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

被引:44
作者
Pan, Shaohua [2 ]
Chen, Jein-Shan [1 ]
机构
[1] Natl Taiwan Normal Univ, Dept Math, Taipei 11677, Taiwan
[2] S China Univ Technol, Sch Math Sci, Guangzhou 510640, Guangdong, Peoples R China
关键词
Symmetric cone complementarity problem; Merit function; Jordan algebra; Smoothness; Lipschitz continuity; Cartesian P-properties; INTERIOR-POINT ALGORITHMS; JORDAN ALGEBRAS; NONLINEAR COMPLEMENTARITY; SPECTRAL FUNCTIONS; SQUARED NORM; P-PROPERTIES; TRANSFORMATIONS; PROPERTY;
D O I
10.1016/j.jmaa.2009.01.064
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
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. Ciptim. 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 SCCR 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 R-02-property. All of these results generalize some recent important works in [J.-S. Chen, R 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; R Tseng, Merit function for semidefinite complementarity problems, Math. Program. 83 (1998) 159-185] under a unified framework. (c) 2009 Elsevier Inc. All rights reserved.
引用
收藏
页码:195 / 215
页数:21
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