mathbf{4.85\times10^{-2}} / \textbf{57} /24 6.49\times10^{-2} /100/28 2.92\times10^{-1} /676/ \textbf{23}
    a_5 (100) \mathbf{4.61\times10^{-2}} / \textbf{57} /24 5.92\times10^{-2} /100/28 2.79\times10^{-1} /676/ \textbf{23}
    a_6 (100) 3.35\times10^{-2} /39/16 \mathbf{2.28\times10^{-2}} / \textbf{33} / \textbf{13} 2.64\times10^{-1} /673/22
    a_7 (100) \mathbf{4.73\times10^{-2}} / \textbf{57} /24 6.31\times10^{-2} /100/28 2.89\times10^{-1} /676/ \textbf{23}
    a_8 (100) 7.09\times10^{-2} / \textbf{83} /33 \mathbf{5.68\times10^{-2}} /100/28 3.00\times10^{-1} /676/ \textbf{23}
    a_1 (150) \mathbf{4.56\times10^{-2}} / \textbf{38} / \textbf{16} 6.69\times10^{-2} /83/24 4.36\times10^{-1} /678/24
    a_2 (150) \mathbf{5.06\times10^{-2}} / \textbf{41} / \textbf{17} 1.01\times10^{-1} /125/35 3.39\times10^{-1} /512/39
    a_3 (150) \mathbf{4.98\times10^{-2}} / \textbf{38} / \textbf{16} 1.21\times10^{-1} /154/34 4.10\times10^{-1} /661/52
    a_4 (150) \mathbf{4.84\times10^{-2}} / \textbf{41} / \textbf{17} 1.21\times10^{-1} /125/35 3.36\times10^{-1} /517/41
    a_5 (150) \mathbf{4.28\times10^{-2}} / \textbf{41} / \textbf{17} 1.23\times10^{-1} /125/35 3.83\times10^{-1} /512/39
    a_6 (150) 6.04\times10^{-2} /39/16 \mathbf{3.64\times10^{-2}} / \textbf{35} / \textbf{14} 3.85\times10^{-1} /674/22
    a_7 (150) \mathbf{4.13\times10^{-2}} / \textbf{41} / \textbf{17} 9.58\times10^{-2} /125/35 3.00\times10^{-1} /512/39
    a_8 (150) \mathbf{4.37\times10^{-2}} / \textbf{41} / \textbf{17} 1.17\times10^{-1} /129/36 3.59\times10^{-1} /597/33

     | Show Table
    DownLoad: CSV
    Table 7.  Numerical results for Problem 7.
    Inti( n ) ILR VRMILP DFPRPMHS
    CPUT/Nfunc/Niter CPUT/Nfunc/Niter CPUT/Nfunc/Niter
    a_1 (5) 1.75\times10^{-2} /111/18 \mathbf{6.37\times10^{-3}} / \textbf{73} / \textbf{12} 4.54\times10^{-2} /752/32
    a_2 (5) \mathbf{9.71\times10^{-3}} / \textbf{118} / \textbf{19} 1.35\times10^{-2} /207/29 3.64\times10^{-2} /811/55
    a_3 (5) 5.77\times10^{-3} /104/17 \mathbf{3.63\times10^{-3}} / \textbf{73} / \textbf{12} 3.20\times10^{-2} /829/17
    a_4 (5) 6.72\times10^{-3} /126/20 \mathbf{5.25\times10^{-3}} / \textbf{112} / \textbf{17} 3.93\times10^{-2} /934/46
    a_5 (5) \mathbf{6.59\times10^{-3}} / \textbf{118} / \textbf{19} 9.84\times10^{-3} /207/29 3.90\times10^{-2} /891/51
    a_6 (5) 5.47\times10^{-3} /96/17 \mathbf{3.57\times10^{-3}} / \textbf{72} / \textbf{13} 4.04\times10^{-2} /1024/21
    a_7 (5) \mathbf{6.59\times10^{-3}} / \textbf{118} / \textbf{19} 1.00\times10^{-2} /207/29 3.93\times10^{-2} /892/51
    a_8 (5) \mathbf{6.74\times10^{-3}} / \textbf{128} / \textbf{20} 9.35\times10^{-3} /207/29 4.15\times10^{-2} /927/45
    a_1 (10) 1.70\times10^{-2} /113/18 \mathbf{8.21\times10^{-3}} / \textbf{73} / \textbf{12} 6.53\times10^{-2} /735/43
    a_2 (10) 1.72\times10^{-2} /148/22 \mathbf{1.30\times10^{-2}} / \textbf{137} / \textbf{20} 8.32\times10^{-2} /1047/35
    a_3 (10) 1.36\times10^{-2} /133/20 \mathbf{6.79\times10^{-3}} / \textbf{73} / \textbf{12} 6.49\times10^{-2} /830/17
    a_4 (10) 1.71\times10^{-2} /138/22 \mathbf{1.12\times10^{-2}} / \textbf{112} / \textbf{17} 7.33\times10^{-2} /913/30
    a_5 (10) 1.73\times10^{-2} /148/22 \mathbf{1.28\times10^{-2}} / \textbf{137} / \textbf{20} 1.07\times10^{-1} /1047/35
    a_6 (10) 1.19\times10^{-2} /96/17 \mathbf{8.66\times10^{-3}} / \textbf{72} / \textbf{13} 8.01\times10^{-2} /1024/21
    a_7 (10) 1.66\times10^{-2} /148/22 \mathbf{1.44\times10^{-2}} / \textbf{137} / \textbf{20} 8.95\times10^{-2} /1047/35
    a_8 (10) \mathbf{1.54\times10^{-2}} / \textbf{131} / \textbf{21} 2.27\times10^{-2} /241/33 7.95\times10^{-2} /914/43
    a_1 (50) 4.83\times10^{-2} /118/19 \mathbf{2.48\times10^{-2}} / \textbf{79} / \textbf{13} 2.08\times10^{-1} /836/16
    a_2 (50) \mathbf{4.34\times10^{-2}} / \textbf{133} / \textbf{21} 4.56\times10^{-2} /185/26 2.22\times10^{-1} /988/27
    a_3 (50) 3.22\times10^{-2} /105/17 \mathbf{2.33\times10^{-2}} / \textbf{79} / \textbf{13} 1.81\times10^{-1} /772/37
    a_4 (50) 3.98\times10^{-2} /133/21 \mathbf{2.78\times10^{-2}} / \textbf{112} / \textbf{17} 2.34\times10^{-1} /999/52
    a_5 (50) \mathbf{4.44\times10^{-2}} / \textbf{133} / \textbf{21} 4.96\times10^{-2} /185/26 2.44\times10^{-1} /988/27
    a_6 (50) 3.19\times10^{-2} /102/18 \mathbf{2.16\times10^{-2}} / \textbf{78} / \textbf{14} 2.29\times10^{-1} /1024/21
    a_7 (50) \mathbf{4.29\times10^{-2}} / \textbf{133} / \textbf{21} 4.79\times10^{-2} /185/26 2.28\times10^{-1} /988/27
    a_8 (50) \mathbf{4.32\times10^{-2}} / \textbf{133} / \textbf{21} 5.52\times10^{-2} /209/29 2.13\times10^{-1} /931/27
    a_1 (100) 6.02\times10^{-2} /118/19 \mathbf{3.75\times10^{-2}} / \textbf{79} / \textbf{13} 2.51\times10^{-1} /838/16
    a_2 (100) \mathbf{5.16\times10^{-2}} / \textbf{133} / \textbf{21} 6.32\times10^{-2} /205/29 2.91\times10^{-1} /1018/31
    a_3 (100) 4.38\times10^{-2} /117/19 \mathbf{2.48\times10^{-2}} / \textbf{79} / \textbf{13} 2.37\times10^{-1} /849/18
    a_4 (100) 5.98\times10^{-2} /133/21 \mathbf{4.46\times10^{-2}} / \textbf{112} / \textbf{17} 2.75\times10^{-1} /1018/31
    a_5 (100) \mathbf{5.15\times10^{-2}} / \textbf{133} / \textbf{21} 6.67\times10^{-2} /205/29 2.78\times10^{-1} /1018/31
    a_6 (100) 4.26\times10^{-2} /102/18 \mathbf{2.68\times10^{-2}} / \textbf{78} / \textbf{14} 2.74\times10^{-1} /1024/21
    a_7 (100) \mathbf{5.38\times10^{-2}} / \textbf{133} / \textbf{21} 8.67\times10^{-2} /205/29 3.08\times10^{-1} /1018/31
    a_8 (100) \mathbf{5.41\times10^{-2}} / \textbf{130} / \textbf{21} 9.41\times10^{-2} /241/33 3.10\times10^{-1} /1058/36
    a_1 (150) 5.88\times10^{-2} /111/18 \mathbf{4.44\times10^{-2}} / \textbf{79} / \textbf{13} 3.23\times10^{-1} /838/16
    a_2 (150) \mathbf{6.85\times10^{-2}} / \textbf{139} / \textbf{22} 7.94\times10^{-2} /168/24 4.13\times10^{-1} /1048/35
    a_3 (150) 7.80\times10^{-2} /142/22 \mathbf{3.71\times10^{-2}} / \textbf{79} / \textbf{13} 3.38\times10^{-1} /787/49
    a_4 (150) 6.78\times10^{-2} /132/21 \mathbf{5.61\times10^{-2}} / \textbf{127} / \textbf{19} 3.81\times10^{-1} /1018/31
    a_5 (150) \mathbf{7.81\times10^{-2}} / \textbf{139} / \textbf{22} 8.21\times10^{-2} /168/24 4.24\times10^{-1} /1048/35
    a_6 (150) 5.85\times10^{-2} /108/19 \mathbf{4.74\times10^{-2}} / \textbf{78} / \textbf{14} 4.21\times10^{-1} /1024/21
    a_7 (150) \mathbf{7.66\times10^{-2}} / \textbf{139} / \textbf{22} 8.18\times10^{-2} /168/24 3.88\times10^{-1} /1048/35
    a_8 (150) 8.15\times10^{-2} /149/24 \mathbf{5.84\times10^{-2}} / \textbf{118} / \textbf{18} 4.22\times10^{-1} /1056/36

     | Show Table
    DownLoad: CSV

    To provide a clearer characterization of the performance differences among the three algorithms, we adopt the performance profiles proposed by Dolan and Moré [28]. These profiles evaluate algorithmic behavior based on several key performance indicators, specifically the CPU time in seconds, the number of function evaluations, and the number of iterations. By plotting these indicators, the profiles offer a visual and comparative summary of algorithm efficiency. In these plots, a higher performance curve corresponds to better overall performance, making interpretation both intuitive and informative. By drawing these performance profiles for these three algorithms, we can visually assess and compare their efficiency, as shown in Figures 13. According to Figure 1, the ILR algorithm demonstrates significant efficiency, solving approximately 56% of the test problems with the lowest CPUT compared to the VRMILP and DFPRPMHS algorithms, which solve around 44% and 4% of the test problems, respectively. Similarly, Figure 2 shows that the ILR algorithm maintains its superior performance, solving approximately 75% of the test problems with the fewest Nfunc. In contrast, the VRMILP and DFPRPMHS algorithms solve about 26% and 7% of the test problems, respectively, with the least number of function evaluations. Lastly, Figure 3 further confirms the ILR algorithm's efficiency, solving approximately 53% of the test problems with the fewest Niter, while the VRMILP and DFPRRMHS algorithms solve around 37% and 20% of the test problems, respectively, with the fewest iterations.

    Figure 1.  Performance profiles on CPUT.
    Figure 2.  Performance profiles on Nfunc.
    Figure 3.  Performance profiles on Niter.

    Overall, these performance profiles highlight the ILR algorithm's effectiveness in solving large-scale nonlinear systems of equations with convex constraints, outperforming the VRMILP and DFPRPMHS algorithms across multiple performance metrics.

    In this section, we extend the evaluation of the proposed ILR algorithm to impulse noise image restoration problems. To validate the effectiveness of the ILR algorithm, we apply it to benchmark grayscale images subjected to varying levels of impulse noise.

    Impulse noise image restoration is a critical topic in the field of image processing, particularly due to its importance in improving the quality of images corrupted by noise. Noise in images can be introduced through various sources, such as malfunctioning pixels in camera sensors, faulty memory locations in hardware, or transmission errors in communication channels. Common types of noise include Gaussian noise and impulse noise, with the latter often manifesting as salt-and-pepper noise. To address the challenge of removing impulse noise, Chan et al. [29] proposed a two-phase denoising scheme. This scheme combines the adaptive median filter (AMF) method with a variational method to effectively detect and restore noisy pixels.

    Let m \times n denote the pixel size of an original image. The pixel locations are indexed by the set \mathcal{M} = \{1, 2, \ldots, m \} \times \{1, 2, \ldots, n\} . We denote the noise candidate set by \mathcal{N} \subset \mathcal{M} , and |\mathcal{N}| represents the number of elements in \mathcal{N} . In the first phase, noise detection is performed using an AMF. For a pixel located at (i, j) \in \mathcal{M} , the observed pixel value is denoted by y_{ij} , and the neighborhood of pixel (i, j) is defined as \mathcal{V}_{ij} = \{(i, j - 1), (i, j + 1), (i - 1, j), (i + 1, j)\} . The AMF detects noise by considering these neighborhood values. Once the noisy pixels are detected, the second phase involves the restoration of these pixels. This is achieved by minimizing the following regularization function:

    \min\limits_{\mathbf{x}} \sum\limits_{(i,j) \in \mathcal{N}} \left[ |x_{i,j} - y_{i,j}| + \frac{\beta}{2} \left( 2\Phi_{i,j}^1 + \Phi_{i,j}^2 \right) \right],

    where

    \Phi_{i,j}^1 = \sum\limits_{(m,n) \in \mathcal{V}_{ij} \setminus \mathcal{N}} \varphi_{\alpha}(x_{i,j} - y_{m,n}), \quad \Phi_{i,j}^2 = \sum\limits_{(m,n) \in \mathcal{V}_{ij} \setminus \mathcal{N}} \varphi_{\alpha}(x_{i,j} - u_{m,n}).

    Here, \beta is a regularization parameter, and \varphi_{\alpha}(\cdot) is an even edge-preserving potential function with parameter \alpha > 0 . The vector \mathbf{x} = [x_{i, j}]_{(i, j) \in \mathcal{N}} is optimized lexicographically to achieve denoising. The regularization problem posed in the second phase is nonsmooth due to the data-fitting term |x_{i, j} - y_{i, j}| . To address this, Cai et al. [30] proposed removing the nonsmooth term and instead solving the following smooth unconstrained optimization problem:

    \min\limits_{\mathbf{x}} f_{\alpha}(\mathbf{x}) : = \sum\limits_{(i,j) \in \mathcal{N}} \left( 2\Phi_{i,j}^1 + \Phi_{i,j}^2 \right).

    The potential function \varphi_{\alpha}(\cdot) plays a crucial role in preserving edges while smoothing the image. A commonly used potential function is the Huber function, defined as:

    \varphi_{\alpha}(t) = \begin{cases} \frac{t^2}{2\alpha}, & \text{for } |t| \leq \alpha, \\ |t| - \frac{\alpha}{2}, & \text{for } |t| > \alpha. \end{cases}

    This function is convex and first-order Lipschitz continuous, making it suitable for the minimization problems described above. Let \nabla f_{\alpha}(\mathbf{x}) denote the gradient of the function f_{\alpha}(\mathbf{x}) . In alignment with Proposition 6 in [30], if \varphi_\alpha is convex, then \nabla f_{\alpha}(\mathbf{x}) is monotone.

    In this section, all parameters for these three algorithms are set as described in Section 4. The stopping criteria for these three algorithms are defined as follows:

    \begin{equation*} \frac{\|\mathbf{x}_k - \mathbf{x}_{k-1}\|}{\|\mathbf{x}_k\|} \leq \tau \quad \text{or} \quad \frac{|f_{\alpha}(\mathbf{x}_k)-f_{\alpha}(\mathbf{x}_{k-1})|}{|f_{\alpha}(\mathbf{x}_k)|} \leq \tau. \end{equation*}

    For this experiment, we utilize the well-known grayscale test images Man ( 1024 \times 1024 ) and Tank2 ( 512 \times 512 ), which are sourced from the website https://www.hlevkin.com. We examine the performance of these three algorithms by applying them to images corrupted with 30% and 70% impulse noise. The noisy images, as well as the images recovered by these three algorithms, are shown in Figures 4 and 5. The corresponding numerical results are provided in Table 8. Based on these figures and the table, we can draw the following conclusions: (1) Figures 4 and 5 demonstrate that all three algorithms successfully recover the images affected by 30% and 70% impulse noise; (2) Recovering an image with 30% impulse noise requires less CPU time and fewer iterations compared to recovering an image with 70% impulse noise; (3) Among these three algorithms, the ILR algorithm generally requires less CPU time and fewer iterations than the VRMILP and DFPRPMHS algorithms for a given level of impulse noise.

    Figure 4.  From left to right: A noisy image with 30% impulse noise and recovered images obtained by the ILR, VRMILP, and DFPRPMHS algorithms.
    Figure 5.  From left to right: A noisy image with 70% impulse noise and recovered images obtained by the ILR, VRMILP, and DFPRPMHS algorithms.
    Table 8.  The numerical results of the ILR, VRMILP, and DFPRPMHS algorithms.
    Algorithm Man Tank
    Noise: 30% Noise: 70% Noise: 30% Noise: 70%
    Niter/CPUT Niter/CPUT Niter/CPUT Niter/CPUT
    ILR 14/13.38 29/29.04 8/1.46 16/3.66
    VRMILP 22/28.26 39/55.89 18/5.47 20/6.52
    DFPRPMHS 27/38.08 42/46.95 17/2.59 18/3.92

     | Show Table
    DownLoad: CSV

    In this paper, we presented an improved LS-RMIL-type conjugate gradient projection algorithm aimed at efficiently solving systems of nonlinear equations with convex constraints. The proposed algorithm demonstrates several key advantages, including the ability to generate search directions that satisfy sufficient descent and trust-region properties independently of the line search approach. Additionally, the proposed algorithm only requires continuous and monotone assumptions for systems of nonlinear equations, which makes it applicable under less restrictive conditions compared to existing methods. We established the global convergence of the proposed algorithm without relying on the Lipschitz continuity assumption, further relaxing the conditions that need to be satisfied for successful implementation. Extensive numerical simulations, including large-scale systems of nonlinear equations and impulse noise image restoration problems, have shown that the proposed algorithm exhibits superior efficiency and stability compared to existing algorithms. These results indicate that the proposed algorithm is a promising and competitive approach, with significant potential for practical applications, such as image restoration.

    Yan Xia: Conceptualization, Investigation, Writing–original draft, Writing–review and editing, Funding acquisition; Xuejie Ma: Writing–review and editing, Funding acquisition; Dandan Li: Conceptualization, Funding acquisition, Writing–review and editing. All authors have read and approved the final version of the manuscript for publication.

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

    This work is supported by the Guangzhou Huashang College Daoshi Project (2024HSDS28).

    The authors declare that there are no conflicts of interest regarding the publication of this paper.

    [1] Kultz D (2005) Molecular and evolutionary basis of the cellular stress response. Annu Rev Physiol 67: 225-257. doi: 10.1146/annurev.physiol.67.040403.103635
    [2] Nassif M, Matus S, Castillo K, et al. (2010) Amyotrophic lateral sclerosis pathogenesis: a journey through the secretory pathway. Antioxid Redox Sign 13: 1955-1989. doi: 10.1089/ars.2009.2991
    [3] Schapira AH, Olanow CW, Greenamyre JT, et al. (2014) Slowing of neurodegeneration in Parkinson's disease and Huntington's disease: future therapeutic perspectives. Lancet 384: 545-555.
    [4] Massano J, Bhatia KP (2012) Clinical approach to Parkinson's disease: features, diagnosis, and principles of management. Cold Spring Harbor Perspect Med 2: a008870.
    [5] Chaudhuri KR, Odin P, Antonini A, et al. (2011) Parkinson's disease: the non-motor issues. Parkinsonism Relat D 17: 717-723. doi: 10.1016/j.parkreldis.2011.02.018
    [6] Greenamyre JT, Hastings TG (2004) Biomedicine. Parkinson's--divergent causes, convergent mechanisms. Science 304: 1120-1122.
    [7] Spillantini MG, Schmidt ML, Lee VM, et al. (1997) Alpha-synuclein in Lewy bodies. Nature 388: 839-840.
    [8] Baba M, Nakajo S, Tu PH, et al. (1998) Aggregation of alpha-synuclein in Lewy bodies of sporadic Parkinson's disease and dementia with Lewy bodies. Am J Pathol 152: 879-884.
    [9] Cox D, Carver JA, Ecroyd H (2014) Preventing alpha-synuclein aggregation: the role of the small heat-shock molecular chaperone proteins. BBA 1842: 1830-1843.
    [10] Bonifati V, Rizzu P, van Baren MJ, et al. (2003) Mutations in the DJ-1 gene associated with autosomal recessive early-onset parkinsonism. Science 299: 256-259.
    [11] Andersson FI, Werrell EF, McMorran L, et al. (2011) The effect of Parkinson's-disease-associated mutations on the deubiquitinating enzyme UCH-L1. J Mol Biol 407: 261-272. doi: 10.1016/j.jmb.2010.12.029
    [12] Dauer W, Przedborski S (2003) Parkinson's disease: mechanisms and models. Neuron 39: 889-909. doi: 10.1016/S0896-6273(03)00568-3
    [13] Dawson TM, Dawson VL (2010) The role of parkin in familial and sporadic Parkinson's disease. Movement Disord 25: S32-39. doi: 10.1002/mds.22798
    [14] Sidransky E, Lopez G (2012) The link between the GBA gene and parkinsonism. Lancet Neurol 11: 986-998.
    [15] Al-Chalabi A, Jones A, Troakes C, et al. (2012) The genetics and neuropathology of amyotrophic lateral sclerosis. Acta neuropathol 124: 339-352.
    [16] Rosen DR, Siddique T, Patterson D, et al. (1993) Mutations in Cu/Zn superoxide dismutase gene are associated with familial amyotrophic lateral sclerosis. Nature 362: 59-62. doi: 10.1038/362059a0
    [17] Neumann M, Sampathu DM, Kwong LK, et al. (2006) Ubiquitinated TDP-43 in frontotemporal lobar degeneration and amyotrophic lateral sclerosis. Science 314: 130-133.
    [18] Arai T, Hasegawa M, Akiyama H, et al. (2006) TDP-43 is a component of ubiquitin-positive tau-negative inclusions in frontotemporal lobar degeneration and amyotrophic lateral sclerosis. Biochem Biophys Res Commun 351: 602-611. doi: 10.1016/j.bbrc.2006.10.093
    [19] Deng HX, Zhai H, Bigio EH, et al. (2010) FUS-immunoreactive inclusions are a common feature in sporadic and non-SOD1 familial amyotrophic lateral sclerosis. Annals Neurol 67: 739-748.
    [20] Nishimura AL, Mitne-Neto M, Silva HC, et al. (2004) A mutation in the vesicle-trafficking protein VAPB causes late-onset spinal muscular atrophy and amyotrophic lateral sclerosis. Am J Hum Genet 75: 822-831.
    [21] Parkinson N, Ince PG, Smith MO, et al. (2006) ALS phenotypes with mutations in CHMP2B (charged multivesicular body protein 2B). Neurology 67: 1074-1077. doi: 10.1212/01.wnl.0000231510.89311.8b
    [22] Deng HX, Chen W, Hong ST, et al. (2011) Mutations in UBQLN2 cause dominant X-linked juvenile and adult-onset ALS and ALS/dementia. Nature 477: 211-215. doi: 10.1038/nature10353
    [23] Johnson JO, Mandrioli J, Benatar M, et al. (2010) Exome sequencing reveals VCP mutations as a cause of familial ALS. Neuron 68: 857-864. doi: 10.1016/j.neuron.2010.11.036
    [24] Maruyama H, Morino H, Ito H, et al. (2010) Mutations of optineurin in amyotrophic lateral sclerosis. Nature 465: 223-226. doi: 10.1038/nature08971
    [25] Fecto F, Yan J, Vemula SP, et al. (2011) SQSTM1 mutations in familial and sporadic amyotrophic lateral sclerosis. Arch Neurol 68:1440-1446. doi: 10.1001/archneurol.2011.250
    [26] Rubino E, Rainero I, Chio A, et al. (2012) SQSTM1 mutations in frontotemporal lobar degeneration and amyotrophic lateral sclerosis. Neurology 79: 1556-1562. doi: 10.1212/WNL.0b013e31826e25df
    [27] Teyssou E, Takeda T, Lebon V, et al. (2013) Mutations in SQSTM1 encoding p62 in amyotrophic lateral sclerosis: genetics and neuropathology. Acta Neuropathol 125: 511-522. doi: 10.1007/s00401-013-1090-0
    [28] Li J, Li W, Jiang ZG, et al. (2013) Oxidative stress and neurodegenerative disorders. Int J Mol Sci 14: 24438-24475. doi: 10.3390/ijms141224438
    [29] Ayala A, Munoz MF, Arguelles S (2014) Lipid peroxidation: production, metabolism, and signaling mechanisms of malondialdehyde and 4-hydroxy-2-nonenal. Oxidative Med Cell Longev: 360438.
    [30] Gandhi S, Abramov AY (2012) Mechanism of oxidative stress in neurodegeneration. Oxidative Med Cell Longev: 428010.
    [31] Halliwell B (2001) Role of free radicals in the neurodegenerative diseases. Drug Aging 18: 685-716 doi: 10.2165/00002512-200118090-00004
    [32] Halliwell B (2006) Oxidative stress and neurodegeneration: where are we now? J Neurochem 97: 1634-1658. doi: 10.1111/j.1471-4159.2006.03907.x
    [33] Milani P, Ambrosi G, Gammoh O, et al. (2013) SOD1 and DJ-1 converge at Nrf2 pathway: a clue for antioxidant therapeutic potential in neurodegeneration. Oxidative Med Cell Longev:836760.
    [34] Parakh S, Spencer DM, Halloran MA, et al. (2013) Redox regulation in amyotrophic lateral sclerosis. Oxidative Med Cell Longev: 408681.
    [35] Streck EL, Czapski GA, Goncalves et al. (2013) Neurodegeneration, mitochondrial dysfunction, and oxidative stress. Oxidative Med Cell Longev: 826046.
    [36] Varcin M, Bentea E, Michotte Y, et al. (2012) Oxidative stress in genetic mouse models of Parkinson's disease. Oxidative Med Cell Longev: 624925.
    [37] Navarro A, Boveris A, Bandez MJ, et al. (2009) Human brain cortex: mitochondrial oxidative damage and adaptive response in Parkinson disease and in dementia with Lewy bodies. Free Radical Biol Med 46: 1574-1580. doi: 10.1016/j.freeradbiomed.2009.03.007
    [38] Alam ZI, Jenner A, Daniel SE, et al. (1997) Oxidative DNA damage in the parkinsonian brain: an apparent selective increase in 8-hydroxyguanine levels in substantia nigra. J Neurochem 69: 1196-1203.
    [39] Abe T, Isobe C, Murata T, et al. (2003) Alteration of 8-hydroxyguanosine concentrations in the cerebrospinal fluid and serum from patients with Parkinson's disease. Neurosci Lett 336: 105-108. doi: 10.1016/S0304-3940(02)01259-4
    [40] Kikuchi A, Takeda A, Onodera H, et al. (2002) Systemic increase of oxidative nucleic acid damage in Parkinson's disease and multiple system atrophy. Neurobiol Dis 9: 244-248. doi: 10.1006/nbdi.2002.0466
    [41] Isobe C, Abe T, Terayama Y (2010) Levels of reduced and oxidized coenzyme Q-10 and 8-hydroxy-2'-deoxyguanosine in the cerebrospinal fluid of patients with living Parkinson's disease demonstrate that mitochondrial oxidative damage and/or oxidative DNA damage contributes to the neurodegenerative process. Neurosci Lett 469: 159-163. doi: 10.1016/j.neulet.2009.11.065
    [42] Nikam S, Nikam P, Ahaley SK, et al. (2009) Oxidative stress in Parkinson's disease. Indian J Clin Biochem 24: 98-101. doi: 10.1007/s12291-009-0017-y
    [43] Barber SC, Mead RJ, Shaw PJ (2006) Oxidative stress in ALS: a mechanism of neurodegeneration and a therapeutic target. Biochim Biophys Acta 1762: 1051-1067. doi: 10.1016/j.bbadis.2006.03.008
    [44] Barber SC, Shaw PJ (2010) Oxidative stress in ALS: key role in motor neuron injury and therapeutic target. Free Radical Boil Med 48: 629-641. doi: 10.1016/j.freeradbiomed.2009.11.018
    [45] Ferrante RJ, Browne SE, Shinobu LA, et al. (1997) Evidence of increased oxidative damage in both sporadic and familial amyotrophic lateral sclerosis. J Neurochem 69: 2064-2074.
    [46] Cutler RG, Pedersen WA, Camandola S (2002) Evidence that accumulation of ceramides and cholesterol esters mediates oxidative stress-induced death of motor neurons in amyotrophic lateral sclerosis. Ann Neurol 52: 448-457. doi: 10.1002/ana.10312
    [47] Pedersen WA, Fu W, Keller JN, et al. (1998) Protein modification by the lipid peroxidation product 4-hydroxynonenal in the spinal cords of amyotrophic lateral sclerosis patients. Ann Neurol 44: 819-824.
    [48] Abe K, Pan LH, Watanabe M, et al. (1995) Induction of nitrotyrosine-like immunoreactivity in the lower motor neuron of amyotrophic lateral sclerosis. Neurosci Lett 199: 152-154. doi: 10.1016/0304-3940(95)12039-7
    [49] Beal MF, Ferrante RJ, Browne SE, et al. (1997) Increased 3-nitrotyrosine in both sporadic and familial amyotrophic lateral sclerosis. Ann Neurol 42: 644-654. doi: 10.1002/ana.410420416
    [50] Shaw PJ, Ince PG, Falkous G, et al. (1995) Oxidative damage to protein in sporadic motor neuron disease spinal cord. Ann Neurol 38: 691-695. doi: 10.1002/ana.410380424
    [51] Fitzmaurice PS, Shaw IC, Kleiner HE, et al. (1996) Evidence for DNA damage in amyotrophic lateral sclerosis. Muscle Nerve 19: 797-798.
    [52] Said Ahmed M, Hung WY, Zu JS, et al. (2000) Increased reactive oxygen species in familial amyotrophic lateral sclerosis with mutations in SOD1. J neurol Sci 176: 88-94. doi: 10.1016/S0022-510X(00)00317-8
    [53] Milani P, Amadio M, Laforenza U, et al. (2013) Posttranscriptional regulation of SOD1 gene expression under oxidative stress: Potential role of ELAV proteins in sporadic ALS. Neurobiol Dis 60: 51-60.
    [54] Cereda C, Leoni E, Milani P, et al. (2013) Altered intracellular localization of SOD1 in leukocytes from patients with sporadic amyotrophic lateral sclerosis. PlOS One 8: e75916. doi: 10.1371/journal.pone.0075916
    [55] Smith RG, Henry YK, Mattson MP, et al. (1998) Presence of 4-hydroxynonenal in cerebrospinal fluid of patients with sporadic amyotrophic lateral sclerosis. Ann Neurol 44: 696-699. doi: 10.1002/ana.410440419
    [56] Ihara Y, Nobukuni K, Takata H, et al. (2005) Oxidative stress and metal content in blood and cerebrospinal fluid of amyotrophic lateral sclerosis patients with and without a Cu, Zn-superoxide dismutase mutation. Neurol Res 27: 105-108. doi: 10.1179/016164105X18430
    [57] Kirby J, Halligan E, Baptista MJ, et al. (2005) Mutant SOD1 alters the motor neuronal transcriptome: implications for familial ALS. Brain 128: 1686-1706.
    [58] Mimoto T, Miyazaki K, Morimoto N, et al. (2012) Impaired antioxydative Keap1/Nrf2 system and the downstream stress protein responses in the motor neuron of ALS model mice. Brain Res 1446: 109-118. doi: 10.1016/j.brainres.2011.12.064
    [59] Petri S, Korner S, Kiaei M (2012) Nrf2/ARE Signaling Pathway: Key Mediator in Oxidative Stress and Potential Therapeutic Target in ALS. Neurol Res Int: 878030.
    [60] Sarlette A, Krampfl K, Grothe C, et al. (2008) Nuclear erythroid 2-related factor 2-antioxidative response element signaling pathway in motor cortex and spinal cord in amyotrophic lateral sclerosis. J Neuropath Exp Neurol 67: 1055-1062. doi: 10.1097/NEN.0b013e31818b4906
    [61] Cao SS, Kaufman RJ (2014) Endoplasmic reticulum stress and oxidative stress in cell fate decision and human disease. Antioxid Redox Signaling 21: 396-413. doi: 10.1089/ars.2014.5851
    [62] Begum G, Harvey L, Dixon CE, et al. (2013) ER stress and effects of DHA as an ER stress inhibitor. Translational Stroke Res 4: 635-642. doi: 10.1007/s12975-013-0282-1
    [63] Bellucci A, Navarria L, Zaltieri M, et al. (2011) Induction of the unfolded protein response by alpha-synuclein in experimental models of Parkinson's disease. J Neurochem 116: 588-605. doi: 10.1111/j.1471-4159.2010.07143.x
    [64] Colla E, Jensen PH, Pletnikova O, et al. (2012) Accumulation of toxic alpha-synuclein oligomer within endoplasmic reticulum occurs in alpha-synucleinopathy in vivo. J Neurosci 32: 3301-3305. doi: 10.1523/JNEUROSCI.5368-11.2012
    [65] Nishitoh H, Kadowaki H, Nagai A, et al. (2008) ALS-linked mutant SOD1 induces ER stress- and ASK1-dependent motor neuron death by targeting Derlin-1. Genes Dev 22: 1451-1464. doi: 10.1101/gad.1640108
    [66] Atkin JD, Farg MA, Soo KY, et al. (2014) Mutant SOD1 inhibits ER-Golgi transport in amyotrophic lateral sclerosis. J Neurochem 129: 190-204. doi: 10.1111/jnc.12493
    [67] Hetz C, Mollereau B (2014) Disturbance of endoplasmic reticulum proteostasis in neurodegenerative diseases. Nat Rev Neurosci 15: 233-249.
    [68] Mercado G, Valdes P, Hetz C (2013) An ERcentric view of Parkinson's disease. Trends Mol Med 19: 165-175. doi: 10.1016/j.molmed.2012.12.005
    [69] Hoozemans JJ, van Haastert ES, Eikelenboom P, et al. (2007) Activation of the unfolded protein response in Parkinson's disease. Biochem Bioph Res Commun 354: 707-711. doi: 10.1016/j.bbrc.2007.01.043
    [70] Slodzinski H, Moran LB, Michael GJ, et al. (2009) Homocysteine-induced endoplasmic reticulum protein (herp) is up-regulated in parkinsonian substantia nigra and present in the core of Lewy bodies. Clin Neuropathol 28: 333-343.
    [71] Holtz WA, Turetzky JM, Jong YJ, et al. (2006) Oxidative stress-triggered unfolded protein response is upstream of intrinsic cell death evoked by parkinsonian mimetics. J Neurochem 99: 54-69. doi: 10.1111/j.1471-4159.2006.04025.x
    [72] Dukes AA, Van Laar VS, Cascio M, et al. (2008) Changes in endoplasmic reticulum stress proteins and aldolase A in cells exposed to dopamine. J Neurochem 106: 333-346. doi: 10.1111/j.1471-4159.2008.05392.x
    [73] Tinsley RB, Bye CR, Parish CL, et al. (2009) Dopamine D2 receptor knockout mice develop features of Parkinson disease. Ann Neurol 66: 472-484. doi: 10.1002/ana.21716
    [74] Mercado G, Castillo V, Soto P, et al. (2016) ER stress and Parkinson's disease: Pathological inputs that converge into the secretory pathway. Brain Res 1648: 626-632. doi: 10.1016/j.brainres.2016.04.042
    [75] Walker AK, Atkin JD (2011) Stress signaling from the endoplasmic reticulum: A central player in the pathogenesis of amyotrophic lateral sclerosis. IUBMB Life 63: 754-763.
    [76] Hetz C, Thielen P, Matus S, et al. (2009) XBP-1 deficiency in the nervous system protects against amyotrophic lateral sclerosis by increasing autophagy. Genes Dev 23: 2294-2306. doi: 10.1101/gad.1830709
    [77] Wang L, Popko B, Roos RP (2014) An enhanced integrated stress response ameliorates mutant SOD1-induced ALS. Hum Mol Genet 23: 2629-2638. doi: 10.1093/hmg/ddt658
    [78] Carreras-Sureda A, Pihan P, Hetz C (2017) The Unfolded Protein Response: At the Intersection between Endoplasmic Reticulum Function and Mitochondrial Bioenergetics. Front Oncol 7: 55.
    [79] Erpapazoglou Z, Mouton-Liger F, Corti O (2017) From dysfunctional endoplasmic reticulum-mitochondria coupling to neurodegeneration. Neurochem Int.
    [80] Eletto D, Chevet E, Argon Y, et al. (2014) Redox controls UPR to control redox. J Cell Sci 127: 3649-3658. doi: 10.1242/jcs.153643
    [81] Zhang K, Kaufman RJ (2008) From endoplasmic-reticulum stress to the inflammatory response. Nature 454: 455-462.
    [82] Tu BP, Weissman JS (2004) Oxidative protein folding in eukaryotes: mechanisms and consequences. J Cell Biol 164: 341-346. doi: 10.1083/jcb.200311055
    [83] Cuozzo JW, Kaiser CA (1999) Competition between glutathione and protein thiols for disulphide-bond formation. Nat Cell Biol 1: 130-135. doi: 10.1038/11047
    [84] Perri E, Parakh S, Atkin J (2017) Protein Disulphide Isomerases: emerging roles of PDI and ERp57 in the nervous system and as therapeutic targets for ALS. Exp Opin Targets 21: 37-49. doi: 10.1080/14728222.2016.1254197
    [85] Perri ER, Thomas CJ, Parakh S, et al. (2015) The Unfolded Protein Response and the Role of Protein Disulfide Isomerase in Neurodegeneration. Front Cell Dev Biol 3: 80.
    [86] Chaudhari N, Talwar P, Parimisetty A, et al. (2014) A molecular web: endoplasmic reticulum stress, inflammation, and oxidative stress. Front Cell Neurosci 8: 213.
    [87] Chiribau CB, Gaccioli F, Huang CC, et al. (2010) Molecular symbiosis of CHOP and C/EBP beta isoform LIP contributes to endoplasmic reticulum stress-induced apoptosis. Mol Cell Biol 30: 3722-3731. doi: 10.1128/MCB.01507-09
    [88] Yamaguchi H, Wang HG (2004) CHOP is involved in endoplasmic reticulum stress-induced apoptosis by enhancing DR5 expression in human carcinoma cells. J Biol Chem 279: 45495-45502. doi: 10.1074/jbc.M406933200
    [89] Lu M, Lawrence DA, Marsters S, et al. (2014) Cell death. Opposing unfolded-protein-response signals converge on death receptor 5 to control apoptosis. Science 345: 98-101.
    [90] Li G, Mongillo M, Chin KT, et al (2009) Role of ERO1-alpha-mediated stimulation of inositol 1,4,5-triphosphate receptor activity in endoplasmic reticulum stress-induced apoptosis. J Cell biol 186: 783-792.
    [91] Marciniak SJ, Yun CY, Oyadomari S, et al. (2004) CHOP induces death by promoting protein synthesis and oxidation in the stressed endoplasmic reticulum. Gene Dev 18: 3066-3077. doi: 10.1101/gad.1250704
    [92] Chen G, Bower KA, Ma C, et al. (2004) Glycogen synthase kinase 3beta (GSK3beta) mediates 6-hydroxydopamine-induced neuronal death. FASEB J 18: 1162-1164.
    [93] McNeill A, Magalhaes J, Shen C, et al. (2014) Ambroxol improves lysosomal biochemistry in glucocerebrosidase mutation-linked Parkinson disease cells. Brain 137: 1481-1495.
    [94] Prell T, Lautenschlager J, Weidemann L, et al. (2014) Endoplasmic reticulum stress is accompanied by activation of NF-kappaB in amyotrophic lateral sclerosis. J Neuroimmunol 270: 29-36. doi: 10.1016/j.jneuroim.2014.03.005
    [95] Yang W, Tiffany-Castiglioni E, Koh HC, et al. (2009) Paraquat activates the IRE1/ASK1/JNK cascade associated with apoptosis in human neuroblastoma SH-SY5Y cells. Toxicol Lett 191: 203-210. doi: 10.1016/j.toxlet.2009.08.024
    [96] Chang L, Karin M (2001) Mammalian MAP kinase signalling cascades. Nature 410: 37-40. doi: 10.1038/35065000
    [97] Darling NJ, Cook SJ (2014) The role of MAPK signalling pathways in the response to endoplasmic reticulum stress. BBA 1843: 2150-2163.
    [98] Davis RJ (2000): Signal transduction by the JNK group of MAP kinases. Cell 103: 239-252.
    [99] Abais JM, Xia M, Zhang Y, et al. (2014) Redox Regulation of NLRP3 Inflammasomes: ROS as Trigger or Effector? Antioxid Redox Signaling 22: 1111-1129.
    [100] Jope RS, Yuskaitis CJ, Beurel E (2007) Glycogen synthase kinase-3 (GSK3): inflammation, diseases, and therapeutics. Neurochem Res 32: 577-595. doi: 10.1007/s11064-006-9128-5
    [101] Nijholt DA, Nolle A, van Haastert ES, et al. (2013) Unfolded protein response activates glycogen synthase kinase-3 via selective lysosomal degradation. Neurobiol Aging 34: 1759-1771. doi: 10.1016/j.neurobiolaging.2013.01.008
    [102] Meares GP, Mines MA, Beurel E, et al. (2011) Glycogen synthase kinase-3 regulates endoplasmic reticulum (ER) stress-induced CHOP expression in neuronal cells. Exp Cell Res 317: 1621-1628. doi: 10.1016/j.yexcr.2011.02.012
    [103] Giordano S, Darley-Usmar V, Zhang J (2014) Autophagy as an essential cellular antioxidant pathway in neurodegenerative disease. Redox Biol 2: 82-90. doi: 10.1016/j.redox.2013.12.013
    [104] Loos B, Engelbrecht AM, Lockshin RA, et al. (2013) The variability of autophagy and cell death susceptibility: Unanswered questions. Autophagy 9: 1270-1285. doi: 10.4161/auto.25560
    [105] Scheper W, Nijholt DA, Hoozemans JJ (2011) The unfolded protein response and proteostasis in Alzheimer disease: preferential activation of autophagy by endoplasmic reticulum stress. Autophagy 7: 910-911. doi: 10.4161/auto.7.8.15761
    [106] Deegan S, Saveljeva S, Logue SE, et al. (2014) Deficiency in the mitochondrial apoptotic pathway reveals the toxic potential of autophagy under ER stress conditions. Autophagy 10: 1921-1936. doi: 10.4161/15548627.2014.981790
    [107] Madeo F, Eisenberg T, Kroemer G (2009) Autophagy for the avoidance of neurodegeneration. Gene Dev 23: 2253-2259. doi: 10.1101/gad.1858009
    [108] Cai Y, Arikkath J, Yang L, et al. (2016) Interplay of endoplasmic reticulum stress and autophagy in neurodegenerative disorders. Autophagy 12: 225-244.
    [109] Cortes CJ, Miranda HC, Frankowski H, et al. (2014) Polyglutamine-expanded androgen receptor interferes with TFEB to elicit autophagy defects in SBMA. Nat Neurosci 17: 1180-1189. doi: 10.1038/nn.3787
    [110] Palmieri M, Impey S, Kang H, et al. (2011) Characterization of the CLEAR network reveals an integrated control of cellular clearance pathways. Hum Mol Genet 20: 3852-3866. doi: 10.1093/hmg/ddr306
    [111] Brehme M, Voisine C, Rolland T, et al. (2014) A chaperome subnetwork safeguards proteostasis in aging and neurodegenerative disease. Cell Rep 9: 1135-1150. doi: 10.1016/j.celrep.2014.09.042
    [112] Genereux JC, Qu S, Zhou M, et al. (2014) Unfolded protein response-induced ERdj3 secretion links ER stress to extracellular proteostasis. EMBO J.
    [113] Montane J, Cadavez L, Novials A (2014) Stress and the inflammatory process: a major cause of pancreatic cell death in type 2 diabetes. Diabetes, metab syndrome obesity: targets ther 7: 25-34.
    [114] Song W, Wang F, Savini M, et al. (2013) TFEB regulates lysosomal proteostasis. Hum Mol Genet 22: 1994-2009.
    [115] Tan YL, Genereux JC, Pankow S, et al. (2014) ERdj3 is an endoplasmic reticulum degradation factor for mutant glucocerebrosidase variants linked to Gaucher's disease. Chem Biol 21: 967-976. doi: 10.1016/j.chembiol.2014.06.008
    [116] Wei H, Kim SJ, Zhang Z, et al. (2008) ER and oxidative stresses are common mediators of apoptosis in both neurodegenerative and non-neurodegenerative lysosomal storage disorders and are alleviated by chemical chaperones. Hum Mol Genet 17: 469-477.
    [117] Sybertz E, Krainc D (2014) Development of targeted therapies for Parkinson's disease and related synucleinopathies. J Lipid Res 55: 1996-2003. doi: 10.1194/jlr.R047381
    [118] Duplan E, Giaime E, Viotti J, et al. (2013) ER-stress-associated functional link between Parkin and DJ-1 via a transcriptional cascade involving the tumor suppressor p53 and the spliced X-box binding protein XBP-1. J Cell Sci 126: 2124-2133. doi: 10.1242/jcs.127340
    [119] Yokota T, Sugawara K, Ito K, et al. (2003) Down regulation of DJ-1 enhances cell death by oxidative stress, ER stress, and proteasome inhibition. Biochem Biophys Res Commun 312: 1342-1348. doi: 10.1016/j.bbrc.2003.11.056
    [120] Sajjad MU, Green EW, Miller-Fleming L, et al. (2014) DJ-1 modulates aggregation and pathogenesis in models of Huntington's disease. Hum Mol Genet 23: 755-766.
    [121] Shendelman S, Jonason A, Martinat C, et al. (2004) DJ-1 is a redox-dependent molecular chaperone that inhibits alpha-synuclein aggregate formation. PLOS Biol 2: e362. doi: 10.1371/journal.pbio.0020362
    [122] Jarvela TS, Lam HA, Helwig M, et al. (2016) The neural chaperone proSAAS blocks alpha-synuclein fibrillation and neurotoxicity. P Natl Acad Sci UAS 113: E4708-4715. doi: 10.1073/pnas.1601091113
    [123] Carra S, Rusmini P, Crippa V, et al. (2013) Different anti-aggregation and pro-degradative functions of the members of the mammalian sHSP family in neurological disorders. Phil Trans R Soc B 368: 20110409.
    [124] Chaari A, Hoarau-Vechot J, Ladjimi M (2013) Applying chaperones to protein-misfolding disorders: molecular chaperones against alpha-synuclein in Parkinson's disease. Int J Boil macromolecules 60: 196-205. doi: 10.1016/j.ijbiomac.2013.05.032
    [125] Fontaine SN, Martin MD, Dickey CA (2016) Neurodegeneration and the Heat Shock Protein 70 Machinery: Implications for Therapeutic Development. Curr Top Med Chem 16: 2741-2752. doi: 10.2174/1568026616666160413140741
    [126] Lindberg I, Shorter J, Wiseman RL (2015) Chaperones in Neurodegeneration. J Neurosci 35: 13853-13859. doi: 10.1523/JNEUROSCI.2600-15.2015
    [127] Chen S, Brown IR (2007) Neuronal expression of constitutive heat shock proteins: implications for neurodegenerative diseases. Cell Stress Chaperon 12: 51-58. doi: 10.1379/CSC-236R.1
    [128] Galbiati M, Crippa V, Rusmini P, et al. (2014) ALS-related misfolded protein management in motor neurons and muscle cells. Neurochem Int 79: 70-78. doi: 10.1016/j.neuint.2014.10.007
    [129] Papsdorf K, Richter K (2014) Protein folding, misfolding and quality control: the role of molecular chaperones. Essays Biochem 56: 53-68. doi: 10.1042/bse0560053
    [130] Baluchnejadmojarad T, Roghani M, Nadoushan MR, et al. (2009) Neuroprotective effect of genistein in 6-hydroxydopamine hemi-parkinsonian rat model. Phytother Res 23: 132-135. doi: 10.1002/ptr.2564
    [131] Choi BS, Kim H, Lee HJ, et al. (2014) Celastrol from 'Thunder God Vine' protects SH-SY5Y cells through the preservation of mitochondrial function and inhibition of p38 MAPK in a rotenone model of Parkinson's disease. Neurochem Res 39: 84-96. doi: 10.1007/s11064-013-1193-y
    [132] Inden M, Kitamura Y, Takeuchi H, et al. (2007) Neurodegeneration of mouse nigrostriatal dopaminergic system induced by repeated oral administration of rotenone is prevented by 4-phenylbutyrate, a chemical chaperone. J Neurochem 101: 1491-1504. doi: 10.1111/j.1471-4159.2006.04440.x
    [133] Jiang HQ, Ren M, Jiang HZ, et al. (2014) Guanabenz delays the onset of disease symptoms, extends lifespan, improves motor performance and attenuates motor neuron loss in the SOD1 G93A mouse model of amyotrophic lateral sclerosis. Neurosci 277: 132-138.
    [134] Mortiboys H, Aasly J, Bandmann O (2013) Ursocholanic acid rescues mitochondrial function in common forms of familial Parkinson's disease. Brain 136: 3038-3050.
    [135] Ono K, Ikemoto M, Kawarabayashi T, et al. (2009) A chemical chaperone, sodium 4-phenylbutyric acid, attenuates the pathogenic potency in human alpha-synuclein A30P + A53T transgenic mice. Parkinsonism Relat D 15: 649-654.
    [136] Ozsoy O, Seval-Celik Y, Hacioglu G, et al. (2011) The influence and the mechanism of docosahexaenoic acid on a mouse model of Parkinson's disease. Neurochem Int 59: 664-670. doi: 10.1016/j.neuint.2011.06.012
    [137] Richter F, Fleming SM, Watson M, et al. (2014) A GCase chaperone improves motor function in a mouse model of synucleinopathy. Neurotherapeutics 11: 840-856. doi: 10.1007/s13311-014-0294-x
    [138] Saxena S, Cabuy E, Caroni P (2009) A role for motoneuron subtype-selective ER stress in disease manifestations of FALS mice. Nature Neurosci 12: 627-636. doi: 10.1038/nn.2297
    [139] Kameta N, Masuda M, Shimizu T (2012) Soft nanotube hydrogels functioning as artificial chaperones. ACS Nano 6: 5249-5258. doi: 10.1021/nn301041y
    [140] Song W, Soo Lee S, Savini M, et al. (2014) Ceria nanoparticles stabilized by organic surface coatings activate the lysosome-autophagy system and enhance autophagic clearance. ACS Nano 8: 10328-10342. doi: 10.1021/nn505073u
    [141] Wang W, Sreekumar PG, Valluripalli V, et al. (2014) Protein polymer nanoparticles engineered as chaperones protect against apoptosis in human retinal pigment epithelial cells. J Controlled release 191: 4-14. doi: 10.1016/j.jconrel.2014.04.028
    [142] Liao YH, Chang YJ, Yoshiike Y, et al. (2012) Negatively charged gold nanoparticles inhibit Alzheimer's amyloid-beta fibrillization, induce fibril dissociation, and mitigate neurotoxicity. Small 8: 3631-3639. doi: 10.1002/smll.201201068
    [143] Palmal S, Maity AR, Singh BK, et al. (2014) Inhibition of amyloid fibril growth and dissolution of amyloid fibrils by curcumin-gold nanoparticles. Chemistry 20: 6184-6191. doi: 10.1002/chem.201400079
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