### 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

### Cite this

**Analysis of nonsmooth vector-valued functions associated with infinite-dimensional second-order cones.** / Yang, Ching Yu; Chang, Yu-Lin; Chen, Jein-Shan.

Research output: Contribution to journal › Article

*Nonlinear Analysis, Theory, Methods and Applications*, vol. 74, no. 16, pp. 5766-5783. https://doi.org/10.1016/j.na.2011.05.068

}

TY - JOUR

T1 - Analysis of nonsmooth vector-valued functions associated with infinite-dimensional second-order cones

AU - Yang, Ching Yu

AU - Chang, Yu-Lin

AU - Chen, Jein-Shan

PY - 2011/11/1

Y1 - 2011/11/1

N2 - 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.

AB - 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.

KW - Hilbert space

KW - Infinite-dimensional second-order cone

KW - Strong semismoothness

UR - http://www.scopus.com/inward/record.url?scp=79959704821&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79959704821&partnerID=8YFLogxK

U2 - 10.1016/j.na.2011.05.068

DO - 10.1016/j.na.2011.05.068

M3 - Article

AN - SCOPUS:79959704821

VL - 74

SP - 5766

EP - 5783

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

IS - 16

ER -