Global optimality condition and fixed point continuation algorithm for non-Lipschitz ℓ p regularized matrix minimization

Abstract

Regularized minimization problems with nonconvex, nonsmooth, even non-Lipschitz penalty functions have attracted much attention in recent years, owing to their wide applications in statistics, control, system identification and machine learning. In this paper, the non-Lipschitz $\ell_p~(0<p<1)$ regularized matrix minimization problem is studied. A global necessary optimality condition for this non-Lipschitz optimization problem is firstly obtained, specifically, the global optimal solutions for the problem are fixed points of the so-called $p$-thresholding operator which is matrix-valued and set-valued. Then a fixed point iterative scheme for the non-Lipschitz model is proposed, and the convergence analysis is also addressed in detail. Moreover, some acceleration techniques are adopted to improve the performance of this algorithm. The effectiveness of the proposed $p$-thresholding fixed point continuation ($p$-FPC) algorithm is demonstrated by numerical experiments on randomly generated and real matrix completion problems.