Computing minimum norm solution of linear systems of equations by the generalized Newton method

Saeed Ketabchi; Hossein Moosaei; M. Parandegan; Hamidreza Navidi

doi:10.3934/naco.2017008

Article Contents

2017, Volume 7, Issue 2: 113-119. Doi: 10.3934/naco.2017008

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Computing minimum norm solution of linear systems of equations by the generalized Newton method

1.
Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
2.
Department of Mathematics, Faculty of Science, University of Bojnord, Bojnord, Iran
3.
Shahed University, Tehran, Iran

^* Corresponding author: Saeed Ketabchi
^* Corresponding author: Saeed Ketabchi

Received: July 2015

Published online: June 2017

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Abstract

The aim of this paper is to find the minimum norm solution of a linear system of equations. The proposed method is based on presenting a view of solution on the dual exterior penalty problem of primal quadratic programming. To solve the unconstrained minimization problem, the generalized Newton method was employed and to guarantee its finite global convergence, the Armijo step size regulation was adopted. This method was tested on all systems selected in NETLIB ¹. Numerical results were compared with the MOSEK Optimization Software ² on linear systems in NETLIB (Table 1) and on linear systems generated by the Linear systems generator (Table 2).

¹www.netlib.org

² www.mosek.com

Keywords:

Mathematics Subject Classification: Primary: 90C06; Secondary: 90C20.

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Table 1. Comparison of ssGNewton and cqpMosek on systems in NETLIB. In all solved examples $\|(-x^*)_+\|=0 $

$Systems$	Method	$ time(sec)$	$\\|{x^*}\\|$	$\\|Ax^*-b\\|_{\infty}$
$25fv47$	ssGNewton	$1.51$	$3.31\times 10^3 $	$3.43\times 10^{-9}$
$25fv47$	cqpMosek	$1.36$	$3.31\times 10^3$	$1.91\times 10^{-11}$
$80bau3b$	ssGNewton	$0.95$	$4.13\times 10^3 $	$1.18\times 10^{-12}$
$80bau3b$	cqpMosek	$0.80$	$4.13\times 10^3$	$2.90\times 10^{-7}$
$addlittle$	ssGNewton	$0.05$	$4.3\times 10^2$	$2.27\times 10^{-13}$
$addlittle$	cqpMosek	0.35	$4.3\times 10^2$	$7.18\times 10^{-11}$
$ afiro $	ssGNewton	$0.06$	$6.34\times 10^2$	$6.39\times 10^{-14}$
$ afiro $	cqpMosek	$0.31$	$6.34\times 10^2$	$1.13\times 10^{-13}$
$creb$	ssGNewton	$13.20$	$6.24\times 10^{2}$	$1.62\times 10^{-9}$
$creb$	cqpMosek	$ 2.31$	$6.24\times 10^{2}$	$1.61\times 10^{-6}$
$maroser7$	ssGNewton	$2.86$	$1.41\times 10^5 $	$2.54\times 10^{-11}$
$maroser7$	cqpMosek	$55.20$	$1.41\times 10^5$	$3.27\times 10^{-11}$
$pds02$	ssGNewton	$2.30$	$1.61\times 10^5$	$1.40\times 10^{-7}$
$pds02$	cqpMosek	$0.51$	$1.61\times 10^5$	$8.2\times 10^{-6}$
$ken13$	ssGNewton	$9.09$	$2.53\times 10^4 $	$4.39\times 10^{-9}$
$ken13$	cqpMosek	$2.09$	$2.53\times 10^4 $	$1.71\times 10^{-9}$
$osa14$	ssGNewton	$60.10$	$1.19\times 10^5$	$4.10\times10^{-8}$
$osa14$	cqpMosek	$4.40$	$1.19\times 10^5$	$7.82\times 10^{-5}$
$agg3$	ssGNewton	$0.27$	$7.65\times 10^5$	$3.59\times10^{-8}$
$agg3$	cqpMosek	$ 0.40$	$7.65\times 10^5$	$2.32\times10^{-10}$
$crea $	ssGNewton	$1.25$	$1.62\times10^{3}$	$4.13\times10^{-6}$
$crea $	cqpMosek	0.61	$1.62\times10^{3}$	$2.15\times10^{-10}$

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Table 2. Comparison of ssGNewton and cqpMosek on randomly generated systems. 'OOM' denotes out of memory. In all solved examples $\|(-x^*)_+\|=0 $

$m, n, d$	Method	$ time(sec)$	$\\|{x^*}\\|$	$\\|Ax^*-b\\|_{\infty}$
$1000\times 1200\times 0.1$	ssGNewton	$6.92$	$1.87\times 10^0 $	$7.73\times 10^{-12}$
$1000\times 1200\times 0.1$	cqpMosek	$14.70$	$1.87\times 10^0 $	$5.45\times 10^{-12}$
$1000\times 1500\times 0.01$	ssGNewton	$3.94$	$1.90\times 10^2 $	$1.13\times 10^{-12}$
$1000\times 1500\times 0.01$	cqpMosek	$3.10$	$1.90\times 10^2$	$6.80\times 10^{-13}$
$2000\times 10000\times 0.1$	ssGNewton	$29.10$	$2.94\times 10^2$	$4.54\times 10^{-11}$
$2000\times 10000\times 0.1$	cqpMosek	122.00	$2.94\times 10^2$	$6.18\times 10^{-11}$
$ 4500\times 5000\times 0.1 $	ssGNewton	OOM	$-$	$-$
$ 4500\times 5000\times 0.1 $	cqpMosek	$339.00$	$3.95\times 10^2$	$3.89\times 10^{-8}$
$1000\times (5\times 10^6)\times 10^{-6} $	ssGNewton	$19.20$	$2.02\times 10^{2}$	$2.27\times 10^{-13}$
$1000\times (5\times 10^6)\times 10^{-6} $	cqpMosek	OOM	$-$	$-$
$1000\times 100000\times 0.01$	ssGNewton	$4.54$	$2.45\times 10^2 $	$3.27\times 10^{-11}$
$1000\times 100000\times 0.01$	cqpMosek	$9.10$	$2.45\times 10^2$	$3.63\times 10^{-11}$
$100\times (2\times 10^6)\times 0.001$	ssGNewton	$4.35$	$8.13\times 10$	$5.09\times 10^{-11}$
$100\times (2\times 10^6)\times 0.001$	cqpMosek	$30.40$	$8.13\times 10$	$4.69\times 10^{-10}$
$(2.5\times10^5) \times(3\times 10^5)\times 10^{-6}$	ssGNewton	$70.12$	$1.41\times 10^3 $	$4.10\times 10^{-3}$
$(2.5\times10^5) \times(3\times 10^5)\times 10^{-6}$	cqpMosek	$5.32$	$1.41\times 10^3 $	$4.90\times 10^{-13}$
$10^5\times (2\times10^5)\times 10^{-6}$	ssGNewton	$9.85$	$7.65\times 10^2$	$2.87\times10^{-8}$
$10^5\times (2\times10^5)\times 10^{-6}$	cqpMosek	$3.26$	$7.65\times 10^2$	$2.27\times 10^{-13}$

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