Numerical Analysis and Optimization

1. Introduction

This Editorial introduces this Special Issue of Axioms, which collates 10 articles showcasing the latest research related to problems in the two major scientific fields named in its title: “Numerical Analysis and Optimization”. The presented scholarly papers address various topics, primarily relating to generating appropriate numerical methods designed to solve the problems posed by the authors, which concern, for example, inclusion problems, gradient neural network models, differential evolution methods, conjugate gradient methods including the projection technique, multi-product and multi-criteria supply–demand network equilibrium models, and ZNN dynamical systems. This Special Issue’s principal aim is to disseminate the proposed algorithms and their applications in finding optimal solutions to the presented problems to researchers from related fields, thereby directly contributing to the further advancement of these two significant mathematical areas.

2. An Overview of the Published Papers

Regarding adequate conditions, in contribution 1, the strong convergence of a parameterized variable metric three-operator algorithm is established. In order to accelerate the operation of the algorithm, which is used for solving the monotone inclusion problem, the authors propose a multi-step inertial process. Through numerical investigations, the efficiency of the method is illustrated.

In contribution 2, the author introduces several definitions of generalized affine functions, including affinelikeness, preaffinelikeness, subaffinelikeness, and presubaffinelikeness. Through chosen examples, the author confirms that all the proposed definitions differ from each other. In this contribution, necessary and sufficient conditions for weakly solutions of vector optimization problems in real linear topological spaces are established, and practical generalizations of a number of previously published results are obtained. Additionally, the author details various optimality conditions and proves a strong duality theorem.

Using the gradient of the Frobenius norm of the traditional error function as a basis of a gradient neural network (GNN), the authors of contribution 3 propose the GGNN model for solving the general matrix equation $AXB=D$. From a theoretical perspective, the authors establish that for an arbitrary initial state matrix, $V\left(0\right)$, the neural state matrix $V\left(t\right)$ of the defined GGNN(A,B,D) model asymptotically converges to the solution of the matrix equation and coincides with the general solution of the linear matrix equation. Practically, several applications of the given method are demonstrated, and their global convergence is proved; further, its implementation in calculating various classes of generalized inverses is presented. Extensive numerical examples confirm the effectiveness of the GGNN model compared to the GNN algorithm.

In contribution 4, the authors use a hyper-heuristic approach to design parameter adaptation methods for the scaling factor parameter in differential evolution (DE), which constitutes a black-box numerical optimization method. To define the adaptation of the scaling factor F, the authors use two Taylor expansions to obtain the mean of the random distribution for sampling F and its standard deviation. This process is compared with the L-NTADE algorithm, and the superiority of the designed model is confirmed using two sets of benchmark problems. Furthermore, the results of the numerical experiments demonstrate that the efficiency of the novel method increases at higher dimensions.

The primary achievement of contribution 5 is its combination of the subspace minimization conjugate gradient method with the projection technique for solving nonlinear monotone equations with convex constraints. The presented model is well defined and globally convergent under the posed assumptions. Numerical test results and obtained performance metrics confirm the effectiveness of the generated conjugate gradient scheme.

Using a logarithmic approach, in combination with several new approximate functions, in contribution 6, the author presents a penalty method for solving nonlinear optimization problems. Instead of utilizing line search techniques, the author focuses on a specific algorithm for calculating the displacement step according to the direction. The vector direction of the proposed model is determined on the basis of Newton’s method, and numerical simulations illustrate various characteristics of the algorithm.

In contribution 7, the authors introduce a multi-product, multi-criteria supply–demand network equilibrium model with capacity constraints and uncertain demands, which are assumed to be in a closed interval. As the optimal performance of the network is obtained, in this contribution, the reader may find a significant theoretical framework for solving this type of optimization problem.

Solving a nonlinear optimization problem regarding the packing of different spheres, without mutual overlapping, into a container of a minimal height, bounded by a parabolic surface, is detailed in contribution 8. To solve this problem, the authors use a feasible direction approach combined with the hot-start technique, and containment constraints are described using $\mathsf{\Phi }$-function features. Included numerical experiments, provided for various relevant parameters, confirm the effectiveness of the proposed nonlinear programming model.

In contribution 9, the authors solve the equivalence and partial k-equivalence problems between WFAs. They provide an answer to whether two WFAs generate word functions that are the same or coincide for all input words whose lengths are less then a positive integer value, k. By utilizing two scientific approaches, ZNN neuro-dynamical systems and the existence of approximate heterotypic bisimulations between WFAs over $\mathbb{R}$, the authors present the ZNNL-hbfb and ZNNL-hfbb models. These two dynamical systems are implemented for solving matrix–vector equations involved in the considered heterotypic bisimulations, and extensive numerical simulations including various initial states are presented. The developed models are compared with the Matlab linear programming solver linprog and the pseudoinverse solution generated by the standard function pinv; the superiority of the presented ZNN models is confirmed.

Using the XOR and XNOR operations, the authors in contribution 10 explain the existence of solutions for a generalized Cayley variational inclusion problem. Through a fixed-point approach, they present effective methods for solving the posed problems. Numerical experiments demonstrate the features’ fast convergence and efficient performance in obtaining optimal solutions.