TY - JOUR

T1 - A one-parametric class of merit functions for the symmetric cone complementarity problem

AU - Pan, Shaohua

AU - Chen, Jein Shan

N1 - Funding Information:
E-mail addresses: shhpan@scut.edu.cn (S. Pan), jschen@math.ntnu.edu.tw (J.-S. Chen). 1 Partially supported by the Doctoral Starting-up Foundation (B13B6050640) of GuangDong Province. 2 Member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. Partially supported by National Science Council of Taiwan.

PY - 2009/7/1

Y1 - 2009/7/1

N2 - In this paper, we extend the one-parametric class of merit functions proposed by Kanzow and Kleinmichel [C. Kanzow, H. Kleinmichel, A new class of semismooth Newton-type methods for nonlinear complementarity problems, Comput. Optim. Appl. 11 (1998) 227-251] for the nonnegative orthant complementarity problem to the general symmetric cone complementarity problem (SCCP). We show that the class of merit functions is continuously differentiable everywhere and has a globally Lipschitz continuous gradient mapping. From this, we particularly obtain the smoothness of the Fischer-Burmeister merit function associated with symmetric cones and the Lipschitz continuity of its gradient. In addition, we also consider a regularized formulation for the class of merit functions which is actually an extension of one of the NCP function classes studied by [C. Kanzow, Y. Yamashita, M. Fukushima, New NCP functions and their properties, J. Optim. Theory Appl. 97 (1997) 115-135] to the SCCP. By exploiting the Cartesian P-properties for a nonlinear transformation, we show that the class of regularized merit functions provides a global error bound for the solution of the SCCP, and moreover, has bounded level sets under a rather weak condition which can be satisfied by the monotone SCCP with a strictly feasible point or the SCCP with the joint Cartesian R02-property. All of these results generalize some recent important works in [J.-S. Chen, P. Tseng, An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Program. 104 (2005) 293-327; C.-K. Sim, J. Sun, D. Ralph, A note on the Lipschitz continuity of the gradient of the squared norm of the matrix-valued Fischer-Burmeister function, Math. Program. 107 (2006) 547-553; P. Tseng, Merit function for semidefinite complementarity problems, Math. Program. 83 (1998) 159-185] under a unified framework.

AB - In this paper, we extend the one-parametric class of merit functions proposed by Kanzow and Kleinmichel [C. Kanzow, H. Kleinmichel, A new class of semismooth Newton-type methods for nonlinear complementarity problems, Comput. Optim. Appl. 11 (1998) 227-251] for the nonnegative orthant complementarity problem to the general symmetric cone complementarity problem (SCCP). We show that the class of merit functions is continuously differentiable everywhere and has a globally Lipschitz continuous gradient mapping. From this, we particularly obtain the smoothness of the Fischer-Burmeister merit function associated with symmetric cones and the Lipschitz continuity of its gradient. In addition, we also consider a regularized formulation for the class of merit functions which is actually an extension of one of the NCP function classes studied by [C. Kanzow, Y. Yamashita, M. Fukushima, New NCP functions and their properties, J. Optim. Theory Appl. 97 (1997) 115-135] to the SCCP. By exploiting the Cartesian P-properties for a nonlinear transformation, we show that the class of regularized merit functions provides a global error bound for the solution of the SCCP, and moreover, has bounded level sets under a rather weak condition which can be satisfied by the monotone SCCP with a strictly feasible point or the SCCP with the joint Cartesian R02-property. All of these results generalize some recent important works in [J.-S. Chen, P. Tseng, An unconstrained smooth minimization reformulation of the second-order cone complementarity problem, Math. Program. 104 (2005) 293-327; C.-K. Sim, J. Sun, D. Ralph, A note on the Lipschitz continuity of the gradient of the squared norm of the matrix-valued Fischer-Burmeister function, Math. Program. 107 (2006) 547-553; P. Tseng, Merit function for semidefinite complementarity problems, Math. Program. 83 (1998) 159-185] under a unified framework.

KW - Cartesian P-properties

KW - Jordan algebra

KW - Lipschitz continuity

KW - Merit function

KW - Smoothness

KW - Symmetric cone complementarity problem

UR - http://www.scopus.com/inward/record.url?scp=60749099696&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=60749099696&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2009.01.064

DO - 10.1016/j.jmaa.2009.01.064

M3 - Article

AN - SCOPUS:60749099696

VL - 355

SP - 195

EP - 215

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -