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We study a hybrid conditional gradient - smoothing algorithm (HCGS) forsolving composite convex optimization problems which contain several terms overa bounded set. Examples of these include regularization problems with severalnorms as penalties and a norm constraint. HCGS extends conditional gradientmethods to cases with multiple nonsmooth terms, in which standard conditionalgradient methods may be difficult to apply. The HCGS algorithm borrowstechniques from smoothing proximal methods and requires first-ordercomputations (subgradients and proximity operations). Unlike proximal methods,HCGS benefits from the advantages of conditional gradient methods, which renderit more efficient on certain large scale optimization problems. We demonstratethese advantages with simulations on two matrix optimization problems:regularization of matrices with combined $\ell_1$ and trace norm penalties; anda convex relaxation of sparse PCA.

Hybrid Conditional Gradient - Smoothing Algorithms with Applications to Sparse and Low Rank Regularization