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Numerical Algebra, Control & Optimization

September 2018 , Volume 8 , Issue 3

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Preface
Chuei Yee Chen and Lai Soon Lee
2018, 8(3): ⅰ-ⅰ doi: 10.3934/naco.201803i +[Abstract](205) +[HTML](78) +[PDF](72.58KB)
Abstract:
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](274) +[HTML](85) +[PDF](379.85KB)
Abstract:

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](369) +[HTML](149) +[PDF](397.76KB)
Abstract:

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](512) +[HTML](147) +[PDF](1473.25KB)
Abstract:

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](486) +[HTML](129) +[PDF](770.98KB)
Abstract:

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](395) +[HTML](105) +[PDF](382.49KB)
Abstract:

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](508) +[HTML](96) +[PDF](388.34KB)
Abstract:

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](423) +[HTML](91) +[PDF](1027.29KB)
Abstract:

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](322) +[HTML](83) +[PDF](411.83KB)
Abstract:

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.

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