### Abstract

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).

Original language | English |
---|---|

Pages (from-to) | 167-191 |

Number of pages | 25 |

Journal | Journal of Optimization Theory and Applications |

Volume | 141 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2009 Apr 1 |

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### Keywords

- Coerciveness
- EP merit functions
- Implicit Lagrangian function
- Jordan algebra
- Symmetric cone complementarity problem

### ASJC Scopus subject areas

- Control and Optimization
- Management Science and Operations Research
- Applied Mathematics

### Cite this

**Growth behavior of two classes of merit functions for symmetric cone complementarity problems.** / Pan, S. H.; Chen, Jein-Shan.

Research output: Contribution to journal › Article

*Journal of Optimization Theory and Applications*, vol. 141, no. 1, pp. 167-191. https://doi.org/10.1007/s10957-008-9495-y

}

TY - JOUR

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

AU - Pan, S. H.

AU - Chen, Jein-Shan

PY - 2009/4/1

Y1 - 2009/4/1

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

UR - http://www.scopus.com/inward/record.url?scp=62949186379&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=62949186379&partnerID=8YFLogxK

U2 - 10.1007/s10957-008-9495-y

DO - 10.1007/s10957-008-9495-y

M3 - Article

AN - SCOPUS:62949186379

VL - 141

SP - 167

EP - 191

JO - Journal of Optimization Theory and Applications

JF - Journal of Optimization Theory and Applications

SN - 0022-3239

IS - 1

ER -