Adaptive Linearized Alternating Direction Method of Multipliers for Non-Convex Compositely Regularized Optimization Problems

We consider a wide range of non-convex regularized minimization problems, where the non-convex regularization term is composite with a linear function engaged in sparse learning. Recent theoretical investigations have demonstrated their superiority over their convex counterparts. The computational challenge lies in the fact that the proximal mapping associated with non-convex regularization is not easily obtained due to the imposed linear composition. Fortunately, the problem structure allows one to introduce an auxiliary variable and reformulate it as an optimization problem with linear constraints, which can be solved using the Linearized Alternating Direction Method of Multipliers （LADMM）. Despite the success of LADMM in practice, it remains unknown whether LADMM is convergent in solving such non-convex compositely regularized optimizations. In this research, we first present a detailed convergence analysis of the LADMM algorithm for solving a non-convex compositely regularized optimization problem with a large class of non-convex penalties. Furthermore, we propose an Adaptive LADMM （AdaLADMM） algorithm with a line-search criterion. Experimental results on different genres of datasets validate the efficacy of the proposed algorithm.

Weighted Compact Commutator of Bilinear Fourier Multiplier Operator

Let T_σ be the bilinear Fourier multiplier operator with associated multiplier σ satisfying the Sobolev regularity that sup κ∈Z∥σ_κ∥W^s(R^(2n))< ∞ for some s ∈ (n, 2n]. In this paper, it is proved that the commutator generated by T_σ and CMO(R^n) functions is a compact operator from L^(p1)(R^n, w_1) × L^(p2)(R^n, w_2) to L^p(R^n, ν_w) for appropriate indices p_1, p_2, p ∈ (1, ∞) with1 p=1/ p_1 +1/ p_2 and weights w_1, w_2 such that w = (w_1, w_2) ∈ A_(p/t)(R^(2n)).