Friday, December 7, 2007 



Gradient Methods for Minimizing Composite Objective Function 

In this talk we present several methods for solving optimization problems with the objective function formed as a sum of two convex terms: one is smooth and given by a blackbox oracle, and another is general but simple and its structure is known. It appears that, despite to the bad properties of the sum, such problems can be solved with efficiency typical for the good part of the objective. For these problems, we consider primal and dual variants of the gradient method (converges as $O\left({1 \over k}\right)$), and an accelerated multistep version, which converges as $O\left({1 \over k2}\right)$, where $k$ is the iteration counter. For all methods, we present very efficient "line search" procedures and show that the additional computational work necessary for estimating the unknown problem class parameters can only double the complexity of each iteration. 