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    The Non-convex Sparse Problem with Nonnegative Constraint for Signal Reconstruction

    Access Status
    Fulltext not available
    Authors
    Wang, Y.
    Zhou, Guanglu
    Zhang, X.
    Liu, W.
    Caccetta, Louis
    Date
    2016
    Type
    Journal Article
    
    Metadata
    Show full item record
    Citation
    Wang, Y. and Zhou, G. and Zhang, X. and Liu, W. and Caccetta, L. 2016. The Non-convex Sparse Problem with Nonnegative Constraint for Signal Reconstruction. Journal of Optimization Theory and Applications. 170 (3): pp. 1009-1025.
    Source Title
    Journal of Optimization Theory and Applications
    DOI
    10.1007/s10957-016-0869-2
    ISSN
    0022-3239
    School
    Department of Mathematics and Statistics
    URI
    http://hdl.handle.net/20.500.11937/15730
    Collection
    • Curtin Research Publications
    Abstract

    The problem of finding a sparse solution for linear equations has been investigated extensively in recent years. This is an NP-hard combinatorial problem, and one popular method is to relax such combinatorial requirement into an approximated convex problem, which can avoid the computational complexity. Recently, it is shown that a sparser solution than the approximated convex solution can be obtained by solving its non-convex relaxation rather than by solving its convex relaxation. However, solving the non-convex relaxation is usually very costive due to the non-convexity and non-Lipschitz continuity of the original problem. This difficulty limits its applications and possible extensions. In this paper, we will consider the non-convex relaxation problem with the nonnegative constraint, which has many applications in signal processing with such reasonable requirement. First, this optimization problem is formulated and equivalently transformed into a Lipschitz continuous problem, which can be solved by many existing optimization methods. This reduces the computational complexity of the original problem significantly. Second, we solve the transformed problem by using an efficient and classical limited-memory Broyden–Fletcher–Goldfarb–Shanno algorithm. Finally, some numerical results show that the proposed method can effectively find a nonnegative sparse solution for the given linear equations with very low computational cost.

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