### Abstract

Given a Hilbert space H, the infinite-dimensional Lorentz/second-order cone K is introduced. For any x∈H, a spectral decomposition is introduced, and for any function f:R→R, we define a corresponding vector-valued function ^{fH}(x) on Hilbert space H by applying f to the spectral values of the spectral decomposition of x∈H with respect to K. We show that this vector-valued function inherits from f the properties of continuity, Lipschitz continuity, differentiability, smoothness, as well as s-semismoothness. These results can be helpful for designing and analyzing solution methods for solving infinite-dimensional second-order cone programs and complementarity problems.

Original language | English |
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Pages (from-to) | 5766-5783 |

Number of pages | 18 |

Journal | Nonlinear Analysis, Theory, Methods and Applications |

Volume | 74 |

Issue number | 16 |

DOIs | |

Publication status | Published - 2011 Nov 1 |

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### Keywords

- Hilbert space
- Infinite-dimensional second-order cone
- Strong semismoothness

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics