range space property

Equivalence and Strong Equivalence between Sparsest and Least $\ell_1hBcNorm Nonnegative Solutions of Linear Systems and Their Application

  • Yun-Bin Zhao
    • Categories Applications - Science and Engineering, Convex Optimization, Linear Programming Tags , , , ,

    Many practical problems can be formulated as $\ell_0$-minimization problems with nonnegativity constraints, which seek the sparsest nonnegative solutions to underdetermined linear systems. Recent study indicates that $\ell_1$-minimization is efficient for solving some classes of $\ell_0$-minimization problems. From a mathematical point of view, however, the understanding of the relationship between $\ell_0$- and $\ell_1$-minimization remains incomplete. In … Read more

    Equivalence and Strong Equivalence between Sparsest and Least l1-Norm Nonnegative Solutions of Linear Systems and Their Application

  • Yun-Bin Zhao
    • Categories Convex Optimization, Global Optimization, Linear Programming Tags , , , ,

    Many practical problems can be formulated as $\ell_0$-minimization problems with nonnegativity constraints, which seek the sparsest nonnegative solutions to underdetermined linear systems. Recent study indicates that $\ell_1$-minimization is efficient for solving some classes of $\ell_0$-minimization problems. From a mathematical point of view, however, the understanding of the relationship between $\ell_0$- and $\ell_1$-minimization remains incomplete. In … Read more

    RSP-Based Analysis for Sparsest and Least $\ell_1hBcNorm Solutions to Underdetermined Linear Systems

  • Yun-Bin Zhao
    • Categories Global Optimization, Linear Programming, Linear, Cone and Semidefinite Programming Tags , , , , ,

    Recently, the worse-case analysis, probabilistic analysis and empirical justification have been employed to address the fundamental question: When does $\ell_1$-minimization find the sparsest solution to an underdetermined linear system? In this paper, a deterministic analysis, rooted in the classic linear programming theory, is carried out to further address this question. We first identify a necessary … Read more