# American Institute of Mathematical Sciences

ISSN:
2155-3289

eISSN:
2155-3297

## Journal Home

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### Volume 1, 2011

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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2019, 9(1) : 1-13 doi: 10.3934/naco.2019001 +[Abstract](720) +[HTML](276) +[PDF](397.82KB)
Abstract:

In this paper, we present a brief survey of methods for solving nonlinear least-squares problems. We pay specific attention to methods that take into account the special structure of the problems. Most of the methods discussed belong to the quasi-Newton family (i.e. the structured quasi-Newton methods (SQN)). Our survey comprises some of the traditional and modern developed methods for nonlinear least-squares problems. At the end, we suggest a few topics for further research.

2019, 9(1) : 15-22 doi: 10.3934/naco.2019002 +[Abstract](388) +[HTML](198) +[PDF](327.08KB)
Abstract:

This work is devoted to establish the extremal inertias ofthe two submatrices $X_{1}$ and $X_{4}$ in a Hermitian least rank solution $X$of the matrix equation $AXA^{*}=B$. From these formulas, necessary andsufficient conditions for these submatrices to be positive (nonpositive,negative, nonnegative) definite are achieved.

2019, 9(1) : 23-44 doi: 10.3934/naco.2019003 +[Abstract](303) +[HTML](195) +[PDF](838.1KB)
Abstract:

We consider linear dynamical systems consisting of ordinary differential equations with high dimensionality. The aim of model order reduction is to construct an approximating system of a much lower dimension. Therein, the reduced system may be unstable, even though the original system is asymptotically stable. We focus on projection-based model order reduction of Galerkin-type. A transformation of the original system guarantees an asymptotically stable reduced system. This transformation requires the numerical solution of a high-dimensional Lyapunov equation. We specify an approximation of the solution, which allows for an efficient iterative treatment of the Lyapunov equation under a certain assumption. Furthermore, we generalize this strategy to preserve the asymptotic stability of stationary solutions in model order reduction of nonlinear dynamical systems. Numerical results for high-dimensional examples confirm the computational feasibility of the stability-preserving approach.

2019, 9(1) : 45-52 doi: 10.3934/naco.2019004 +[Abstract](330) +[HTML](168) +[PDF](284.38KB)
Abstract:

The purpose of this paper is to investigate the structure of the solution sets in parametric linear fractional programming problems. It is shown that the solution set of a parametric linear fractional programming problem with smooth data has a local smooth representation. As a consequence, the corresponding marginal function is differentiable and the solution map admits a differentiable selection. We also give an example to illustrate the result.

2019, 9(1) : 53-69 doi: 10.3934/naco.2019005 +[Abstract](453) +[HTML](185) +[PDF](678.7KB)
Abstract:

In this paper we study spectral properties of graphs which are constructed from two given invertible graphs by bridging them over a bipartite graph. We analyze the so-called HOMO-LUMO spectral gap which is the difference between the smallest positive and largest negative eigenvalue of the adjacency matrix of a graph. We investigate its dependence on the bridging bipartite graph and we construct a mixed integer semidefinite program for maximization of the HOMO-LUMO gap with respect to the bridging bipartite graph. We also derive upper and lower bounds for the optimal HOMO-LUMO spectral graph by means of semidefinite relaxation techniques. Several computational examples are also presented in this paper.

2019, 9(1) : 71-84 doi: 10.3934/naco.2019006 +[Abstract](329) +[HTML](196) +[PDF](3582.26KB)
Abstract:

In this paper, we derive an implicit symmetric, symplectic and exponentially fitted Runge-Kutta-Nyström (ISSEFRKN) method. The new integrator ISSEFRKN2 is of fourth order and integrates exactly differential systems whose solutions can be expressed as linear combinations of functions from the set $\{\exp(λ t), \exp(-λ t)|λ∈ \mathbb{C}\}$, or equivalently $\{\sin(ω t), \cos(ω t)|λ = iω, ~ω∈ \mathbb{R}\}$. We analysis the periodicity stability of the derived method ISSEFRKN2. Some the existing implicit RKN methods in the literature are used to compare with ISSEFRKN2 for several oscillatory problems. Numerical results show that the method ISSEFRKN2 possess a more accuracy among them.

2019, 9(1) : 85-99 doi: 10.3934/naco.2019007 +[Abstract](457) +[HTML](183) +[PDF](405.18KB)
Abstract:

Newton-type methods have gained much attention in the past decades, especially for the semilocal convergence based on no information around the solution $x_*$ of the target nonlinear equation. For large sparse non-Hermitian positive definite systems of nonlinear equation, assuming that the nonlinear operator satisfies the center Lipschitz condition, which is wider than usual Lipschtiz condition and H$ö$lder continuous condition, we establish a new Newton-Kantorovich convergence theorem for the Newton-HSS method. Once the convergence criteria is satisfied, the iteration sequence $\{x_k\}_{k = 0}^∞$ generated by the Newton-HSS method is well defined, and converges to the solution $x_*$. Numerical results illustrate the effect.

2019, 9(1) : 101-112 doi: 10.3934/naco.2019008 +[Abstract](424) +[HTML](215) +[PDF](359.78KB)
Abstract:

In this paper, we derive the operational matrices of integration, derivative and production of Hermite wavelets and use a direct numerical method based on Hermite wavelet, for solving optimal control problems. The properties of Hermite polynomials are used for finding these matrices. First, we approximate the state and control variables by Hermite wavelets basis; then, the operational matrices is used to transfer the given problem into a linear system of algebraic equations. In fact, operational matrices of Hermite wavelet are employed to achieve a linear algebraic equation, in place of the dynamical system in terms of the unknown coefficients. The solution of this system gives us the solution of the original problem. Numerical examples with time varying and time invariant coefficient are given to demonstrate the applicability of these matrices.

2011, 1(1) : 15-34 doi: 10.3934/naco.2011.1.15 +[Abstract](1631) +[PDF](445.3KB) Cited By(27)
2012, 2(3) : 571-599 doi: 10.3934/naco.2012.2.571 +[Abstract](1238) +[PDF](325.0KB) Cited By(18)
2011, 1(1) : 71-82 doi: 10.3934/naco.2011.1.71 +[Abstract](1088) +[PDF](194.9KB) Cited By(16)
2012, 2(1) : 69-90 doi: 10.3934/naco.2012.2.69 +[Abstract](1137) +[PDF](602.3KB) Cited By(15)
2012, 2(3) : 601-617 doi: 10.3934/naco.2012.2.601 +[Abstract](1046) +[PDF](342.2KB) Cited By(13)
2012, 2(3) : 547-570 doi: 10.3934/naco.2012.2.547 +[Abstract](1122) +[PDF](298.8KB) Cited By(13)
2012, 2(3) : 619-630 doi: 10.3934/naco.2012.2.619 +[Abstract](1025) +[PDF](199.0KB) Cited By(12)
2012, 2(4) : 695-711 doi: 10.3934/naco.2012.2.695 +[Abstract](1385) +[PDF](810.5KB) Cited By(12)
2013, 3(3) : 583-599 doi: 10.3934/naco.2013.3.583 +[Abstract](1117) +[PDF](442.5KB) Cited By(12)
2013, 3(2) : 247-260 doi: 10.3934/naco.2013.3.247 +[Abstract](1224) +[PDF](461.8KB) Cited By(11)
2018, 8(4) : 413-440 doi: 10.3934/naco.2018026 +[Abstract](986) +[HTML](218) +[PDF](716.28KB) PDF Downloads(140)
2019, 9(1) : 1-13 doi: 10.3934/naco.2019001 +[Abstract](720) +[HTML](276) +[PDF](397.82KB) PDF Downloads(132)
2018, 8(1) : 1-16 doi: 10.3934/naco.2018001 +[Abstract](1221) +[HTML](365) +[PDF](300.52KB) PDF Downloads(110)
2017, 7(4) : 379-401 doi: 10.3934/naco.2017024 +[Abstract](1718) +[HTML](297) +[PDF](381.9KB) PDF Downloads(90)
2018, 8(1) : 63-80 doi: 10.3934/naco.2018004 +[Abstract](1518) +[HTML](537) +[PDF](271.19KB) PDF Downloads(84)
2018, 8(4) : 389-412 doi: 10.3934/naco.2018025 +[Abstract](573) +[HTML](168) +[PDF](474.37KB) PDF Downloads(73)
2019, 9(1) : 15-22 doi: 10.3934/naco.2019002 +[Abstract](388) +[HTML](198) +[PDF](327.08KB) PDF Downloads(70)
2018, 8(2) : 135-155 doi: 10.3934/naco.2018008 +[Abstract](831) +[HTML](140) +[PDF](310.66KB) PDF Downloads(68)
2018, 8(3) : 315-326 doi: 10.3934/naco.2018020 +[Abstract](702) +[HTML](131) +[PDF](770.98KB) PDF Downloads(64)
2019, 9(1) : 101-112 doi: 10.3934/naco.2019008 +[Abstract](424) +[HTML](215) +[PDF](359.78KB) PDF Downloads(63)