  Conditional gradient methods have attracted much attention in both machine
learning and optimization communities recently. These simple methods can
guarantee the generation of sparse solutions. In addition, without the
computation of full gradients, they can handle huge-scale problems sometimes
even with an exponentially increasing number of decision variables. This paper
aims to significantly expand the application areas of these methods by
presenting new conditional gradient methods for solving convex optimization
problems with general affine and nonlinear constraints. More specifically, we
first present a new constraint extrapolated condition gradient (CoexCG) method
that can achieve an ${\cal O}(1/\epsilon^2)$ iteration complexity for both
smooth and structured nonsmooth function constrained convex optimization. We
further develop novel variants of CoexCG, namely constraint extrapolated and
dual regularized conditional gradient (CoexDurCG) methods, that can achieve
similar iteration complexity to CoexCG but allow adaptive selection for
algorithmic parameters. We illustrate the effectiveness of these methods for
solving an important class of radiation therapy treatment planning problems
arising from healthcare industry. To the best of our knowledge, all the
algorithmic schemes and their complexity results are new in the area of
projection-free methods.
