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Numerical Algebra, Control and Optimization

June 2020 , Volume 10 , Issue 2

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Resource allocation and target setting based on virtual profit improvement
Jafar Sadeghi, Mojtaba Ghiyasi and Akram Dehnokhalaji
2020, 10(2): 127-142 doi: 10.3934/naco.2019043 +[Abstract](2339) +[HTML](591) +[PDF](361.67KB)

One application of Data Envelopment Analysis (DEA) is the resource allocation and target setting among homogeneous Decision Making Units (DMUs). In this paper, we assume that all units are under the supervision and control of a central decision making unit, for instance chain stores, banks, schools, etc. The aim is to allocate available resources among units in a way that the so-called organisational overall "virtual profit" is maximized. Our method is highly flexible in decision making to achieve the goals of the Decision Maker (DM). The resulting production plans maintain the following characteristics: (1) the virtual profit of each unit is calculated with a common set of weights; (2) the selected weights for calculating the virtual profit prevent the virtual profit of the system from getting worse; (3) the virtual profits of less profitable units are improved as much as possible. The proposed method is illustrated with a simple numerical example and a real life application.

Linear optimal control of time delay systems via Hermite wavelet
Akram Kheirabadi, Asadollah Mahmoudzadeh Vaziri and Sohrab Effati
2020, 10(2): 143-156 doi: 10.3934/naco.2019044 +[Abstract](1965) +[HTML](628) +[PDF](419.26KB)

To solve the time delay optimal control problem with quadratic performance index, a direct numerical method based on Hermite wavelet has been proposed in the present study. The idea is to convert the time delay optimal control problem into a quadratic programming problem. To do so, various time functions in the system are expanded as their truncated series and the properties of the operational matrices of integration, delay and product of two Hermite wavelet vectors are used as well. These matrices are utilized to reduce the solution of optimal control with time delay system, to the solution of a quadratic programming with linear constraints. Finally, three examples of time varying and time invariant coefficients are given to compare the results with some of the existing methods.

Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation
Yu-Feng Sun, Zheng Zeng and Jie Song
2020, 10(2): 157-164 doi: 10.3934/naco.2019045 +[Abstract](2317) +[HTML](587) +[PDF](327.75KB)

In this paper, the existence and uniqueness of solution for a class of boundary value problems of nonlinear fractional order differential equations involving the Caputo fractional derivative are studied. The estimation of error between the approximate solution and the solution for such equation is presented by employing the quasilinear iterative method, and an example is given to demonstrate the application of our main result.

Quadratic optimization with two ball constraints
Saeid Ansary Karbasy and Maziar Salahi
2020, 10(2): 165-175 doi: 10.3934/naco.2019046 +[Abstract](2169) +[HTML](660) +[PDF](385.88KB)

In this paper, the minimization of a general quadratic function subject to two ball constraints, called two ball trust-region subproblem (TBTRS), is studied. It is shown that the global optimal solution can be found by solving two extended trust-region subproblems. Strong duality conditions for two special cases are discussed. Finally, a comparison of results of the new algorithm with the other two recently proposed algorithms and CVX software are presented for several classes of randomly generated test problems.

Formal analysis of the Schulz matrix inversion algorithm: A paradigm towards computer aided verification of general matrix flow solvers
Vassilios A. Tsachouridis, Georgios Giantamidis, Stylianos Basagiannis and Kostas Kouramas
2020, 10(2): 177-206 doi: 10.3934/naco.2019047 +[Abstract](2876) +[HTML](774) +[PDF](1003.84KB)

This paper pilots Schulz generalised matrix inverse algorithm as a paradigm in demonstrating how computer aided reachability analysis and theoretical numerical analysis can be combined effectively in developing verification methodologies and tools for matrix iterative solvers. It is illustrated how algorithmic convergence to computed solutions with required accuracy is mathematically quantified and used within computer aided reachability analysis tools to formally verify convergence over predefined sets of multiple problem data. In addition, some numerical analysis results are used to form computational reliability monitors to escort the algorithm on-line and monitor the numerical performance, accuracy and stability of the entire computational process. For making the paper self-contained, formal verification preliminaries and background on tools and approaches are reported together with the detailed numerical analysis in basic mathematical language. For demonstration purposes, a custom made reachability analysis program based on affine arithmetic is applied to numerical examples.

Optimal control of an HIV model with CTL cells and latently infected cells
Jaouad Danane and Karam Allali
2020, 10(2): 207-225 doi: 10.3934/naco.2019048 +[Abstract](2001) +[HTML](582) +[PDF](627.91KB)

This paper deals with an optimal control problem for an human immunodeficiency virus (HIV) infection model with cytotoxic T-lymphocytes (CTL) immune response and latently infected cells. The model under consideration describes the interaction between the uninfected cells, the latently infected cells, the productively infected cells, the free viruses and the CTL cells. The two treatments represent the efficiency of drug treatment in inhibiting viral production and preventing new infections. Existence of the optimal control pair is established and the Pontryagin's minimum principle is used to characterize these two optimal controls. The optimality system is derived and solved numerically using the forward and backward difference approximation. Finally, numerical simulations are performed in order to show the role of optimal therapy in controlling the infection severity.

A new type of quasi-newton updating formulas based on the new quasi-newton equation
Basim A. Hassan
2020, 10(2): 227-235 doi: 10.3934/naco.2019049 +[Abstract](2212) +[HTML](574) +[PDF](364.64KB)

The quasi-Newton equation is the very foundation of an assortment of the quasi-Newton methods. Therefore, by using the offered alternative equation, we derive the modified BFGS quasi-Newton updating formulas. In this paper, a new y-technique has been introduced to modify the secant equation of the quasi-Newton methods. Prove the global convergence of this algorithm is associated with a line search rule. The numerical results explain that the offered method is effectual for the known test problems.

Stabilization on input time-varying delay for linear switched systems with truncated predictor control
K. Aruna Sakthi and A. Vinodkumar
2020, 10(2): 237-247 doi: 10.3934/naco.2019050 +[Abstract](2453) +[HTML](608) +[PDF](386.61KB)

This study is concerned with the stabilization problem for input time-varying delay switched system under the truncated predictor control scheme. The delay in the prediction feedback, is subjected by predicting the future trajectory of the states by system equations and initial conditions, which is known as truncated prediction feedback (TPF). The TPF is used to construct the state feedback law for stabilizing the linear switched system. By constructing Lyapunov-Krasovskii functions and, the stability condition is derived to ensure the globally asymptotically stable of the state feedback stabilization at the origin. When switching system is unstable, truncated predictor control method and Hurwitz convex combination makes the system stable. Finally, a numerical example and their simulation results are given to show the effectiveness of the proposed approach.

2021 CiteScore: 1.9




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