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We propose a conditional gradient framework for a composite convexminimization template with broad applications. Our approach combines smoothingand homotopy techniques under the CGM framework, and provably achieves theoptimal $\mathcal{O}(1/\sqrt{k})$ convergence rate. We demonstrate that thesame rate holds if the linear subproblems are solved approximately withadditive or multiplicative error. In contrast with the relevant work, we areable to characterize the convergence when the non-smooth term is an indicatorfunction. Specific applications of our framework include the non-smoothminimization, semidefinite programming, and minimization with linear inclusionconstraints over a compact domain. Numerical evidence demonstrates the benefitsof our framework.

A Conditional Gradient Framework for Composite Convex Minimization with Applications to Semidefinite Programming