Smooth Alternating Direction Methods for Nonsmooth Constrained Convex Optimization
This work addresses optimization challenges in fields like machine learning and engineering by providing improved algorithms for nonsmooth problems, though it appears incremental as it builds upon classical alternating direction methods.
The authors tackled the problem of solving fully nonsmooth constrained convex optimization by proposing two new alternating direction methods, achieving the best known worst-case iteration-complexity guarantees for both objective residual and feasibility gap under mild assumptions.
We propose two new alternating direction methods to solve "fully" nonsmooth constrained convex problems. Our algorithms have the best known worst-case iteration-complexity guarantee under mild assumptions for both the objective residual and feasibility gap. Through theoretical analysis, we show how to update all the algorithmic parameters automatically with clear impact on the convergence performance. We also provide a representative numerical example showing the advantages of our methods over the classical alternating direction methods using a well-known feasibility problem.