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A generalized conditional gradient method for minimizing the sum of twoconvex functions, one of them differentiable, is presented. This iterativemethod relies on two main ingredients: First, the minimization of a partiallylinearized objective functional to compute a descent direction and, second, astepsize choice based on an Armijo-like condition to ensure sufficient descentin every iteration. We provide several convergence results. Under mildassumptions, the method generates sequences of iterates which converge, onsubsequences, towards minimizers. Moreover a sublinear rate of convergence forthe objective functional values is derived. Second, we show that the methodenjoys improved rates of convergence if the partially linearized problemfulfills certain growth estimates. Most notably these results do not requirestrong convexity of the objective functional. Numerical tests on a variety ofchallenging PDE-constrained optimization problems confirm the practicalefficiency of the proposed algorithm.

On fast convergence rates for generalized conditional gradient methods with backtracking stepsize