A primal-dual interior point method for $ P_{\ast }\left( \kappa \right)  $-HLCP based on a class of parametric kernel functions

Nadia Hazzam; Zakia Kebbiche

doi:10.3934/naco.2020053

Article Contents

2021, Volume 11, Issue 4: 513-531. Doi: 10.3934/naco.2020053

This issue Previous Article Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable Next Article A new methodology for solving bi-criterion fractional stochastic programming

A primal-dual interior point method for $ P_{\ast }\left( \kappa \right) $-HLCP based on a class of parametric kernel functions

Nadia Hazzam and
Zakia Kebbiche

Laboratoire de Mathématiques Fondamentales et Numériques, Université Ferhat Abbas Sétif1. Sétif. Algérie

Received: May 31, 2020

Revised: September 30, 2020

Early access: November 2020

Published: December 2021

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Abstract

In an attempt to improve theoretical complexity of large-update methods, in this paper, we propose a primal-dual interior-point method for $ P_{\ast}\left( \kappa \right) $-horizontal linear complementarity problem. The method is based on a class of parametric kernel functions. We show that the corresponding algorithm has $ O\left( \left( 1+2\kappa \right) p^{2}n^{\frac{2+p}{2\left( 1+p\right) }}\log \frac{n}{\epsilon }\right) $ iteration complexity for large-update methods and we match the best known iteration bounds with special choice of the parameter $ p $ for $ P_{\ast }\left(\kappa \right) $-horizontal linear complementarity problem that is $ O\left(\left( 1+2\kappa \right) \sqrt{n}\log n\log \frac{n}{\epsilon }\right) $. We illustrate the performance of the proposed kernel function by some comparative numerical results that are derived by applying our algorithm on five kernel functions.

Keywords:

Horizontal linear complementarity problem,
P_*(κ)-matrix,
interior-point method,
kernel function,
complexity bound.

Mathematics Subject Classification: Primary: 90C33; Secondary: 90C51.

Citation:

Full Text(HTML)

Table 1. Some kernel functions

${\psi _1}(t) = p\left( {\frac{{{t^2} - 1}}{2}} \right) + \frac{4}{\pi }\left( {{{\tan }^p}\left( {\frac{\pi }{{2t + 2}}} \right) - 1} \right),t > 0,p \ge \sqrt 2 ,$ $\alpha_{1}=\frac{1}{(1+2 \kappa)\left(9 p+4 \pi p^{2}\right)\left(\frac{8}{p}+2\right)^{\frac{p+2}{p+1}}}.$
$\psi_{2}(t)=\frac{t^{2}-1}{2}-\log t+\frac{1}{8} \tan ^{2}\left(\frac{\pi(1-t)}{2+4 t}\right), t>0,[25]$ $\alpha_{2}=\frac{1}{(1+2 \kappa) 5020 \delta^{\frac{4}{3}}}.$
$\psi_{3}(t)=t-1+\frac{t^{1-q}-1}{q-1}, t>0, q \geq 2,[8]$ $\alpha_{3}=\frac{1}{(1+2 \kappa) q(2 \delta+1)^{\frac{1}{q}}(4 \delta+1)}.$
$\psi_{4}(t)=\frac{t^{2}-1}{2}+\frac{q^{\frac{1}{t}-1}}{q \log q}-\frac{q-1}{q}(t-1), q>1,[2]$ $\alpha_{4}=\frac{1}{(1+2 \kappa)(\log q+2)(1+4 \delta)\left(2+\frac{\log (1+4 \delta)}{\log q}\right)}.$
$\psi_{5}(t)=\frac{t^{2}-1}{2}+\frac{4}{p \pi}\left(\tan ^{p}\left(\frac{\pi}{2 t+2}\right)-1\right), t>0, p \geq 2,[9]$ $\alpha_{5}=\frac{1}{(1+2 \kappa)(9+4 p \pi)(8 \delta+2)^{\frac{p+2}{p+1}}}.$

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Table 2.

	$ \psi _{1}\left( t\right) $	$ \psi _{2}\left( t\right) $	$ \psi _{3}\left( t\right) $	$ \psi _{4}\left( t\right) $	$ \psi _{5}\left( t\right) $
Inn	$ 5 $	$ 282 $	$ 10 $	$ 16 $	$ 5 $
Time	$ 0.07 $	$ 0.18 $	$ 0.062 $	$ 0.06 $	$ 0.06 $

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Table 3.

	$ \psi _{1}\left( t\right) $	$ \psi _{2}\left( t\right) $	$ \psi _{3}\left( t\right) $	$ \psi _{4}\left( t\right) $	$ \psi _{5}\left( t\right) $
Inn	$ 7 $	$ 167 $	$ 12 $	$ 19 $	$ 10 $
Time	$ 0.06 $	$ 0.10 $	$ 0.06 $	$ 0.063 $	$ 0.07 $

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Table 4.

	$ n=3 $		$ n=100 $		$ n=150 $
	Inn	Time	Inn	Time	Inn	Time
$ \psi _{1}\left( t\right) $	$ 4 $	$ 0.06 $	$ 12 $	$ 0.45 $	$ 18 $	$ 3.06 $
$ \psi _{2}\left( t\right) $	$ 179 $	$ 0.11 $	$ 70 $	$ 9.07 $	$ 1447 $	$ 11.6 $
$ \psi _{3}\left( t\right) $	$ 15 $	$ 0.06 $	$ 52 $	$ 0.19 $	$ 77 $	$ 0.91 $
$ \psi _{4}\left( t\right) $	$ 14 $	$ 0.063 $	$ - $	$ - $	$ - $	$ - $
$ \psi _{5}\left( t\right) $	$ 8 $	$ 0.068 $	$ 23 $	$ 0.63 $	$ 28 $	$ 2.30 $

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Table 5.

	$ n=3 $		$ n=50 $		$ n=150 $
	Inn	Time	Inn	Time	Inn	Time
$ \psi _{1}\left( t\right) $	$ 3 $	$ 0.07 $	$ 10 $	$ 0.12 $	$ 19 $	$ 1.73 $
$ \psi _{2}\left( t\right) $	$ 259 $	$ 0.15 $	$ 615 $	$ 0.67 $	$ 1301 $	$ 13.96 $
$ \psi _{3}\left( t\right) $	$ 16 $	$ 0.05 $	$ 215 $	$ 0.45 $	$ 71 $	$ 0.64 $
$ \psi _{4}\left( t\right) $	$ 18 $	$ 0.05 $	$ - $	$ - $	$ - $	$ - $
$ \psi _{5}\left( t\right) $	$ 12 $	$ 0.07 $	$ 18 $	$ 0.15 $	$ 39 $	$ 2.47 $

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