2016/05/01 by Tianxiang Liu, Ting Kei Pong, Liu, Tianxiang +1 · 1 citation
Computer Science · Engineering · Mathematics · #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (stat.ML) #Numerical methods in inverse problems #Optimization and Control (math.OC) #Sparse and Compressive Sensing Techniques #Stochastic Gradient Optimization Techniques
paper · pdf · doi:10.48550/arxiv.1605.00201
openalex publication_date 2016/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we further study the forward-backward envelope first introduced in [28] and [30] for problems whose objective is the sum of a proper closed convex function and a twice continuously differentiable possibly nonconvex function with Lipschitz continuous gradient. We derive sufficient conditions on the original problem for the corresponding forward-backward envelope to be a level-bounded and Kurdyka-Łojasiewicz function with an exponent of \frac12; these results are important for the efficient minimization of the forward-backward envelope by classical optimization algorithms. In addition, we demonstrate how to minimize some difference-of-convex regularized least squares problems by minimizing a suitably constructed forward-backward envelope. Our preliminary numerical results on randomly generated instances of large-scale ℓ1-2 regularized least squares problems [37] illustrate that an implementation of this approach with a limited-memory BFGS scheme usually outperforms standard first-order methods such as the nonmonotone proximal gradient method in [35].