  For the general problem of minimizing a convex function over a compact convex
domain, we will investigate a simple iterative approximation algorithm based on
the method by Frank & Wolfe 1956, that does not need projection steps in order
to stay inside the optimization domain. Instead of a projection step, the
linearized problem defined by a current subgradient is solved, which gives a
step direction that will naturally stay in the domain. Our framework
generalizes the sparse greedy algorithm of Frank & Wolfe and its primal-dual
analysis by Clarkson 2010 (and the low-rank SDP approach by Hazan 2008) to
arbitrary convex domains. We give a convergence proof guaranteeing
{\epsilon}-small duality gap after O(1/{\epsilon}) iterations.
  The method allows us to understand the sparsity of approximate solutions for
any l1-regularized convex optimization problem (and for optimization over the
simplex), expressed as a function of the approximation quality. We obtain
matching upper and lower bounds of {\Theta}(1/{\epsilon}) for the sparsity for
l1-problems. The same bounds apply to low-rank semidefinite optimization with
bounded trace, showing that rank O(1/{\epsilon}) is best possible here as well.
As another application, we obtain sparse matrices of O(1/{\epsilon}) non-zero
entries as {\epsilon}-approximate solutions when optimizing any convex function
over a class of diagonally dominant symmetric matrices.
  We show that our proposed first-order method also applies to nuclear norm and
max-norm matrix optimization problems. For nuclear norm regularized
optimization, such as matrix completion and low-rank recovery, we demonstrate
the practical efficiency and scalability of our algorithm for large matrix
problems, as e.g. the Netflix dataset. For general convex optimization over
bounded matrix max-norm, our algorithm is the first with a convergence
guarantee, to the best of our knowledge.
