Gradient Support Projection Algorithm for Affine Feasibility Problem with Sparsity and Nonnegativity

Let $A$ be a real $M \times N$ measurement matrix and $b\in \mathbb{R}^M$ be an observations vector. The affine feasibility problem with sparsity and nonnegativity ($AFP_{SN}$ for short) is to find a sparse and nonnegative vector $x\in \mathbb{R}^N$ with $Ax=b$ if such $x$ exists. In this paper, we focus on establishment of optimization approach to solving the $AFP_{SN}$. By discussing tangent cone and normal cone of sparse constraint, we give the first necessary optimality conditions, $α$-Stability, T-Stability and N-Stability, and the second necessary and sufficient optimality conditions for the related minimization problems with the $AFP_{SN}$. By adopting Armijo-type stepsize rule, we present a framework of gradient support projection algorithm for the $AFP_{SN}$ and prove its full convergence when matrix $A$ is $s$-regular. By doing some numerical experiments, we show the excellent performance of the new algorithm for the $AFP_{SN}$ without and with noise.

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