Parallel and Hybrid Soft-Thresholding Algorithms with Line Search for Sparse Nonlinear Regression

In this paper, we propose a convergent iterative algorithm for nondifferentiable nonconvex nonlinear regression problems. The proposed parallel algorithm consists in optimizing a sequence of successively refined approximate functions. Compared with the popular iterative soft-thresholding algorithm commonly known as ISTA, which is the benchmark algorithm for such problems, it has two attractive features which lead to a notable reduction in the algorithm's complexity: the proposed approximate function does not have to be a global upper bound of the original function, and the stepsize can be efficiently computed by the line search scheme which is carried out over a properly constructed differentiable function. Furthermore, when the parallel algorithm cannot be fully parallelized due to memory/processor constraints, we propose a hybrid updating scheme that divides the whole set of variables into blocks which are updated sequentially. Since the stepsize is obtained by performing the line search along the coordinate of each block variable, the proposed hybrid algorithm converges faster than state-of-the-art hybrid algorithms based on constant stepsizes and/or decreasing stepsizes. Finally, the proposed algorithms are numerically tested.