This work presents a new iterative method for solving single-variable nonlinear equations. The method achieves ninth-order convergence with just three derivative evaluations per step, offering both accuracy and lower computational cost. Unlike slower bracketing methods, it builds on faster open methods, though these may sometimes fail to converge. By blending ideas from Newton's and Halley's methods, the new approach provides strong performance, as shown by a detailed convergence analysis and MATLAB tests. Compared to existing techniques, it finds solutions in fewer steps and less time, making it especially effective for difficult nonlinear problems

This study introduces a new spectral conjugate gradient (SCG) method for solving large-scale nonlinear unconstrained optimisation problems in structural engineering. The method guarantees sufficient descent directions and global convergence under standard assumptions. Numerical tests show its improved efficiency over the classic Dai-Yuan algorithm. The proposed method is applied to train a feedforward neural network to predict the seismic vulnerability of existing multistorey buildings with masonry infills in Zakho City, Kurdistan Region of Iraq. Using key structural and geotechnical parameters, the network trained with the new SCG method converges faster with the reduction in execution time by more than 35% compared to Dai-Yuan algorithm. This work demonstrates the potential of combining advanced optimisation with artificial neural networks (ANN) for effective seismic risk assessment in earthquake-prone regions. New ANN model is formulated as a straightforward expression in terms properties of the buildings without using a tedious process of codes for building vulnerability.

This study proposes a new Conjugate Gradient (CG) method that is both theoretically sound and computationally efficient for solving large-scale unconstrained optimization problems. The method integrates the Dai-Liao (DL) parameter with a gradient-difference vector formulation inspired by Powell (1978) and incorporates the Barzilai-Borwein step size to enhance convergence performance. Under standard assumptions, the proposed method is proven to satisfy the sufficient descent condition and to guarantee global convergence. Extensive numerical experiments demonstrate that the new method outperforms classical CG algorithms including the Hestenes-Stiefel (HS) and DL methods in terms of iteration count, function evaluations, and computational time. Furthermore, the method is applied to train a feedforward neural network aimed at predicting the seismic vulnerability of 641 multistory buildings in Zakho City, Iraq. Using a dataset of structural and geotechnical features, the neural network trained with the proposed CG method achieves faster convergence and maintains high prediction accuracy, confirming the method's practical utility in real-world applications.

This work examines novel conjugate gradient methods. To address unconstrained optimization problems and optimize the training of neural networks. The methodologies employ advanced derivative-based techniques to enhance optimization outcomes. The novel approach employs any line search to guarantee adequate descent. Moreover, we prove that, given specific assumptions, our proposed method converges universally. Experimental evidence has demonstrated that our proposed approach is superior in terms of efficiency and robustness compared to traditional conjugate gradient approaches for training neural networks and solving unconstrained optimization issues.

The development of a new conjugate gradient method, derived from the modified difference gradient vector and based on an approximation of the Hessian matrix, marks a significant advancement in optimization algorithms. This study shows that the robustness of the proposed method, particularly in applications such as compressive sensing and image processing. Its global convergence properties and sufficient descent conditions ensure reliable performance even in computationally challenging scenarios. Extensive numerical experiments in signal recovery, image restoration, and unconstrained optimization clearly illustrate the superiority of the proposed method over existing approaches, including traditional conjugate gradient techniques. The proposed CG method not only boosts computational efficiency but also significantly enhances the quality of solutions for both signal recovery and image restoration. These promising results emphasize the method's potential as a versatile and powerful tool for solving a wide range of practical problems within computational mathematics, optimization, and signal processing.

Deep neural networks (DNNs) are widely used for large-scale learning tasks because of their ability to model complex relationships within data. The Adaptive Moment Estimation (Adam) optimizer is a popular choice for training DNNs; however, its generalization performance can be suboptimal on challenging datasets. To address this limitation, we propose three modified Adam variants (Adam-V1, Adam-V2, and Adam-V3) that incorporate a projection-based gradient-correction mechanism inspired by quasi-Newton and conjugate gradient methods. This correction introduces curvature awareness without requiring full Hessian computations, improving convergence stability and reducing the tendency to settle at sharp or poorly generalizing minima. The proposed methods were systematically evaluated on both low- and high-dimensional tasks, including one- and two-variable non-convex functions, two-dimensional image segmentation, image classification using CNNs on MNIST, CIFAR- 10, and the more challenging CIFAR-100 datasets, as well as ResNet-based architectures on CIFAR-10. In addition, robustness on non-stationary real-world signals was assessed through ECG beat classification using the MIT-BIH Arrhythmia dataset. Experimental results demonstrate consistent improvements over baseline Adam. On CNN models trained on MNIST, Adam-V2 achieved the highest accuracy of 97.93 %, surpassing standard Adam (96.48 %) and highlighting the benefit of combining gradient correction with adaptive step-size adjustment in lower-dimensional settings. For CNNs trained on CIFAR-10, Adam-V3 attained a validation accuracy of 73.59 %, improving generalization relative to Adam (72.44 %). On the more complex CIFAR-100 dataset, the proposed variants consistently outperformed baseline Adam and recent adaptive optimizers in terms of accuracy and F1-score. Using a ResNet-50 model on CIFAR-10, Adam-V1 reached the highest accuracy of 79.9 %, while Adam-V3 achieved the best F1-score of 0.704, demonstrating strong performance in deeper network architectures. These results show that curvature-aware gradient corrections enhance convergence speed, stability, and generalization in deep learning tasks with minimal additional computational overhead. The proposed optimizers offer practical advantages for both shallow and deep architectures, providing a simple and effective improvement to existing adaptive optimization methods.

This paper presents a new conjugate gradient algorithm developed to address nonlinear unconstrained optimization problems. The proposed algorithm demonstrates substantial efficacy by fulfilling sufficient descent conditions, ensuring global convergence under mild assumptions. Furthermore, the new method is applied to effective signal recovery in compressive sensing, where it reconstructs sparse signals from limited measurements. The method enhances solution sparsity by reformulating the recovery problem as a convex optimization problem with a regularization component. The performance of our method is compared with established conjugate gradient methods, both for solving unconstrained optimization problems and for signal recovery. The results show that the new method significantly improves the efficiency and accuracy of sparse signal recovery, making it particularly suitable for real-time signal processing tasks. This research offers valuable insights into optimization techniques, highlighting the considerable potential of the proposed method to tackle nonlinear optimization problems and facilitate innovative applications in practical contexts.

In this paper, a three-term conjugate gradient method is proposed as an advanced optimization technique for enhancing ECG signal classification using deep learning. This method is designed for solving unconstrained optimization problems by leveraging a modified gradient difference vector while ensuring sufficient descent and convergence properties. The results demonstrate that the proposed method outperforms traditional techniques in terms of the number of iterations, function evaluations, and computational time required to reach a solution. It was also applied to train deep neural networks for ECG signal classification, achieving an accuracy of 96.32%, which is a significant improvement over existing models. This integration of deep learning and advanced gradient-based optimization techniques leads to higher classification accuracy for ECG signals, with a notable reduction in mean squared error, indicating better generalization capability and reduced prediction errors compared to other methods. These findings highlight the potential of the proposed approach in improving ECG classification accuracy, contributing to more efficient and precise diagnosis of heart diseases, ultimately assisting healthcare professionals in making faster and more accurate clinical decisions.

This work presents a new iterative method for solving single-variable nonlinear equations. The method achieves ninth-order convergence with just three derivative evaluations per step, offering both accuracy and lower computational cost. Unlike slower bracketing methods, it builds on faster open methods, though these may sometimes fail to converge. By blending ideas from Newton's and Halley's methods, the new approach provides strong performance, as shown by a detailed convergence analysis and MATLAB tests. Compared to existing techniques, it finds solutions in fewer steps and less time, making it especially effective for difficult nonlinear problems

Conjugate gradient methods are widely recognized for their efficiency in solving large-scale optimization problems with minimal memory requirements. This paper introduces a new CG algorithm derived from a modified secant condition, utilizing a redefined difference gradient vector to improve numerical stability and optimization performance. By integrating this modified vector into the computation of the conjugacy parameter we propose a new parmeter, which effectively resolves the issues of non-negativity and convergence that are encountered in existing methods. Numerical experiments confirm the superior performance of the proposed method on unconstrained optimization problems across various dimensions, demonstrating faster convergence rates, fewer iterations, and reduced computational times compared to traditional CG (HS, DL, and LS) methods. Additionally, the algorithm's practical effectiveness is validated in medical image restoration tasks, specifically in reducing salt-and-pepper noise in grayscale and color images. When evaluated against established methods like HS, FHTTCGM-N, and HZ, the proposed algorithm achieves notable improvements in metrics such as iteration count, computational time, mean squared error, peak signal-to-noise ratio, and structural similarity index. These results underscore the algorithms significant contributions to both the theoretical and practical advancements of CG methods in optimization and image restoration.

This study presents a new optimization technique, the three-term conjugate gradient method, which enhances the efficiency of solving unconstrained optimization problems and its application in training artificial neural networks to predict heart diseases. It incorporates a gradient condition and a modified gradient difference vector. The method is validated through its global convergence and sufficient condition, ensuring both mathematical accuracy and computational stability. Compared to traditional conjugate gradient methods and three-term conjugate gradient methods, the proposed method demonstrates superior performance, characterized by fewer iterations, reduced function evaluations, and lower computational time. The method is also used to teach artificial neural networks how to predict heart disease, with an 88.15% success rate and precision, recall, and F1-Score values of 0.87, 0.86, and 0.87, respectively. These results confirm the method’s effectiveness in enhancing the predictive power of artificial neural networks making it a valuable tool for clinical applications. This study underscores the potential of the new method to improve decision-support systems in healthcare, aiding early detection and better patient outcomes. Future research may focus on refining the method and integrating it with other machine learning techniques to create advanced real-time predictive systems that further advance intelligent healthcare solutions.

In this study, we present a novel conjugate gradient method specifically designed for addressing with unconstrained optimization problems. Traditional conjugate gradient methods have shown effectiveness in solving optimization problems, but they may encounter challenges when dealing with unconstrained problems. Our method addresses this issue by introducing modifications that enhance its performance in the unconstrained setting. We demonstrate that, under certain conditions, our method satisfies both the descent and the sufficient descent criteria and establishes global convergence, ensuring progress towards the optimal solution at each iteration. Moreover, we establish the global convergence of our method, providing confidence in its ability to find the global optimum. To showcase the practical applicability of our approach, we apply this novel method to a dataset, applying a feed-forward neural network value estimation for continuous trigonometric function value estimation. To evaluate the efficiency and effectiveness of our modified approach, we conducted numerical experiments on a set of well-known test functions. These experiments reveal that our algorithm significantly reduces computational time due to its faster convergence rates and increased speed in directional minimization. These compelling results highlight the advantages of our approach over traditional conjugate gradient methods in the context of unconstrained optimization problems.

This work introduces a new variation of the Hestenes and Stiefel nonlinear conjugate gradient (HS) method by combining the advantages of the spectral conjugate gradient method and the conjugacy condition of the quasi-Newton method. The proposed method incorporates inexact line searches and categorizing it as a descent method. By employing line searches that satisfy the Wolfe conditions, we establish sufficient descent properties and global convergence condition, assuming that the appropriate conditions are met. Additionally, we perform numerical experiments utilizing benchmark functions frequently used in optimization assignments to evaluate the effectiveness of the proposed method. The results demonstrate that our method outperforms the traditional HS method. Furthermore, we successfully implement the newly developed technique to train neural networks (NNs), demonstrating its practicality for non-traditional optimization tasks.

The method of optimization is used to determine the most precise value for certain functions within a certain domain; it is mostly studied and employed in the fields of mathematics, computer science, and physics. This work presents a novel three-term conjugate gradient (CG) approach for unconstrained optimization problems. Both the descending criteria and the sufficient descent criterion were met by the new approach. The novel method that has been proposed has been evaluated for global convergence. The outcomes of numerical trials on a few well-known test functions demonstrated how highly successful our new modified method is, depending on the number of iterations (NOI) and the number of functions to be evaluated (NOF).

Optimization refers to the process of finding the best possible solution to a problem within a given set of constraints. It involves maximizing or minimizing a specific objective function while adhering to specific constraints. Optimization is used in various fields, including mathematics, engineering, economics, computer science, and data science, among others. The objective function can be a simple equation, a complex algorithm, or a mathematical model that describes a system or process. There are various optimization techniques available, including linear programming, nonlinear programming, genetic algorithms, simulated annealing, and particle swarm optimization, among others. These techniques use different algorithms to search for the optimal solution to a problem. In this paper, the main goal of unconstrained optimization is to minimize an objective function that uses real variables and has no value restrictions. In this study, based on the modified conjugacy condition, we offer a new conjugate gradient (CG) approach for nonlinear unconstrained problems in optimization. The new method satisfied the descent condition and the sufficient descent condition. We compare the numerical results of the new method with the Hestenes-Stiefel (HS) method. Our novel method is quite effective according to the number of iterations (NOI) and the number of functions (NOF) evaluated, as demonstrated by the numerical results on certain well-known non-linear test functions.

The major stationary iterative method used to solve nonlinear optimization problems is the quasi-Newton (QN) method. Symmetric Rank-One (SR1) is a method in the quasi-Newton family. This algorithm converges towards the true Hessian fast and has computational advantages for sparse or partially separable problems [1]. Thus, investigating the efficiency of the SR1 algorithm is significant. It's possible that the matrix generated by SR1 update won't always be positive. The denominator may also vanish or become zero. To overcome the drawbacks of the SR1 method, resulting in better performance than the standard SR1 method, in this work, we derive a new vector 𝑦𝑦𝑘𝑘 ∗ depending on the Barzilai-Borwein step size to obtain a new SR1 method. Then using this updating formula with preconditioning conjugate gradient (PCG) method is presented. With the aid of inexact line search procedure by strong Wolfe conditions, the new SR1 method is proposed and its performance is evaluated in comparison to the conventional SR1 method. It is proven that the updated matrix of the new SR1 method, 𝐻𝐻𝑘𝑘+1 𝑛𝑛𝑛𝑛𝑛𝑛, is symmetric matrix and positive definite matrix, given 𝐻𝐻𝑘𝑘 is initialized to identity matrix. In this study, the proposed method solved 13 problems effectively in terms of the number of iterations (NI) and the number of function evaluations (NF). Regarding NF, the new SR1 method also outperformed the classic SR1 method. The proposed method is shown to be more efficient in solving relatively large-scale problems (5,000 variables) compared to the original method. From the numerical results, the proposed method turned out to be significantly faster, effective and suitable for solving large dimension nonlinear equations

In this paper, we propose two hybrid gradient based methods and genetic algorithm for solving systems of linear equations with fast convergence. The first proposed hybrid method is obtained by using the steepest descent method and the second one by the Cauchy-Barzilai-Borwein method. These algorithms are based on minimizing the residual of solution which has genetic characteristics. They are compared with the normal genetic algorithm and standard gradient based methods in order to show the accuracy and the convergence speed of them. Since the conjugate gradient method is recommended for solving large sparse and symmetric positive definite matrices, we also compare the numerical results of our proposed algorithms with this method. The numerical results demonstrate the robustness and efficiency of the proposed algorithms. Moreover, we observe that our hybridization of the CBB method and genetic algorithm gives more accurate results with faster convergence than other mentioned methods in all given cases

The steepest descent (SD) method is well-known as the simplest method in optimization. In this paper, we propose a new SD search direction for solving system of linear equations Ax= b. We also prove that the proposed SD method with exact line search satisfies descent condition and possesses global convergence properties. This proposed method is motivated by previous work on the SD method by Zubai’ah-Mustafa-Rivaie-Ismail (ZMRI)[2]. Numerical comparisons with a classical SD algorithm and ZMRI algorithm show that this algorithm is very effective depending on the number of iterations (NOI) and CPU time.

In this paper a new conjugate gradient method for unconstrained optimization is suggested, the new method is based on Conjugate Descent (CD) formula which is a modified of the CD formula and which is also sufficiently descent and globally convergent. Numerical evidence shows that this new conjugate gradient algorithm is considered as one of the competitive conjugate gradient methods.

In this paper, we propose a modification of the self-scaling quasi-Newton (DFP) method for unconstrained optimization using logistic mapping. We shoe that it produces a positive definite matrix. Numerical results demonstrate that the new algorithm is superior to standard DFP method with respect to the NOI and NOF.

The conjugate gradient method is one of the most important algorithms for solving large and sparse symmetric positive definite (SPD) matrices. In this paper, the sufficient condition for convergence of the Fletcher-Reeves conjugate gradient method for almost symmetric matrices when the Euclidean norm of the nonsymmetric part of a positive definite matrix is less than the value of which is related to the smallest and the largest eigenvalues of the symmetric part of the matrix under consideration is presented.

Due to its simplicity and numerical efficiency, the Barzilai and Borwein (BB) gradient method has received numerous attentions in different scientific fields. In this paper, the sufficient condition for convergence of the BB method when the coefficient matrix of linear algebraic equations is slightly unsymmetric with positive definite symmetric part is presented.

In this paper, we proposed a new conjugate gradient method for unconstrained optimization problems by using Logistics Equation, One of the remarkable properties of the conjugate gradient method is its ability to generate so they are widely used for large scale unconstrained optimization problems.