  A generalized conditional gradient method for minimizing the sum of two
convex functions, one of them differentiable, is presented. This iterative
method relies on two main ingredients: First, the minimization of a partially
linearized objective functional to compute a descent direction and, second, a
stepsize choice based on an Armijo-like condition to ensure sufficient descent
in every iteration. We provide several convergence results. Under mild
assumptions, the method generates sequences of iterates which converge, on
subsequences, towards minimizers. Moreover a sublinear rate of convergence for
the objective functional values is derived. Second, we show that the method
enjoys improved rates of convergence if the partially linearized problem
fulfills certain growth estimates. Most notably these results do not require
strong convexity of the objective functional. Numerical tests on a variety of
challenging PDE-constrained optimization problems confirm the practical
efficiency of the proposed algorithm.
