We study a hybrid conditional gradient - smoothing algorithm (HCGS) for
solving composite convex optimization problems which contain several terms over
a bounded set. Examples of these include regularization problems with several
norms as penalties and a norm constraint...
HCGS extends conditional gradient
methods to cases with multiple nonsmooth terms, in which standard conditional
gradient methods may be difficult to apply. The HCGS algorithm borrows
techniques from smoothing proximal methods and requires first-order
computations (subgradients and proximity operations). Unlike proximal methods,
HCGS benefits from the advantages of conditional gradient methods, which render
it more efficient on certain large scale optimization problems. We demonstrate
these advantages with simulations on two matrix optimization problems:
regularization of matrices with combined $\ell_1$ and trace norm penalties; and
a convex relaxation of sparse PCA.