Abstract: We briefly introduce the basic principle and theory of compressive sensing and sparse optimization. Compressive sensing is a new paradigm of signal acquisition, which senses a sparse signal by taking a set of incomplete measurements and  recovers the signal by solving an optimization problem. This article first illustrates the compressive sensing paradigm through a synthetic example. Then we describe two sufficient conditions,  the null space property and restricted isometry principle, for l₁ convex minimization to give the sparsest solution. Finally, we summarize a few typical algorithms for solving the optimization models arising from compressive sensing.

Key words: compressive sensing, sparse optimization, null space property, RIP,shrinkage, prox-linear algorithms, split Bregman/alternating direction augmented Lagragian method, Bregman/augmented Lagragian method