Abstract
When solving an optimization problem over the set of graph Laplacian matrices, one must deal with a large number of constraints as well as the large objective variable size. In this paper we explore first-order methods for optimization over graph Laplacian matrices. These methods include two popular methods for constrained optimization: the mirror descent algorithm and the Frank-Wolfe (conditional gradient) algorithm. We derive efficiently implementable formulations of these algorithms over graph Laplacians, and use existing theory to show their iteration complexity in various regimes. Experiments demonstrate the efficiency of these methods over alternatives like interior point methods.