Abstract: Low-rank and sparse matrix optimization problem is a class of non-convex and non-smooth optimization problems with combinatorial properties. Due to the importance and special structure of the $\ell_0$-norm and rank functions, the models and algorithms of such NP-hard matrix optimization problems have been studied extensively in the past ten years with great success. This paper summarizes the progress from four aspects:sparse matrix optimization problem, low rank matrix optimization problems, low rank plus sparse matrix optimization problem, and low rank tensor optimization problems. For the sparse matrix optimization problem, we mainly focus on the sparse inverse covariance matrix estimation and column sparse matrix optimization problems as examples. For the low-rank matrix optimization problem, we mainly consider the convex relaxation methods and the matrix decomposition methods. Finally, we summarize a few challenging issues in low-rank and sparse matrix optimization, and disucss some interesting topics.

Key words: low-rank and sparse matrix optimization, convex relaxation models, factorization models, exact recovery conditions, convergence