Abstract We propose an adaptive smoothing algorithm based on Nesterov's smoothing technique in [24] for solving "fully" nonsmooth composite convex optimization problems. Our method combines both Nesterov's accelerated proximal gradient scheme and a new homotopy strategy for smoothness parameter. By an appropriate choice of smoothing functions, we develop a new algorithm that has the O 1 ε -worst-case iteration-complexity while preserves the same complexity-per-iteration as in Nesterov's method and allows one to automatically update the smoothness parameter at each iteration. Then, we customize our algorithm to solve four special cases that cover various applications. We also specify our algorithm to solve constrained convex optimization problems and show its convergence guarantee on a primal sequence of iterates. We demonstrate our algorithm through three numerical examples and compare it with other related algorithms. Keywords Nesterov's smoothing technique · accelerated proximal-gradient method · adaptive algorithm · composite convex minimization · nonsmooth convex optimization Mathematics Subject Classification (2000) 90C25 · 90-08 1 Introduction This paper develops new smoothing optimization methods for solving the following "fully" nonsmooth composite convex minimization problem:
REF proposed an adaptive smoothing algorithm by combining Nesterov's accelerated proximal gradient method and a homotopy strategy for smoothing parameter.
Adaptive Smoothing Algorithms for Nonsmooth Composite Convex Minimization
arXiv: Optimization and Control
Mathematics Computer Science