  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.
