Incremental and Parallel Line Search Subgradient Methods for Constrained Nonsmooth Convex Optimization - Numerically accelerated results by multi-core computing
Kazuhiro Hishinuma, Hideaki Iiduka · arXiv (Cornell University) · 2016
The objective of this paper is to accelerate the existing incremental and parallel subgradient methods for constrained nonsmooth convex optimization to solve the problem of minimizing the sum of nonsmooth, convex functionals over a constraint set in a Hilbert space. Our approach to accelerate the existing algorithms is to develop line search methods that can determine a more adequate step-size than the ones used in the existing methods at each iteration. The proposed line search method can determine step-sizes to satisfy weaker conditions than the ones used in the existing methods. This implies that the incremental and parallel subgradient methods based on the line search approach are generalizations of the existing ones. We show that the whole sequence generated by each of the two proposed methods weakly converges to a solution to the problem. The main contribution of this paper is provision of numerical examples showing that the two proposed methods perform better than the existing ones. The numerical results are evaluated with a multi-core computer and show that our parallel method reduces the running time and iterations needed to find an optimal solution compared with other ones.