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Volume 8, 2018

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Numerical Algebra, Control and Optimization (NACO) aims at publishing original papers on any non-trivial interplay between control and optimization, and numerical techniques for their underlying linear and nonlinear algebraic systems. Topics of interest to NACO include the following: original research in theory, algorithms and applications of optimization; numerical methods for linear and nonlinear algebraic systems arising in modelling, control and optimisation; and original theoretical and applied research and development in the control of systems including all facets of control theory and its applications. In the application areas, special interests are on artificial intelligence and data sciences. The journal also welcomes expository submissions on subjects of current relevance to readers of the journal. The publication of papers in NACO is free of charge.

  • AIMS is a member of COPE. All AIMS journals adhere to the publication ethics and malpractice policies outlined by COPE.
  • Publishes 4 issues a year in March, June, September and December.
  • Publishes both online and in print.
  • Indexed in Scopus, MathSciNet, Zentralblatt MATH and Emerging Sources Citation Index.
  • Archived in Portico and CLOCKSS.
  • NACO is a publication of the American Institute of Mathematical Sciences. All rights reserved.

Note: “Most Cited” is by Cross-Ref , and “Most Downloaded” is based on available data in the new website.

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Chuei Yee Chen and Lai Soon Lee
2018, 8(3) : ⅰ-ⅰ doi: 10.3934/naco.201803i +[Abstract](109) +[HTML](63) +[PDF](72.58KB)
Construction and research of adequate computational models for quasilinear hyperbolic systems
Aloev Rakhmatillo, Khudoyberganov Mirzoali and Blokhin Alexander
2018, 8(3) : 277-289 doi: 10.3934/naco.2018017 +[Abstract](119) +[HTML](68) +[PDF](379.85KB)

In the paper, we study a class of three-dimensional quasilinear hyperbolic systems. For such system, we set the initial boundary value problem and construct the energy integral. We construct the difference scheme and obtain an a priori estimate for its solution.

Pricing down-and-out power options with exponentially curved barrier
Teck Wee Ng and Siti Nur Iqmal Ibrahim
2018, 8(3) : 291-297 doi: 10.3934/naco.2018018 +[Abstract](124) +[HTML](111) +[PDF](397.76KB)

Power barrier options are options where the payoff depends on an underlying asset raised to a constant number. The barrier determines whether the option is knocked in or knocked out of existence when the underlying asset hits the prescribed barrier level, or not. This paper derives the analytical solution of the power options with an exponentially curved barrier by utilizing the reflection principle and the change of measure. Numerical results show that prices of power options with exponentially curved barrier are cheaper than those of power barrier options and power options.

A controlled treatment strategy applied to HIV immunology model
Shohel Ahmed, Abdul Alim and Sumaiya Rahman
2018, 8(3) : 299-314 doi: 10.3934/naco.2018019 +[Abstract](188) +[HTML](123) +[PDF](1473.25KB)

Optimal control can be helpful to test and compare different vaccination strategies of a certain disease. This study investigates a mathematical model of HIV infections in terms of a system of nonlinear ordinary differential equations (ODEs) which describes the interactions between the human immune systems and the HIV virus. We introduce chemotherapy in an early treatment setting through a dynamic treatment and then solve for an optimal chemotherapy strategy. The aim is to obtain a new optimal chemotherapeutic strategy where an isoperimetric constraint on the chemotherapy supply plays a crucial role. We outline the steps in formulating an optimal control problem, derive optimality conditions and demonstrate numerical results of an optimal control for the model. Numerical results illustrate how such a constraint alters the optimal vaccination schedule and its effect on cell-virus interactions.

On a two-phase approximate greatest descent method for nonlinear optimization with equality constraints
M. S. Lee, B. S. Goh, H. G. Harno and K. H. Lim
2018, 8(3) : 315-326 doi: 10.3934/naco.2018020 +[Abstract](127) +[HTML](100) +[PDF](770.98KB)

Lagrange multipliers are usually used in numerical methods to solve equality constrained optimization problems. However, when the intersection between a search region for a current point and the feasible set defined by the equality constraints is empty, Lagrange multipliers cannot be used without additional conditions. To cope with this condition, a new method based on a two-phase approximate greatest descent approach is presented in this paper. In Phase-Ⅰ, an accessory function is used to drive a point towards the feasible set and the optimal point of an objective function. It has been observed that for some current points, it may be necessary to maximize the objective function while minimizing the constraint violation function in a current search region in order to construct the best numerical iterations. When the current point is close to or inside the feasible set and when optimality conditions are nearly satisfied, the numerical iterations are switched to Phase-Ⅱ. The Lagrange multipliers are defined and used in this phase. The approximate greatest descent method is then applied to minimize a merit function which is constructed from the optimality conditions. Results of numerical experiments are presented to show the effectiveness of the aforementioned two-phase method.

Approximate greatest descent in neural network optimization
King Hann Lim, Hong Hui Tan and Hendra G. Harno
2018, 8(3) : 327-336 doi: 10.3934/naco.2018021 +[Abstract](106) +[HTML](76) +[PDF](382.49KB)

Numerical optimization is required in artificial neural network to update weights iteratively for learning capability. In this paper, we propose the use of Approximate Greatest Descent (AGD) algorithm to optimize neural network weights using long-term backpropagation manner. The modification and development of AGD into stochastic diagonal AGD (SDAGD) algorithm could improve the learning ability and structural simplicity for deep learning neural networks. It is derived from the operation of a multi-stage decision control system which consists of two phases: (1) when local search region does not contain the minimum point, iteration shall be defined at the boundary of the local search region, (2) when local region contains the minimum point, Newton method is approximated for faster convergence. The integration of SDAGD into Multilayered perceptron (MLP) network is investigated with the goal of improving the learning ability and structural simplicity. Simulation results showed that two-layer MLP with SDAGD achieved a misclassification rate of 9.4% on a smaller mixed national institute of national and technology (MNIST) dataset. MNIST is a database equipped with handwritten digits images suitable for algorithm prototyping in artificial neural networks.

Homotopy perturbation method and Chebyshev polynomials for solving a class of singular and hypersingular integral equations
Zainidin Eshkuvatov
2018, 8(3) : 337-350 doi: 10.3934/naco.2018022 +[Abstract](124) +[HTML](73) +[PDF](388.34KB)

TIn this note, we review homotopy perturbation method (HPM), Discrete HPM, Chebyshev polynomials and its properties. Moreover, the convergences of HPM and error term of Chebyshev polynomials were discussed. Then, linear singular integral equations (SIEs) and hyper-singular integral equations (HSIEs) are solved by combining modified HPM together with Chebyshev polynomials. Convergences of the mixed method for the linear HSIEs are also obtained. Finally, illustrative examples and comparisons with different methods are presented.

Differential evolution with improved sub-route reversal repair mechanism for multiobjective urban transit routing problem
Ahmed Tarajo Buba and Lai Soon Lee
2018, 8(3) : 351-376 doi: 10.3934/naco.2018023 +[Abstract](126) +[HTML](72) +[PDF](1027.29KB)

The urban transit routing problem (UTRP) deals with public transport systems in determining a set of efficient transit routes on existing road networks to meet transit demands. The UTRP is a complex combinatorial optimization problem characterized with a large search space, multi-constraint, and multiobjective nature where the likelihood of generating infeasible route sets is high. In this paper, an improved sub-route reversal repair mechanism is proposed to deal with the infeasibility. A population-based metaheuristic, namely, Differential Evolution (DE) algorithm is then proposed to handle the multiobjective UTRP with the aim of devising an efficient transit route network that optimizes both passengers' and operators' costs. Computational experiments are performed on well-known benchmark instances to evaluate the effectiveness of the proposed repair mechanism and the DE algorithm. The computational results are reported to have better parameter values in most cases when compared to other approaches in the literature.

Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization
Hong Seng Sim, Wah June Leong, Chuei Yee Chen and Siti Nur Iqmal Ibrahim
2018, 8(3) : 377-387 doi: 10.3934/naco.2018024 +[Abstract](138) +[HTML](66) +[PDF](411.83KB)

In this paper, we aim to propose some spectral gradient methods via variational technique under log-determinant norm. The spectral parameters satisfy the modified weak secant relations that inspired by the multistep approximation for solving large scale unconstrained optimization. An executable code is developed to test the efficiency of the proposed method with spectral gradient method using standard weak secant relation as constraint. Numerical results are presented which suggest a better performance has been achieved.

Recent advances in numerical methods for nonlinear equations and nonlinear least squares
Ya-Xiang Yuan
2011, 1(1) : 15-34 doi: 10.3934/naco.2011.1.15 +[Abstract](881) +[PDF](445.3KB) Cited By(27)
Control parameterization for optimal control problems with continuous inequality constraints: New convergence results
Ryan Loxton, Qun Lin, Volker Rehbock and Kok Lay Teo
2012, 2(3) : 571-599 doi: 10.3934/naco.2012.2.571 +[Abstract](617) +[PDF](325.0KB) Cited By(18)
A modified Fletcher-Reeves-Type derivative-free method for symmetric nonlinear equations
Dong-Hui Li and Xiao-Lin Wang
2011, 1(1) : 71-82 doi: 10.3934/naco.2011.1.71 +[Abstract](571) +[PDF](194.9KB) Cited By(16)
Univariate geometric Lipschitz global optimization algorithms
Dmitri E. Kvasov and Yaroslav D. Sergeyev
2012, 2(1) : 69-90 doi: 10.3934/naco.2012.2.69 +[Abstract](608) +[PDF](602.3KB) Cited By(15)
Optimal control strategies for tuberculosis treatment: A case study in Angola
Cristiana J. Silva and Delfim F. M. Torres
2012, 2(3) : 601-617 doi: 10.3934/naco.2012.2.601 +[Abstract](537) +[PDF](342.2KB) Cited By(13)
Error bounds for Euler approximation of linear-quadratic control problems with bang-bang solutions
Walter Alt, Robert Baier, Matthias Gerdts and Frank Lempio
2012, 2(3) : 547-570 doi: 10.3934/naco.2012.2.547 +[Abstract](615) +[PDF](298.8KB) Cited By(13)
Noether's symmetry Theorem for variational and optimal control problems with time delay
Gastão S. F. Frederico and Delfim F. M. Torres
2012, 2(3) : 619-630 doi: 10.3934/naco.2012.2.619 +[Abstract](557) +[PDF](199.0KB) Cited By(12)
Towards globally optimal operation of water supply networks
Ambros M. Gleixner, Harald Held, Wei Huang and Stefan Vigerske
2012, 2(4) : 695-711 doi: 10.3934/naco.2012.2.695 +[Abstract](736) +[PDF](810.5KB) Cited By(12)
An unconstrained optimization approach for finding real eigenvalues of even order symmetric tensors
Lixing Han
2013, 3(3) : 583-599 doi: 10.3934/naco.2013.3.583 +[Abstract](540) +[PDF](442.5KB) Cited By(12)
Linearized alternating direction method of multipliers with Gaussian back substitution for separable convex programming
Bingsheng He and Xiaoming Yuan
2013, 3(2) : 247-260 doi: 10.3934/naco.2013.3.247 +[Abstract](669) +[PDF](461.8KB) Cited By(11)
Linearly-growing reductions of Karp's 21 NP-complete problems
Jerzy A. Filar, Michael Haythorpe and Richard Taylor
2018, 8(1) : 1-16 doi: 10.3934/naco.2018001 +[Abstract](475) +[HTML](358) +[PDF](300.52KB) PDF Downloads(103)
Performance evaluation of four-stage blood supply chain with feedback variables using NDEA cross-efficiency and entropy measures under IER uncertainty
Shiva Moslemi and Abolfazl Mirzazadeh
2017, 7(4) : 379-401 doi: 10.3934/naco.2017024 +[Abstract](836) +[HTML](292) +[PDF](381.9KB) PDF Downloads(83)
Mathematical model of Chimeric Anti-gene Receptor (CAR) T cell therapy with presence of cytokine
Reihaneh Mostolizadeh, Zahra Afsharnezhad and Anna Marciniak-Czochra
2018, 8(1) : 63-80 doi: 10.3934/naco.2018004 +[Abstract](600) +[HTML](526) +[PDF](271.19KB) PDF Downloads(77)
Fuzzy target-environment networks and fuzzy-regression approaches
Erik Kropat and Gerhard Wilhelm Weber
2018, 8(2) : 135-155 doi: 10.3934/naco.2018008 +[Abstract](244) +[HTML](135) +[PDF](310.66KB) PDF Downloads(62)
On a two-phase approximate greatest descent method for nonlinear optimization with equality constraints
M. S. Lee, B. S. Goh, H. G. Harno and K. H. Lim
2018, 8(3) : 315-326 doi: 10.3934/naco.2018020 +[Abstract](127) +[HTML](100) +[PDF](770.98KB) PDF Downloads(56)
Fused LASSO penalized least absolute deviation estimator for high dimensional linear regression
Yanqing Liu, Jiyuan Tao, Huan Zhang, Xianchao Xiu and Lingchen Kong
2018, 8(1) : 97-117 doi: 10.3934/naco.2018006 +[Abstract](489) +[HTML](207) +[PDF](491.02KB) PDF Downloads(49)
Globalizer: A novel supercomputer software system for solving time-consuming global optimization problems
Victor Gergel, Konstantin Barkalov and Alexander Sysoyev
2018, 8(1) : 47-62 doi: 10.3934/naco.2018003 +[Abstract](436) +[HTML](246) +[PDF](224.85KB) PDF Downloads(49)
A study of numerical integration based on Legendre polynomial and RLS algorithm
Hongguang Xiao, Wen Tan, Dehua Xiang, Lifu Chen and Ning Li
2017, 7(4) : 457-464 doi: 10.3934/naco.2017028 +[Abstract](551) +[HTML](249) +[PDF](291.3KB) PDF Downloads(46)
Approximate greatest descent in neural network optimization
King Hann Lim, Hong Hui Tan and Hendra G. Harno
2018, 8(3) : 327-336 doi: 10.3934/naco.2018021 +[Abstract](106) +[HTML](76) +[PDF](382.49KB) PDF Downloads(46)
Numerical method for solving optimal control problems with phase constraints
Alexander Tyatyushkin and Tatiana Zarodnyuk
2017, 7(4) : 481-492 doi: 10.3934/naco.2017030 +[Abstract](877) +[HTML](224) +[PDF](541.6KB) PDF Downloads(45)




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