A damped Gauss-Newton method for the second-order cone complementarity problem

Shaohua Pan; Jein-Shan Chen

doi:10.1007/s00245-008-9054-9

A damped Gauss-Newton method for the second-order cone complementarity problem

Shaohua Pan, Jein-Shan Chen

Department of Mathematics

Research output: Contribution to journal › Article

37 Citations (Scopus)

Abstract

We investigate some properties related to the generalized Newton method for the Fischer-Burmeister (FB) function over second-order cones, which allows us to reformulate the second-order cone complementarity problem (SOCCP) as a semismooth system of equations. Specifically, we characterize the B-subdifferential of the FB function at a general point and study the condition for every element of the B-subdifferential at a solution being nonsingular. In addition, for the induced FB merit function, we establish its coerciveness and provide a weaker condition than Chen and Tseng (Math. Program. 104:293-327, 2005) for each stationary point to be a solution, under suitable Cartesian P-properties of the involved mapping. By this, a damped Gauss-Newton method is proposed, and the global and superlinear convergence results are obtained. Numerical results are reported for the second-order cone programs from the DIMACS library, which verify the good theoretical properties of the method.

Original language	English
Pages (from-to)	293-318
Number of pages	26
Journal	Applied Mathematics and Optimization
Volume	59
Issue number	3
DOIs	https://doi.org/10.1007/s00245-008-9054-9
Publication status	Published - 2009 Jun 1

Fingerprint

Keywords

B-subdifferential
Complementarity
Fischer-Burmeister function
Generalized Newton method
Second-order cones

ASJC Scopus subject areas

Applied Mathematics
Control and Optimization

Cite this

Pan, S., & Chen, J-S. (2009). A damped Gauss-Newton method for the second-order cone complementarity problem. Applied Mathematics and Optimization, 59(3), 293-318. https://doi.org/10.1007/s00245-008-9054-9

@article{30115922d44149bb8a24f3b33f895131,

title = "A damped Gauss-Newton method for the second-order cone complementarity problem",

abstract = "We investigate some properties related to the generalized Newton method for the Fischer-Burmeister (FB) function over second-order cones, which allows us to reformulate the second-order cone complementarity problem (SOCCP) as a semismooth system of equations. Specifically, we characterize the B-subdifferential of the FB function at a general point and study the condition for every element of the B-subdifferential at a solution being nonsingular. In addition, for the induced FB merit function, we establish its coerciveness and provide a weaker condition than Chen and Tseng (Math. Program. 104:293-327, 2005) for each stationary point to be a solution, under suitable Cartesian P-properties of the involved mapping. By this, a damped Gauss-Newton method is proposed, and the global and superlinear convergence results are obtained. Numerical results are reported for the second-order cone programs from the DIMACS library, which verify the good theoretical properties of the method.",

keywords = "B-subdifferential, Complementarity, Fischer-Burmeister function, Generalized Newton method, Second-order cones",

author = "Shaohua Pan and Jein-Shan Chen",

year = "2009",

month = jun

day = "1",

doi = "10.1007/s00245-008-9054-9",

language = "English",

volume = "59",

pages = "293--318",

journal = "Applied Mathematics and Optimization",

issn = "0095-4616",

publisher = "Springer New York",

number = "3",

}

TY - JOUR

T1 - A damped Gauss-Newton method for the second-order cone complementarity problem

AU - Pan, Shaohua

AU - Chen, Jein-Shan

PY - 2009/6/1

Y1 - 2009/6/1

N2 - We investigate some properties related to the generalized Newton method for the Fischer-Burmeister (FB) function over second-order cones, which allows us to reformulate the second-order cone complementarity problem (SOCCP) as a semismooth system of equations. Specifically, we characterize the B-subdifferential of the FB function at a general point and study the condition for every element of the B-subdifferential at a solution being nonsingular. In addition, for the induced FB merit function, we establish its coerciveness and provide a weaker condition than Chen and Tseng (Math. Program. 104:293-327, 2005) for each stationary point to be a solution, under suitable Cartesian P-properties of the involved mapping. By this, a damped Gauss-Newton method is proposed, and the global and superlinear convergence results are obtained. Numerical results are reported for the second-order cone programs from the DIMACS library, which verify the good theoretical properties of the method.

AB - We investigate some properties related to the generalized Newton method for the Fischer-Burmeister (FB) function over second-order cones, which allows us to reformulate the second-order cone complementarity problem (SOCCP) as a semismooth system of equations. Specifically, we characterize the B-subdifferential of the FB function at a general point and study the condition for every element of the B-subdifferential at a solution being nonsingular. In addition, for the induced FB merit function, we establish its coerciveness and provide a weaker condition than Chen and Tseng (Math. Program. 104:293-327, 2005) for each stationary point to be a solution, under suitable Cartesian P-properties of the involved mapping. By this, a damped Gauss-Newton method is proposed, and the global and superlinear convergence results are obtained. Numerical results are reported for the second-order cone programs from the DIMACS library, which verify the good theoretical properties of the method.

KW - B-subdifferential

KW - Complementarity

KW - Fischer-Burmeister function

KW - Generalized Newton method

KW - Second-order cones

UR - http://www.scopus.com/inward/record.url?scp=63049104088&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=63049104088&partnerID=8YFLogxK

U2 - 10.1007/s00245-008-9054-9

DO - 10.1007/s00245-008-9054-9

M3 - Article

AN - SCOPUS:63049104088

VL - 59

SP - 293

EP - 318

JO - Applied Mathematics and Optimization

JF - Applied Mathematics and Optimization

SN - 0095-4616

IS - 3

ER -

Access to Document

10.1007/s00245-008-9054-9