# American Institute of Mathematical Sciences

ISSN:
2155-3289

eISSN:
2155-3297

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## Numerical Algebra, Control & Optimization

December 2018 , Volume 8 , Issue 4

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2018, 8(4): 389-412 doi: 10.3934/naco.2018025 +[Abstract](3272) +[HTML](258) +[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.

2018, 8(4): 413-440 doi: 10.3934/naco.2018026 +[Abstract](5296) +[HTML](539) +[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.

2018, 8(4): 441-449 doi: 10.3934/naco.2018027 +[Abstract](2921) +[HTML](195) +[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.

2018, 8(4): 451-460 doi: 10.3934/naco.2018028 +[Abstract](2633) +[HTML](271) +[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.

2018, 8(4): 461-479 doi: 10.3934/naco.2018029 +[Abstract](3858) +[HTML](354) +[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.

2018, 8(4): 481-492 doi: 10.3934/naco.2018030 +[Abstract](3555) +[HTML](265) +[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.

2018, 8(4): 493-503 doi: 10.3934/naco.2018031 +[Abstract](2738) +[HTML](269) +[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).