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Conditional gradient methods have attracted much attention in both machinelearning and optimization communities recently. These simple methods canguarantee the generation of sparse solutions. In addition, without thecomputation of full gradients, they can handle huge-scale problems sometimeseven with an exponentially increasing number of decision variables. This paperaims to significantly expand the application areas of these methods bypresenting new conditional gradient methods for solving convex optimizationproblems with general affine and nonlinear constraints. More specifically, wefirst present a new constraint extrapolated condition gradient (CoexCG) methodthat can achieve an ${\cal O}(1/\epsilon^2)$ iteration complexity for bothsmooth and structured nonsmooth function constrained convex optimization. Wefurther develop novel variants of CoexCG, namely constraint extrapolated anddual regularized conditional gradient (CoexDurCG) methods, that can achievesimilar iteration complexity to CoexCG but allow adaptive selection foralgorithmic parameters. We illustrate the effectiveness of these methods forsolving an important class of radiation therapy treatment planning problemsarising from healthcare industry. To the best of our knowledge, all thealgorithmic schemes and their complexity results are new in the area ofprojection-free methods.

Conditional Gradient Methods for Convex Optimization with General Affine and Nonlinear Constraints