TY - JOUR

T1 - Growth behavior of two classes of merit functions for symmetric cone complementarity problems

AU - Pan, S. H.

AU - Chen, J. S.

N1 - Funding Information:
The authors would like to thank the two anonymous referees for their helpful comments which improved the presentation of this paper greatly. The research of J.-S. Chen was partially supported by National Science Council of Taiwan.

PY - 2009/4

Y1 - 2009/4

N2 - In the solution methods of the symmetric cone complementarity problem (SCCP), the squared norm of a complementarity function serves naturally as a merit function for the problem itself or the equivalent system of equations reformulation. In this paper, we study the growth behavior of two classes of such merit functions, which are induced by the smooth EP complementarity functions and the smooth implicit Lagrangian complementarity function, respectively. We show that, for the linear symmetric cone complementarity problem (SCLCP), both the EP merit functions and the implicit Lagrangian merit function are coercive if the underlying linear transformation has the P-property; for the general SCCP, the EP merit functions are coercive only if the underlying mapping has the uniform Jordan P-property, whereas the coerciveness of the implicit Lagrangian merit function requires an additional condition for the mapping, for example, the Lipschitz continuity or the assumption as in (45).

AB - In the solution methods of the symmetric cone complementarity problem (SCCP), the squared norm of a complementarity function serves naturally as a merit function for the problem itself or the equivalent system of equations reformulation. In this paper, we study the growth behavior of two classes of such merit functions, which are induced by the smooth EP complementarity functions and the smooth implicit Lagrangian complementarity function, respectively. We show that, for the linear symmetric cone complementarity problem (SCLCP), both the EP merit functions and the implicit Lagrangian merit function are coercive if the underlying linear transformation has the P-property; for the general SCCP, the EP merit functions are coercive only if the underlying mapping has the uniform Jordan P-property, whereas the coerciveness of the implicit Lagrangian merit function requires an additional condition for the mapping, for example, the Lipschitz continuity or the assumption as in (45).

KW - Coerciveness

KW - EP merit functions

KW - Implicit Lagrangian function

KW - Jordan algebra

KW - Symmetric cone complementarity problem

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U2 - 10.1007/s10957-008-9495-y

DO - 10.1007/s10957-008-9495-y

M3 - Article

AN - SCOPUS:62949186379

SN - 0022-3239

VL - 141

SP - 167

EP - 191

JO - Journal of Optimization Theory and Applications

JF - Journal of Optimization Theory and Applications

IS - 1

ER -