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Sparse Solutions of a Class of Constrained Optimization Problems
In this paper, we consider a well-known sparse optimization problem that aims to find a sparse solution of a possibly noisy underdetermined system of linear equations. Mathematically, it can be modeled in a unified manner by minimizing ||x||pp subject to ||Ax−b∥q≤σ for given A∈Rm×n, b∈Rm, σ≥0, 0≤p≤1 and q≥1. We then study various properties of the optimal solutions of this problem. Specifically, without any condition on the matrix A, we provide upper bounds in cardinality and infinity norm for the optimal solutions and show that all optimal solutions must be on the boundary of the feasible set when 0
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