Recent works have developed new projection-free first-order methods based on utilizing linesearches and normal vector computations to maintain feasibility. These oracles can be cheaper than orthogonal projection or linear optimization subroutines but have the drawback of requiring a known strictly feasible point to do these linesearches with respect to. In this work, we develop new theory and algorithms which can operate using these cheaper linesearches while only requiring knowledge of points strictly satisfying each constraint separately. Convergence theory for several resulting ``multiradial'' gradient methods is established. We also provide preliminary numerics showing performance is essentially independent of how one selects the reference points for synthetic quadratically constrained quadratic programs.
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