Conjugate gradient (CG) methods are considered among the most efficient methods for solving optimization problems thanks to their straightforward iterative process and low memory requirements. In the present work, we propose a combined CG method to address large-scale problems, with a particular application to training artificial neural networks (ANNs) for early breast cancer prediction and electrocardiogram (ECG) classification. Under the strong Wolfe line search conditions, the global convergence was demonstrated under mild assumptions and the generated descent direction and the convergence features of the suggested approach are examined. The proposed approach was successfully applied to train neural networks for early breast cancer prediction, achieving an accuracy of 98.24%, with precision, recall, and F1-score values of 0.99, 0.97, and 0.98, respectively. It also reduces the final mean squared error by over 52% and exhibited faster convergence with smoother training dynamics. Furthermore, on the ECG classification dataset, the proposed hybrid Dai-Liao (hDL ${}^{+}$) achieves an accuracy of 80.45%, demonstrating strong generalization performance across different medical diagnostic applications. Comparisons with recent CG methods on a set of test problems from the CUTE library confirmed the robustness and efficiency of the proposed method.

In this paper, we propose a new five-dimensional system that is capable of producing multistability and hyperchaos with three positive Lyapunov exponents (LEs) for specific parameter settings. The proposed system was derived by making an extension of Dong's chaotic system with changes that increase its dimension, complexity, and applicability. Using numerical simulations, we confirmed that the model produces hyperchaos with three positive LEs depending on parameter values, which means that its dynamics expands in three directions of the phase space. Regions of periodicity, chaos, hyperchaos with two positive LEs, and hyperchaos with three positive LEs were identified by LE spectra and verified using phase diagrams. The ability of the system to generate coexisting attractors is also shown, where the system demonstrates various behaviors for various initial conditions at fixed parameters. Furthermore, offset boosting control is presented to show that the attractors can be shifted without altering the internal dynamics of the system. Moreover, a topological complexity optimization framework is proposed to maximize the Kaplan–Yorke dimension (KYD) using Particle Swarm Optimization (PSO) and Differential Evolution (DE), followed by the 0–1 test and Approximate Entropy calculation. With its multistability, properties, hyperchaos with three positive LEs, controllable offset boosting, and optimized complexity, the novel model has excellent potential for practical applications.

This work investigated the computational efficiency of primal-dual interior-point methods for nonlinear convex optimization by refining both the underlying kernel functions and the barrier parameter update mechanisms. We introduced a unified parametric class of self-regular kernels that generalizes several established barrier families while maintaining optimal theoretical iteration complexity. To bridge the gap between theoretical convergence and practical performance, we proposed an adaptive update rule for the barrier parameter and evaluated various heuristics for its dynamic selection. Extensive numerical testing on a diverse benchmark suite demonstrated that the proposed framework significantly outperforms the Interior Point OPTimizer (IPOPT) solver while maintaining high numerical accuracy and minimal stationarity residuals. Moreover, the framework exhibited robust performance even on nonconvex problems, highlighting its practical versatility beyond the theoretical convex setting.

This paper proposed a new iterative method for solving nonlinear monotone equations with convex constraint and its applications in sparse signal reconstruction and image de-blurring problems. The method can be viewed as an improved adaptation of the generalized Hager-Zhang conjugate gradient method for unconstrained optimization. Unlike the latter which only converged globally for strongly convex functions when the Hager-Zhang parameter ${\theta }_k$ lies in the interval $\left(\frac{1}{4},+\mathrm{\infty }\right)$, the new method exhibited this attribute for nonlinear monotone and Lipschitz continuous functions without restriction for ${\theta }_k$ under a more relaxed condition. The derivative-free structure of the method made it suitable for both smooth and non-smooth problems. Numerical experiments on benchmark test problems demonstrated the method's superior performance compared to some state-of-the-art algorithms. Furthermore, the algorithm was successfully applied to sparse signal recovery and image de-blurring problems in compressed sensing, confirming its practical effectiveness.

The hybrid procedure is an efficient technique for improving the global and numerical performance of iterative algorithms for large-scale monotone nonlinear problems. This is achieved by integrating two or more methods into a unified framework. In this study, we present a hybrid of the double step-length method and the Picard-Mann iterative technique for solving monotone nonlinear equations with convex constraints. By combining the Picard-Mann approach with a newly proposed scheme, we obtain an iterative method that reduces computational cost and achieves faster convergence. The acceleration parameter is determined by evaluating the difference between the Broyden update and its approximation using the Frobenius norm. The global convergence of the proposed method is proved, and a Q-linear convergence rate is also established. Numerical experiments demonstrate that the proposed approach is computationally efficient for solving large-scale nonlinear equations compared with existing methods. Finally, the method is applied to signal processing and image restoration problems, highlighting its practical relevance.

Due to their simplicity, low memory requirements, strong convergence properties, and ability to solve problems of high dimensions, the conjugate gradient (CG) methods are widely used to solve linear and non-linear unconstrained optimization problems. The Polak-Ribière-Polyak (PRP) is considered as one of the most efficient CG methods in practical computation. However, theoretically, its convergence properties are poor. Therefore, many variants of PRP with good numerical results and good convergence properties have been developed, such as Gilbert and Nocedal method (PRP ${}^{+}$), Wei-Yau-Liu method (WYL), and Yousif et al. method (OPRP). In this paper, based on PRP ${}^{+}$ and OPRP methods, we proposed another modified version of PRP that inherits all the convergence properties of PRP ${}^{+}$ and OPRP and has improved numerical results. To show the efficiency and robustness of the new modified method in practice, it was compared with PRP ${}^{+}$, WYL, and OPRP when they are all applied under the strong Wolfe line search. At the same time, the new method was applied in deep learning to obtain ideal parameters of some neural network (NN) models during the training process.