In order to recover a signal from its compressive measurements, the compressed sensing theory seeks the sparsest signal that agrees with the measurements, which is actually an $l_{\mathrm{0}}$ norm minimization problem. In this paper, we equivalently transform the $l_{\mathrm{0}}$ norm minimization into a concave continuous piecewise linear programming, and propose an optimization algorithm based on a modified interior point method. Numerical experiments demonstrate that our algorithm improves the sufficient number of measurements, relaxes the restrictions of the sensing matrix to some extent, and performs robustly in the noisy scenarios.

This paper works on a modified simplex algorithm for the local optimization of Continuous PieceWise Linear (CPWL) programming with generalization of hinging hyperplane objective and linear constraints. CPWL programming is popular since it can be equivalently transformed into difference of convex functions programming or concave optimization. Inspired by the concavity of the concave CPWL functions, we propose an Objective Variation Simplex Algorithm (OVSA), which is able to find a local optimum in a reasonable time. Computational results are presented for further insights into the performance of the OVSA compared with two other algorithms on random test problems.