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

N1 - Funding Information:
The authors would like to thank the referees for their helpful suggestions on the revision of this manuscript. The third author’s work is partially supported by the National Science Council of Taiwan .

PY - 2011/11

Y1 - 2011/11

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

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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 -