Nonconvex and Nonsmooth Sparse Optimization via Adaptively Iterative Reweighted Methods
This work addresses sparse optimization for applications in machine learning and signal processing, but it is incremental as it builds on existing reweighted methods.
The authors tackled nonconvex and nonsmooth sparse optimization problems by proposing a general formulation and an adaptively iterative reweighted algorithmic framework, demonstrating effectiveness in numerical experiments.
We propose a general formulation of nonconvex and nonsmooth sparse optimization problems with convex set constraint, which can take into account most existing types of nonconvex sparsity-inducing terms, bringing strong applicability to a wide range of applications. We design a general algorithmic framework of iteratively reweighted algorithms for solving the proposed nonconvex and nonsmooth sparse optimization problems, which solves a sequence of weighted convex regularization problems with adaptively updated weights. First-order optimality condition is derived and global convergence results are provided under loose assumptions, making our theoretical results a practical tool for analyzing a family of various reweighted algorithms. The effectiveness and efficiency of our proposed formulation and the algorithms are demonstrated in numerical experiments on various sparse optimization problems.