The Spectral conjugate gradient (SCG) methods are among the efficient variants of CG algorithms which are obtained by combining the spectral gradient parameter and CG parameter. The success of SCG methods relies on effective choices of the step-size α _k and the search direction d _k . This paper presents an SCG method for unconstrained optimization models. The search directions generated by the new method possess sufficient descent property without the restart condition and independent of the line search procedure used. The global convergence of the new method is proved under the weak Wolfe line search. Preliminary numerical results are presented which show that the method is efficient and promising, particularly for large-scale problems. Also, the method was applied to solve the robotic motion control problem and portfolio selection problem.

The three-term conjugate gradient (CG) algorithms are among the efficient variants of CG algorithms for solving optimization models. This is due to their simplicity and low memory requirements. On the other hand, the regression model is one of the statistical relationship models whose solution is obtained using one of the least square methods including the CG-like method. In this paper, we present a modification of a three-term conjugate gradient method for unconstrained optimization models and further establish the global convergence under inexact line search. The proposed method was extended to formulate a regression model for the novel coronavirus (COVID-19). The study considers the globally infected cases from January to October 2020 in parameterizing the model. Preliminary results have shown that the proposed method is promising and produces efficient regression model for COVID-19 pandemic. Also, the method was extended to solve a motion control problem involving a two-joint planar robot.

The conjugate gradient (CG) method is an optimization method, which, in its application, has a fast convergence. Until now, many CG methods have been developed to improve computational performance and have been applied to real-world problems. In this paper, a new hybrid three-term CG method is proposed for solving unconstrained optimization problems. The search direction is a three-term hybrid form of the Hestenes-Stiefel (HS) and the Polak-Ribiére-Polyak (PRP) CG coefficients, and it satisfies the sufficient descent condition. In addition, the global convergence properties of the proposed method will also be proved under the weak Wolfe line search. By using several test functions, numerical results show that the proposed method is most efficient compared to some of the existing methods. In addition, the proposed method is used in practical application problems for image restoration and portfolio selection.

Nonlinear systems of equations are widely used in science and engineering and, therefore, exploring efficient ways to solve them is paramount. In this paper, a new derivative-free approach for solving a nonlinear system of equations with convex constraints is proposed. The search direction of the proposed method is derived based on a modified conjugate gradient method, in such a way that it is sufficiently descent. It is worth noting that, unlike many existing methods that require a monotonicity assumption to prove the convergence result, our new method needs the underlying function to be pseudomonotone, which is a weaker assumption. The performance of the proposed algorithm is demonstrated on a set of some test problems and applications arising from compressive sensing. The obtained results confirm that the proposed method is effective compared to some existing algorithms in the literature.

The interaction between prey, secondary predator, and primary predator as a mathematical model of the one-prey and two-predator system with constant harvesting in prey population will be introduced in this article. Their interaction might describe as a food pyramid, with the preys is in the lowest level of the pyramid, secondary predators in the middle, and primary predators in the top of the pyramid. Human intervention to controlling prey population is needed and will be analyzed how this will effect on the existence of secondary predator and primary predator population. Equilibrium points and their existence criteria will be analyzed to find a threshold that will guarantee the coexistence of this system. Some numerical simulation will be given to illustrate the analytical results. We find that as long as harvesting rate in prey population is smaller than prey intrinsic growth rate, coexistence might achieve.