A smoothing Newton method based on the generalized Fischer-Burmeister function for MCPs

Jein Shan Chen; Shaohua Pan; Tzu Ching Lin

doi:10.1016/j.na.2010.01.012

A smoothing Newton method based on the generalized Fischer-Burmeister function for MCPs

Jein Shan Chen^*
, Shaohua Pan
, Tzu Ching Lin

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

16 Citations (Scopus)

Abstract

We present a smooth approximation for the generalized Fischer-Burmeister function where the 2-norm in the FB function is relaxed to a general p-norm (p > 1), and establish some favorable properties for it - for example, the Jacobian consistency. With the smoothing function, we transform the mixed complementarity problem (MCP) into solving a sequence of smooth system of equations, and then trace a smooth path generated by the smoothing algorithm proposed by Chen (2000) [28] to the solution set. In particular, we investigate the influence of p on the numerical performance of the algorithm by solving all MCPLIP test problems, and conclude that the smoothing algorithm with p ∈ (1, 2] has better numerical performance than the one with p > 2.

Original language	English
Pages (from-to)	3739-3758
Number of pages	20
Journal	Nonlinear Analysis, Theory, Methods and Applications
Volume	72
Issue number	9-10
DOIs	https://doi.org/10.1016/j.na.2010.01.012
Publication status	Published - 2010 May 1

Keywords

Convergence rate
Mixed complementarity problem
Smoothing approximation
The generalized FB function

ASJC Scopus subject areas

Analysis
Applied Mathematics

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10.1016/j.na.2010.01.012

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title = "A smoothing Newton method based on the generalized Fischer-Burmeister function for MCPs",

abstract = "We present a smooth approximation for the generalized Fischer-Burmeister function where the 2-norm in the FB function is relaxed to a general p-norm (p > 1), and establish some favorable properties for it - for example, the Jacobian consistency. With the smoothing function, we transform the mixed complementarity problem (MCP) into solving a sequence of smooth system of equations, and then trace a smooth path generated by the smoothing algorithm proposed by Chen (2000) [28] to the solution set. In particular, we investigate the influence of p on the numerical performance of the algorithm by solving all MCPLIP test problems, and conclude that the smoothing algorithm with p ∈ (1, 2] has better numerical performance than the one with p > 2.",

keywords = "Convergence rate, Mixed complementarity problem, Smoothing approximation, The generalized FB function",

author = "Chen, \{Jein Shan\} and Shaohua Pan and Lin, \{Tzu Ching\}",

note = "Funding Information: The first author{\textquoteright}s work is partially supported by National Science Council of Taiwan. The second author{\textquoteright}s work is supported by National Young Natural Science Foundation (No. 10901058) and Guangdong Natural Science Foundation (No. 9251802902000001). ",

year = "2010",

month = may,

day = "1",

doi = "10.1016/j.na.2010.01.012",

language = "English",

volume = "72",

pages = "3739--3758",

journal = "Nonlinear Analysis, Theory, Methods and Applications",

issn = "0362-546X",

publisher = "Elsevier Limited",

number = "9-10",

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TY - JOUR

T1 - A smoothing Newton method based on the generalized Fischer-Burmeister function for MCPs

AU - Chen, Jein Shan

AU - Pan, Shaohua

AU - Lin, Tzu Ching

N1 - Funding Information: The first author’s work is partially supported by National Science Council of Taiwan. The second author’s work is supported by National Young Natural Science Foundation (No. 10901058) and Guangdong Natural Science Foundation (No. 9251802902000001).

PY - 2010/5/1

Y1 - 2010/5/1

N2 - We present a smooth approximation for the generalized Fischer-Burmeister function where the 2-norm in the FB function is relaxed to a general p-norm (p > 1), and establish some favorable properties for it - for example, the Jacobian consistency. With the smoothing function, we transform the mixed complementarity problem (MCP) into solving a sequence of smooth system of equations, and then trace a smooth path generated by the smoothing algorithm proposed by Chen (2000) [28] to the solution set. In particular, we investigate the influence of p on the numerical performance of the algorithm by solving all MCPLIP test problems, and conclude that the smoothing algorithm with p ∈ (1, 2] has better numerical performance than the one with p > 2.

AB - We present a smooth approximation for the generalized Fischer-Burmeister function where the 2-norm in the FB function is relaxed to a general p-norm (p > 1), and establish some favorable properties for it - for example, the Jacobian consistency. With the smoothing function, we transform the mixed complementarity problem (MCP) into solving a sequence of smooth system of equations, and then trace a smooth path generated by the smoothing algorithm proposed by Chen (2000) [28] to the solution set. In particular, we investigate the influence of p on the numerical performance of the algorithm by solving all MCPLIP test problems, and conclude that the smoothing algorithm with p ∈ (1, 2] has better numerical performance than the one with p > 2.

KW - Convergence rate

KW - Mixed complementarity problem

KW - Smoothing approximation

KW - The generalized FB function

UR - https://www.scopus.com/pages/publications/76549123306

UR - https://www.scopus.com/pages/publications/76549123306#tab=citedBy

U2 - 10.1016/j.na.2010.01.012

DO - 10.1016/j.na.2010.01.012

M3 - Article

AN - SCOPUS:76549123306

SN - 0362-546X

VL - 72

SP - 3739

EP - 3758

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

IS - 9-10

ER -

A smoothing Newton method based on the generalized Fischer-Burmeister function for MCPs

Abstract

Keywords

ASJC Scopus subject areas

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