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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.

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A projected preconditioned conjugate gradient method for the linear response eigenvalue problem
Xing Li, Chungen Shen and Lei-Hong Zhang
2018, 8(4) : 389-412 doi: 10.3934/naco.2018025 +[Abstract](201) +[HTML](80) +[PDF](474.37KB)
Abstract:

The linear response eigenvalue problem aims at computing a few smallest positive eigenvalues together with the associated eigenvectors of a special Hamiltonian matrix and plays an important role for estimating the excited states of physical systems. A subspace version of the Thouless minimization principle was established by Bai and Li (SIAM J. Matrix Anal. Appl., 33:1075-1100, 2012) which characterizes the desired eigenpairs as its solution. In this paper, we propose a Projected Preconditioned Conjugate Gradient (\begin{document}$\texttt{PPCG_lrep}$\end{document}) method to solve this subspace version of Thouless's minimization directly. We show that \begin{document}$\texttt{PPCG_lrep}$\end{document} is an efficient implementation of the inverse power iteration and can be performed in parallel. It also enjoys several properties including the monotonicity and constraint preservation in the Thouless minimization principle. Convergence of both eigenvalues and eigenvectors are established and numerical experiences on various problems are reported.

Optimization problems with orthogonal matrix constraints
K. T. Arasu and Manil T. Mohan
2018, 8(4) : 413-440 doi: 10.3934/naco.2018026 +[Abstract](425) +[HTML](99) +[PDF](716.28KB)
Abstract:

The optimization problems involving orthogonal matrices have been formulated in this work. A lower bound for the number of stationary points of such optimization problems is found and its connection to the number of possible partitions of natural numbers is also established. We obtained local and global optima of such problems for different orders and showed their connection with the Hadamard, conference and weighing matrices. The application of general theory to some concrete examples including maximization of Shannon, Rény, Tsallis and Sharma-Mittal entropies for orthogonal matrices, minimum distance orthostochastic matrices to uniform van der Waerden matrices, Cressie-Read and K-divergence functions for orthogonal matrices, etc are also discussed. Global optima for all orders has been found for the optimization problems involving unitary matrix constraints.

On the cyclic pseudomonotonicity and the proximal point algorithm
Hadi Khatibzadeh, Vahid Mohebbi and Mohammad Hossein Alizadeh
2018, 8(4) : 441-449 doi: 10.3934/naco.2018027 +[Abstract](151) +[HTML](72) +[PDF](264.45KB)
Abstract:

We introduce various versions of cyclic pseudomonotonicity and study the relations between them. Some examples about the relation between them and monotonicity are also presented. By imposing some assumptions on the cyclic pseudomonotone bifunctions, we study the convergence analysis of the proximal point algorithm which has been studied by Iusem and Sosa [5] for pseudomonotone bifunctions, with better assumptions.

Quantitative stability analysis of stochastic mathematical programs with vertical complementarity constraints
Yongchao Liu
2018, 8(4) : 451-460 doi: 10.3934/naco.2018028 +[Abstract](127) +[HTML](82) +[PDF](311.16KB)
Abstract:

This paper studies the quantitative stability of stochastic mathematical programs with vertical complementarity constraints (SMPVCC) with respect to the perturbation of the underlying probability distribution. We first show under moderate conditions that the optimal solution set-mapping is outer semiconitnuous and optimal value function is Lipschitz continuous with respect to the probability distribution. We then move on to investigate the outer semiconitnuous of the M-stationary points by employing the reformulation of stationary points and some stability results on the stochastic generalized equations. The particular focus is given to discrete approximation of probability distributions, where both cases that the sample is chosen in a fixed procedure and random procedure are considered. The technical results lay a theoretical foundation for approximation schemes to be applied to solve SMPVCC.

Weighted vertices optimizer (WVO): A novel metaheuristic optimization algorithm
Soheil Dolatabadi
2018, 8(4) : 461-479 doi: 10.3934/naco.2018029 +[Abstract](266) +[HTML](72) +[PDF](908.28KB)
Abstract:

This paper introduces a novel optimization algorithm that is based on the basic idea underlying the bisection root-finding method in mathematics. The bisection method is modified for use as an optimizer by weighting each agent or vertex, and the algorithm developed from this process is called the weighted vertices optimizer (WVO). For exploitation and exploration, both swarm intelligence and evolution strategy are used to improve the accuracy and speed of WVO, which is then compared with six other popular optimization algorithms. Results confirm the superiority of WVO in most of the test functions.

A stochastic approach to model housing markets: The US housing market case
Bilgi Yilmaz and A. Sevtap Selcuk-Kestel
2018, 8(4) : 481-492 doi: 10.3934/naco.2018030 +[Abstract](172) +[HTML](73) +[PDF](575.18KB)
Abstract:

This study aims to estimate the price changes in housing markets using a stochastic process, which is defined in the form of stochastic differential equations (SDEs). It proposes a general SDEs system on the price structure in terms of house price index and mortgage rate to establish an effective process. As an empirical analysis, it applies a calibration procedure to an SDE on monthly S&P/Case-Shiller US National Home Price Index (HPI) and 30-year fixed mortgage rate to estimate parameters of differentiable functions defined in SDEs. The prediction power of the proposed stochastic model is justified through a Monte Carlo algorithm for one-year ahead monthly forecasts of the HPI returns. The results of the study show that the stochastic processes are flexible in terms of the choice of structure, compact with respect to the number of exogenous variables involved, and it is a literal method. Furthermore, this approach has a relatively high estimation power in forecasting the national house prices.

Further results on the perturbation estimations for the Drazin inverse
Haifeng Ma and Xiaoshuang Gao
2018, 8(4) : 493-503 doi: 10.3934/naco.2018031 +[Abstract](134) +[HTML](109) +[PDF](298.84KB)
Abstract:

For \begin{document}$n× n$\end{document} complex singular matrix \begin{document}$A$\end{document} with ind\begin{document}$(A) = k>1$\end{document}, let \begin{document}$A^D$\end{document} be the Drazin inverse of \begin{document}$A$\end{document}. If a matrix \begin{document}$B = A+E$ \end{document} with ind\begin{document}$(B) = 1$ \end{document} is said to be an acute perturbation of \begin{document}$A$ \end{document}, if \begin{document}$\|E\|$ \end{document} is small and the spectral radius of \begin{document}$B_gB- A^DA$ \end{document} satisfies

where \begin{document}$B_g$ \end{document} is the group inverse of \begin{document}$B$ \end{document}.

The acute perturbation coincides with the stable perturbation of the group inverse, if the matrix \begin{document}$B$ \end{document} satisfies geometrical condition:

which introduced by Vélez-Cerrada, Robles, and Castro-González, (Error bounds for the perturbation of the Drazin inverse under some geometrical conditions, Appl. Math. Comput., 215 (2009), 2154-2161).

Furthermore, two examples are provided to illustrate the acute perturbation of the Drazin inverse. We prove the correctness of the conjecture in a special case of ind\begin{document}$(B) = 1$ \end{document} by Wei (Acute perturbation of the group inverse, Linear Algebra Appl., 534 (2017), 135-157).

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](1106) +[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](862) +[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](740) +[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](788) +[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](704) +[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](786) +[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](726) +[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](929) +[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](717) +[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](861) +[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](831) +[HTML](362) +[PDF](300.52KB) PDF Downloads(104)
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](1167) +[HTML](296) +[PDF](381.9KB) PDF Downloads(84)
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](977) +[HTML](533) +[PDF](271.19KB) PDF Downloads(79)
Optimization problems with orthogonal matrix constraints
K. T. Arasu and Manil T. Mohan
2018, 8(4) : 413-440 doi: 10.3934/naco.2018026 +[Abstract](425) +[HTML](99) +[PDF](716.28KB) PDF Downloads(65)
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](444) +[HTML](139) +[PDF](310.66KB) PDF Downloads(64)
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](337) +[HTML](129) +[PDF](770.98KB) PDF Downloads(63)
A projected preconditioned conjugate gradient method for the linear response eigenvalue problem
Xing Li, Chungen Shen and Lei-Hong Zhang
2018, 8(4) : 389-412 doi: 10.3934/naco.2018025 +[Abstract](201) +[HTML](80) +[PDF](474.37KB) PDF Downloads(59)
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](676) +[HTML](249) +[PDF](224.85KB) PDF Downloads(51)
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](290) +[HTML](105) +[PDF](382.49KB) PDF Downloads(51)
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](677) +[HTML](210) +[PDF](491.02KB) PDF Downloads(50)

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