Adaptive Smoothing Algorithms for Nonsmooth Composite Convex Minimization

We propose novel adaptive smoothing algorithms based on Nesterov's smoothing technique in [26] for solving nonsmooth composite convex optimization problems. Our methods combine both Nesterov's accelerated proximal gradient scheme and a new homotopy strategy for smoothness parameter. By an appropriate choice of smoothing functions, we develop new algorithms that have upto the $\mathcal{O}\left(\frac{1}{\varepsilon}\right)$-optimal worst-case iteration complexity while allow 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. We also specify our algorithms to solve constrained convex optimization problems and show their convergence guarantee on the primal sequence of iterates. We demonstrate our algorithms through three numerical examples and compare them with the nonadaptive algorithm in [26].