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We develop a novel variant of the classical Frank-Wolfe algorithm, which wecall spectral Frank-Wolfe, for convex optimization over a spectrahedron. Thespectral Frank-Wolfe algorithm has a novel ingredient: it computes a feweigenvectors of the gradient and solves a small-scale SDP in each iteration.Such procedure overcomes slow convergence of the classical Frank-Wolfealgorithm due to ignoring eigenvalue coalescence. We demonstrate that strictcomplementarity of the optimization problem is key to proving linearconvergence of various algorithms, such as the spectral Frank-Wolfe algorithmas well as the projected gradient method and its accelerated version.

Spectral Frank-Wolfe Algorithm: Strict Complementarity and Linear Convergence