TY - JOUR

T1 - A least-square semismooth Newton method for the second-order cone complementarity problem

AU - Pan, Shaohua

AU - Chen, Jein Shan

N1 - Funding Information:
The authors would like to thank the two anonymous referees for their valuable comments and suggestions for this paper. Member of Mathematics Division, National Centre for Theoretical Sciences, Taipei Office. The author’s work is partially supported by National Science Council of Taiwan.

PY - 2011/2

Y1 - 2011/2

N2 - We present a nonlinear least-square formulation for the second-order cone complementarity problem based on the Fischer-Burmeister (FB) function and the plus function. This formulation has two-fold advantages. First, the operator involved in the over-determined system of equations inherits the favourable properties of the FB function for local convergence, for example, the (strong) semi-smoothness; second, the natural merit function of the over-determined system of equations share all the nice features of the class of merit functions fYF studied in [J.-S. Chen and P. Tseng, An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Program. 104 (2005), pp. 293-327] for global convergence. We propose a semi-smooth Levenberg-Marquardt method to solve the arising over-determined system of equations, and establish the global and local convergence results. Among others, the superlinear (quadratic) rate of convergence is obtained under strict complementarity of the solution and a local error bound assumption, respectively. Numerical results verify the advantages of the least-square reformulation for difficult problems.

AB - We present a nonlinear least-square formulation for the second-order cone complementarity problem based on the Fischer-Burmeister (FB) function and the plus function. This formulation has two-fold advantages. First, the operator involved in the over-determined system of equations inherits the favourable properties of the FB function for local convergence, for example, the (strong) semi-smoothness; second, the natural merit function of the over-determined system of equations share all the nice features of the class of merit functions fYF studied in [J.-S. Chen and P. Tseng, An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Program. 104 (2005), pp. 293-327] for global convergence. We propose a semi-smooth Levenberg-Marquardt method to solve the arising over-determined system of equations, and establish the global and local convergence results. Among others, the superlinear (quadratic) rate of convergence is obtained under strict complementarity of the solution and a local error bound assumption, respectively. Numerical results verify the advantages of the least-square reformulation for difficult problems.

KW - Fischer-Burmeister function

KW - Levenberg-Marquardt method

KW - second-order cone complementarity problem

KW - semi-smooth

UR - http://www.scopus.com/inward/record.url?scp=77957316917&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77957316917&partnerID=8YFLogxK

U2 - 10.1080/10556780903180366

DO - 10.1080/10556780903180366

M3 - Article

AN - SCOPUS:77957316917

SN - 1055-6788

VL - 26

SP - 1

EP - 22

JO - Optimization Methods and Software

JF - Optimization Methods and Software

IS - 1

ER -