# Bundle-type methods uniformly optimal for smooth and nonsmooth convex optimization

The bundle-level method and their certain variants are known to exhibit an optimal rate of convergence, i.e., ${\cal O}(1/\sqrt{t})$, and also excellent practical performance for solving general non-smooth convex programming (CP) problems. However, this rate of convergence is significantly worse than the optimal one for solving smooth CP problems, i.e., ${\cal O}(1/t^2)$. In this paper, we present new bundle-type methods which possess the optimal rate of convergence for solving, not only non-smooth, but also smooth CP problems. Interestingly, these optimal rates of convergence can be obtained without even knowing whether a CP problem is smooth or not. Moreover, given that the problem is smooth, the developed methods do not require any smoothness information, such as, the Lipschitz constant. To the best of our knowledge, this is the first time that uniformly optimal algorithms of this type are presented in the literature.

## Citation

Technical Report, Department of Industrial and Systems Engineering, University of Florida, November 2010.