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    Spectral operators of matrices

    254801.pdf (247.6Kb)
    Access Status
    Open access
    Authors
    Ding, C.
    Sun, D.
    Sun, Jie
    Toh, K.
    Date
    2017
    Type
    Journal Article
    
    Metadata
    Show full item record
    Citation
    Ding, C. and Sun, D. and Sun, J. and Toh, K. 2017. Spectral operators of matrices. Mathematical Programming. 168 (1-2): pp. 509-531.
    Source Title
    Mathematical Programming
    DOI
    10.1007/s10107-017-1162-3
    ISSN
    0025-5610
    School
    Department of Mathematics and Statistics
    Remarks

    The final publication is available at Springer via http://dx.doi.org/10.1007/s10107-017-1162-3

    URI
    http://hdl.handle.net/20.500.11937/55609
    Collection
    • Curtin Research Publications
    Abstract

    The class of matrix optimization problems (MOPs) has been recognized in recent years to be a powerful tool to model many important applications involving structured low rank matrices within and beyond the optimization community. This trend can be credited to some extent to the exciting developments in emerging fields such as compressed sensing. The Löwner operator, which generates a matrix valued function via applying a single-variable function to each of the singular values of a matrix, has played an important role for a long time in solving matrix optimization problems. However, the classical theory developed for the Löwner operator has become inadequate in these recent applications. The main objective of this paper is to provide necessary theoretical foundations from the perspectives of designing efficient numerical methods for solving MOPs. We achieve this goal by introducing and conducting a thorough study on a new class of matrix valued functions, coined as spectral operators of matrices. Several fundamental properties of spectral operators, including the well-definedness, continuity, directional differentiability and Fréchet-differentiability are systematically studied. © 2017 Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.

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