# American Institute of Mathematical Sciences

ISSN:
2155-3289

eISSN:
2155-3297

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## Numerical Algebra, Control & Optimization

December 2021 , Volume 11 , Issue 4

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2021, 11(4): 487-493 doi: 10.3934/naco.2020039 +[Abstract](1280) +[HTML](541) +[PDF](308.99KB)
Abstract:

As it is known the equation \begin{document}$A\varphi = f$\end{document} with injective compact operator has a unique solution for all \begin{document}$f$\end{document} in the range \begin{document}$R(A).$\end{document}Unfortunately, the right-hand side \begin{document}$f$\end{document} is never known exactly, so we can take an approximate data \begin{document}$f_{\delta }$\end{document} and used the perturbed problem \begin{document}$\alpha \varphi +A\varphi = f_{\delta }$\end{document} where the solution \begin{document}$\varphi _{\alpha \delta }$\end{document} depends continuously on the data \begin{document}$f_{\delta },$\end{document} and the bounded inverse operator \begin{document}$\left( \alpha I+A \right) ^{-1}$\end{document} approximates the unbounded operator \begin{document}$A^{-1}$\end{document} but not stable. In this work we obtain the convergence of the approximate solution of \begin{document}$\varphi _{\alpha \delta }$\end{document} of the perturbed equation to the exact solution \begin{document}$\varphi$\end{document} of initial equation provided \begin{document}$\alpha$\end{document} tends to zero with \begin{document}$\dfrac{\delta }{\sqrt{\alpha }}.$\end{document}

2021, 11(4): 495-512 doi: 10.3934/naco.2020040 +[Abstract](1548) +[HTML](518) +[PDF](468.75KB)
Abstract:

The direct scheme method is applied to construct an asymptotic approximation of any order to a solution of a singularly perturbed optimal problem with scalar state, controlled via a second-order linear ODE and two fixed end points. The error estimates for state and control variables and for the functional are obtained. An illustrative example is given.

2021, 11(4): 513-531 doi: 10.3934/naco.2020053 +[Abstract](1213) +[HTML](448) +[PDF](449.06KB)
Abstract:

In an attempt to improve theoretical complexity of large-update methods, in this paper, we propose a primal-dual interior-point method for \begin{document}$P_{\ast}\left( \kappa \right)$\end{document}-horizontal linear complementarity problem. The method is based on a class of parametric kernel functions. We show that the corresponding algorithm has \begin{document}$O\left( \left( 1+2\kappa \right) p^{2}n^{\frac{2+p}{2\left( 1+p\right) }}\log \frac{n}{\epsilon }\right)$\end{document} iteration complexity for large-update methods and we match the best known iteration bounds with special choice of the parameter \begin{document}$p$\end{document} for \begin{document}$P_{\ast }\left(\kappa \right)$\end{document}-horizontal linear complementarity problem that is \begin{document}$O\left(\left( 1+2\kappa \right) \sqrt{n}\log n\log \frac{n}{\epsilon }\right)$\end{document}. We illustrate the performance of the proposed kernel function by some comparative numerical results that are derived by applying our algorithm on five kernel functions.

2021, 11(4): 533-554 doi: 10.3934/naco.2020054 +[Abstract](1101) +[HTML](483) +[PDF](443.7KB)
Abstract:

Solving a bi-criterion fractional stochastic programming using an existing multi criteria decision making tool demands sufficient efforts and it is time consuming. There are many cases in financial situations that a nonlinear fractional programming, generated as a result of studying fractional stochastic programming, must be solved. Often management is not in needs of an optimal solution for the problem but rather an approximate solution can give him/her a good starting for the decision making or running a new model to find an intermediate or final solution. To this end, this author introduces a new linear approximation technique for solving a fractional stochastic programming (CCP) problem. After introducing the problem, the equivalent deterministic form of the fractional nonlinear programming problem is developed. To solve the problem, a fuzzy goal programming model of the equivalent deterministic form of the fractional stochastic programming is provided and then, the process of defuzzification and linearization of the problem is presented. A sample test problem is solved for presentation purposes. There are some limitations to the proposed approach: (1) solution obtains from this type of modeling is an approximate solution and, (2) preparation of approximation model of the problem may take some times for the beginners.

2021, 11(4): 555-566 doi: 10.3934/naco.2020055 +[Abstract](988) +[HTML](398) +[PDF](379.71KB)
Abstract:

The aim of this paper is to study the problem of constrained controllability for distributed parabolic linear system evolving in spatial domain \begin{document}$\Omega$\end{document} using the Reverse Hilbert Uniqueness Method (RHUM approach) introduced by Lions in 1988. It consists in finding the control \begin{document}$u$\end{document} that steers the system from an initial state \begin{document}$y_{_{0}}$\end{document} to a state between two prescribed functions. We give some definitions and properties concerning this concept and then we resolve the problem that relays on computing a control with minimum cost in the case of \begin{document}$\omega = \Omega$\end{document} and in the regional case where \begin{document}$\omega$\end{document} is a part of \begin{document}$\Omega$\end{document}.

2021, 11(4): 567-578 doi: 10.3934/naco.2020056 +[Abstract](730) +[HTML](375) +[PDF](430.99KB)
Abstract:

Biometric characteristics have been used since antiquated decades, particularly in the detection of crimes and investigations. The rapid development in image processing made great progress in biometric features recognition that is used in all life directions, especially when these features recognition is constructed as a computer system. The target of this research is to set up a left foot biometric system by hybridization between image processing and artificial bee colony (ABC) for feature choice that is addressed within artificial image processing. The algorithm is new because of the rare availability of hybridization algorithms in the literature of footprint recognition with the artificial bee colony assessment. The suggested system is tested on a live-captured ninety colored footprint images that composed the visual database. Then the constructed database was classified into nine clusters and normalized to be used at the advanced stages. Features database is constructed from the visual database off-line. The system starts with a comparison operation between the foot-tip image features extracted on-line and the visual database features. The outcome from this process is either a reject or an acceptance message. The results of the proposed work reflect the accuracy and integrity of the output. That is affected by the perfect choice of features as well as the use of artificial bee colony and data clustering which decreased the complexity and later raised the recognition rate to 100%. Our outcomes show the precision of our proposed procedures over others' methods in the field of biometric acknowledgment.

2021, 11(4): 579-611 doi: 10.3934/naco.2020057 +[Abstract](1038) +[HTML](362) +[PDF](3088.97KB)
Abstract:

The assimilation of flexible AC transmission (FACTS) controllers to the existing power network outweigh the numerous alternatives in enhancing the damping behavior for the inter-area /intra-area system oscillations of a power network. This paper provides a rigorous analysis in damping of oscillations in a power network. It utilizes a shunt connected voltage source converter (VSC) based FACTS device to enhance the system operating characteristics. A comprehensive system mathematical modelling has been developed for demonstrating the system behavior under different loading conditions. A novel hybrid augmented grey wolf optimization-particle swarm optimization (AGWO-PSO) is proposed for the coordinated design of controllers static synchronous compensator (STATCOM) and power system stabilizers (PSSs). A multi-objective function, comprising damping ratio improvement and drifting the real part to the left-hand side of S-plane of the system poles, has been developed to achieve the objective and the effectiveness of the proposed algorithms have been analyzed by monitoring the system performance under different loading conditions. Eigenvalue analysis and damping nature of the system states under perturbation have been presented for the proposed algorithms under different loading conditions, and the performance evaluation of the proposed algorithms have been done by means of time of execution and the convergence characteristics.

2021, 11(4): 613-631 doi: 10.3934/naco.2020058 +[Abstract](1304) +[HTML](436) +[PDF](428.7KB)
Abstract:

An interior point modified Nelder Mead method for nonlinearly constrained optimization is described. This method neither uses nor estimates objective function or constraint gradients. A modified logarithmic barrier function is used. The method generates a sequence of points which converges to KKT point(s) under mild conditions including existence of a Slater point. Numerical results are presented that show the algorithm performs well in practice.

2021, 11(4): 633-644 doi: 10.3934/naco.2021001 +[Abstract](1002) +[HTML](371) +[PDF](413.37KB)
Abstract:

In this paper, Lyapunov's artificial small parameter method (LASPM) with continuous particle swarm optimization (CPSO) is presented and used for solving nonlinear differential equations. The proposed method, LASPM-CPSO, is based on estimating the \begin{document}$\varepsilon$\end{document} parameter in LASPM through a PSO algorithm and based on a proposed objective function. Three different examples are used to evaluate the proposed method LASPM-CPSO, and compare it with the classical method LASPM through different intervals of the domain. The results from the maximum absolute error (MAE) and mean squared error (MSE) obtained through the given examples show the reliability and efficiency of the proposed LASPM-CPSO method, compared to the classical method LASPM.

2021, 11(4): 645-663 doi: 10.3934/naco.2021002 +[Abstract](1004) +[HTML](373) +[PDF](372.25KB)
Abstract:

In this work, we have proposed a new approach for solving the linear-quadratic optimal control problem, where the quality criterion is a quadratic function, which can be convex or non-convex. In this approach, we transform the continuous optimal control problem into a quadratic optimization problem using the Cauchy discretization technique, then we solve it with the active-set method. In order to study the efficiency and the accuracy of the proposed approach, we developed an implementation with MATLAB, and we performed numerical experiments on several convex and non-convex linear-quadratic optimal control problems. The obtained simulation results show that our method is more accurate and more efficient than the method using the classical Euler discretization technique. Furthermore, it was shown that our method fastly converges to the optimal control of the continuous problem found analytically using the Pontryagin's maximum principle.

2021, 11(4): 665-676 doi: 10.3934/naco.2021003 +[Abstract](843) +[HTML](366) +[PDF](374.52KB)
Abstract:

In this paper, we propose a new alternate gradient (AG) method to solve a class of optimization problems with orthogonal constraints. In particular, our AG method alternately takes several gradient reflection steps followed by one gradient projection step. It is proved that any accumulation point of the iterations generated by the AG method satisfies the first-order optimal condition. Numerical experiments show that our method is efficient.

2021, 11(4): 677-685 doi: 10.3934/naco.2021012 +[Abstract](816) +[HTML](317) +[PDF](333.38KB)
Abstract:

An efficiently preconditioned Newton-like method for the computation of the eigenpairs of large and sparse nonsymmetric matrices is proposed. A sequence of preconditioners based on the Broyden-type rank-one update formula are constructed for the solution of the linearized Newton system. The properties of the preconditioned matrix are investigated. Numerical results are given which reveal that the new proposed algorithms are efficient.

2020 CiteScore: 1.6