Two new Hager-Zhang iterative schemes with improved parameter choices for monotone nonlinear systems and their applications in compressed sensing

Notwithstanding its efficiency and nice attributes, most research on the iterative scheme by Hager and Zhang [Pac. J. Optim. 2(1) (2006) 35-58] are focused on unconstrained minimization problems. Inspired by this and recent works by Waziri et al. [Appl. Math. Comput. 361(2019) 645-660], Sabi’u et al. [Appl. Numer. Math. 153(2020) 217-233], and Sabi’u et al. [Int. J. Comput. Meth, doi:10.1142/S0219876220500437], this paper extends the Hager-Zhang (HZ) approach to nonlinear monotone systems with convex constraint. Two new HZ-type iterative methods are developed by combining the prominent projection method by Solodov and Svaiter [Springer, pp 355-369, 1998] with HZ-type search directions, which are obtained by developing two new parameter choices for the Hager-Zhang scheme. The first choice, is obtained by minimizing the condition number of a modified HZ direction matrix, while the second choice is realized using singular value analysis and minimizing the spectral condition number of the nonsingular HZ search direction matrix. Interesting properties of the schemes include solving non-smooth functions and generating descent directions. Using standard assumptions, the methods’ global convergence are obtained and numerical experiments with recent methods in the literature, indicate that the methods proposed are promising. The schemes effectiveness are further demonstrated by their applications to sparse signal and image reconstruction problems, where they outperform some recent schemes in the literature.

Simple dynamics in non-monotone Kolmogorov systems

In this paper we analyse the global dynamical behaviour of some classical models in the plane. Informally speaking we prove that the folkloric criteria based on the relative positions of the nullclines for Lotka–Volterra systems are also valid in a wide class of discrete systems. The method of proof consists of dividing the plane into suitable positively invariant regions and applying the theory of translation arcs in a subtle manner. Our approach allows us to extend several results of the theory of monotone systems to nonmonotone systems. Applications in models with weak Allee effect, population models for pioneer-climax species, and predator–prey systems are given.

A study on the effects of containment policies and vaccination on the spread of SARS-CoV-2

This paper presents a method to predict the spread of the SARS-CoV-2 in a population with a known age-structure, and then, to quantify the effects of various containment policies, including those policies that affect each age-group differently. The model itself is a compartmental model in which each compartment is divided into a number of age-groups. The parameters of the model are estimated using an optimisation scheme and some known results from the theory of monotone systems such that the model output agrees with some collected data on the spread of SARS-CoV-2. To highlight the strengths of this framework, a few case studies are presented in which different populations are subjected to different containment strategies. They include cases in which the containment policies switch between scenarios with different levels of severity. Then a case study on herd immunity due to vaccination is presented. And then it is shown how we can use this framework to optimally distribute a limited number of vaccine units in a given population to maximise their impact and reduce the total number of infectious individuals.