maximise d(lambda) = -1/2 lambda^t B lambda + U^t lambda subject to lambda_i >= 0 for i not in J
which is the Lagrangian dual of Eq. (9) with H = I. Optimum equals the primal optimum c* = 1/2 ||y* - x||^2; the primal solution is recovered from the KKT conditions (12) as y* = x - sum_i lambda_i (x_i - x).
Reused across every candidate simplex sharing the same base vertex x -- only J changes, which is exactly the observation in the paper that lets B and U be assembled once per vertex rather than once per simplex.
The instance is stateful and not thread safe; give each worker its own.