Adaptive Smoothing Algorithms for Nonsmooth Composite Convex Minimization

We propose a novel adaptive smoothing algorithm based on Nesterov's smoothing technique in \cite{Nesterov2005c} for solving 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 $\mathcal{O}\left(\frac{1}{\varepsilon}\right)$ optimal worst-case iteration-complexity while allows one to automatically update the smoothness parameter at each iteration. We then further exploit the structure of problems to select smoothing functions and develop suitable algorithmic variants that reduce the complexity-per-iteration, while preserve the optimal worst-case iteration-complexity. We also specify our algorithm to solve constrained convex optimization problems and show its convergence guarantee on the primal sequence of iterates. Our preliminarily numerical tests verify the efficiency of our algorithms.