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Stability Analysis for a Class of Sparse Optimization Problems

J.L. XU(JXX168***at***bham.ac.uk)
Yun-Bin ZHAO(y.zhao.2***at***bham.ac.uk)

Abstract: The sparse optimization problems arise in many areas of science and engineering, such as compressed sensing, image processing, statistical and machine learning. The $\ell_{0}$-minimization problem is one of such optimization problems, which is typically used to deal with signal recovery. The $\ell_{1}$-minimization method is one of the plausible approaches for solving the $\ell_{0}$-minimization problems, and thus the stability of such a numerical method is vital for signal recovery. In this paper, we establish a stability result for the $\ell_{1}$-minimization problems associated with a general class of $\ell_{0}$-minimization problems. To this goal, we introduce the concept of restricted weak range space property (RSP) of a transposed sensing matrix, which is a generalized version of the weak RSP of the transposed sensing matrix introduced in [Zhao et al., Math. Oper. Res., 44(2019), 175-193]. The stability result established in this paper includes several existing ones as special cases.

Keywords: Sparsity optimization; $\ell_{1}$-minimization; stability; optimality condition; Hoffman theorem; restricted weak range space property

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Category 2: Nonlinear Optimization (Constrained Nonlinear Optimization )

Category 3: Linear, Cone and Semidefinite Programming (Linear Programming )


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Entry Submitted: 04/29/2019
Entry Accepted: 04/29/2019
Entry Last Modified: 04/29/2019

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