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Numerical Algebra, Control & Optimization

2018 , Volume 8 , Issue 1

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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](53) +[HTML](33) +[PDF](300.52KB)

We address the question of whether it may be worthwhile to convert certain, now classical, NP-complete problems to one of a smaller number of kernel NP-complete problems. In particular, we show that Karp's classical set of 21 NP-complete problems contains a kernel subset of six problems with the property that each problem in the larger set can be converted to one of these six problems with only linear growth in problem size. This finding has potential applications in optimisation theory because the kernel subset includes 0-1 integer programming, job sequencing and undirected Hamiltonian cycle problems.

Fourier-splitting method for solving hyperbolic LQR problems
Petra Csomós and Hermann Mena
2018, 8(1): 17-46 doi: 10.3934/naco.2018002 +[Abstract](54) +[HTML](35) +[PDF](1094.77KB)

We consider the numerical approximation to linear quadratic regulator problems for hyperbolic partial differential equations where the dynamics is driven by a strongly continuous semigroup. The optimal control is given in feedback form in terms of Riccati operator equations. The computational cost relies on solving the associated Riccati equation and computing the optimal state. In this paper we propose a novel approach based on operator splitting idea combined with Fourier's method to efficiently compute the optimal state. The Fourier's method allows to accurately approximate the exact flow making our approach computational efficient. Numerical experiments in one and two dimensions show the performance of the proposed method.

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](45) +[HTML](33) +[PDF](366.99KB)

In this paper, we describe the Globalizer software system for solving the global optimization problems. The system is designed to maximize the use of computational potential of the modern high-performance computational systems in order to solve the most time-consuming optimization problems. The highly parallel computations are facilitated using various distinctive computational schemes: processing several optimization iterations simultaneously, reducing multidimensional optimization problems using multiple Peano space-filling curves, and multi-stage computing based on the nested block reduction schemes. These novelties provide for the use of the supercomputer system capabilities with shared and distributed memory and with large numbers of processors to solve the global optimization problems efficiently.

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](53) +[HTML](81) +[PDF](437.15KB)

In this paper, we reconstruct a mathematical model of therapy by CAR T cells for acute lymphoblastic leukemia (ALL) With injection of modified T cells to body, then some signs such as fever, nausea and etc appear. These signs occur for the sake of cytokine release syndrome (CRS). This syndrome has a direct effect on result and satisfaction of therapy. So, the presence of cytokine will be played an important role in modelling process of therapy (CAR T cells). Therefore, the model will include the CAR T cells, B healthy and cancer cells, other circulating lymphocytes in blood, and cytokine. We analyse stability conditions of therapy Without any control, the dynamic model evidences sub-clinical or clinical decay, chronic destabilization, singularity immediately after a few hours and finally, it depends on the initial conditions. Hence, we try to show by which conditions, therapy will be effective. For this aim, we apply optimal control theory. Since the therapy of CAR T cells affects on both normal and cancer cell; so the optimization dose of CAR T cells will be played an important role and added to system as one controller \begin{document} $ u_{1} $ \end{document}. On the other hand, in order to control of cytokine release syndrome which is a factor for occurrence of singularity, one other controller \begin{document} $ u_{2} $ \end{document} as tocilizumab, an immunosuppressant drug for cytokine release syndrome is added to system. At the end, we apply method of Pontryagin's maximum principle for optimal control theory and simulate the clinical results by Matlab (ode15s and ode45).

Unbounded state-dependent sweeping processes with perturbations in uniformly convex and q-uniformly smooth Banach spaces
Samir Adly and Ba Khiet Le
2018, 8(1): 81-95 doi: 10.3934/naco.2018005 +[Abstract](44) +[HTML](25) +[PDF](345.28KB)

In this paper, the existence of solutions for a class of first and second order unbounded state-dependent sweeping processes with perturbation in uniformly convex and $q$-uniformly smooth Banach spaces are analyzed by using a discretization method. The sweeping process is a particular differential inclusion with a normal cone to a moving set and is of a great interest in many concrete applications. The boundedness of the moving set, which plays a crucial role for the existence of solutions in many works in the literature, is not necessary in the present paper. The compactness assumption on the moving set is also improved.

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](49) +[HTML](35) +[PDF](491.02KB)

The least absolute shrinkage and selection operator (LASSO) has been playing an important role in variable selection and dimensionality reduction for high dimensional linear regression under the zero-mean or Gaussian assumptions of the noises. However, these assumptions may not hold in practice. In this case, the least absolute deviation is a popular and useful method. In this paper, we focus on the least absolute deviation via Fused LASSO, called Robust Fused LASSO, under the assumption that the unknown vector is sparsity for both the coefficients and its successive differences. Robust Fused LASSO estimator does not need any knowledge of standard deviation of the noises or any moment assumptions of the noises. We show that the Robust Fused LASSO estimator possesses near oracle performance, i.e. with large probability, the \begin{document} $\ell_2$ \end{document} norm of the estimation error is of order \begin{document} $O(\sqrt{k(\log p)/n})$ \end{document}. The result is true for a wide range of noise distributions, even for the Cauchy distribution. In addition, we apply the linearized alternating direction method of multipliers to find the Robust Fused LASSO estimator, which possesses the global convergence. Numerical results are reported to demonstrate the efficiency of our proposed method.

$\mathcal{L}_{∞}$-norm computation for large-scale descriptor systems using structured iterative eigensolvers
Peter Benner, Ryan Lowe and Matthias Voigt
2018, 8(1): 119-133 doi: 10.3934/naco.2018007 +[Abstract](66) +[HTML](34) +[PDF](472.96KB)

In this article, we discuss a method for computing the \begin{document} $\mathcal{L}_∞$ \end{document}-norm for transfer functions of descriptor systems using structured iterative eigensolvers. In particular, the algorithm computes some desired imaginary eigenvalues of an even matrix pencil and uses them to determine an upper and lower bound to the \begin{document} $\mathcal{L}_∞$ \end{document}-norm. Finally, we compare our method to a previously developed algorithm using pseudopole sets. Numerical examples demonstrate the reliability and accuracy of the new method along with a significant drop in the runtime.




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