Linear Convergence of a Frank-Wolfe-type Method over the Spectrahedron without Strict Complementarity

Abstract

We consider smooth convex minimization over the spectrahedron using Frank-Wolfe-type methods based only on extreme-eigenvector computations. In our recent work \cite{garber2026randomized} we presented the first ambient-dimension-independent linear convergence rate under quadratic growth. However, the method makes an additional strong strict complementarity assumption, it is randomized, its linear rate holds only after a burn-in phase and in expectation, and it requires the objective smoothness constant. We show that these limitations can be removed. Assuming quadratic growth and that all optimal solutions have the same rank, but without assuming strict complementarity, we give a deterministic and parameter-free Frank-Wolfe-type method with a global ambient-dimension-independent linear convergence rate.