  We consider the best approximation problem (BAP) of projecting a point onto
the intersection of a number of convex sets. It is known that Dykstra's
algorithm is alternating minimization on the dual problem. We extend Dykstra's
algorithm so that it can be enhanced by the SHQP strategy of using quadratic
programming to project onto the intersection of supporting halfspaces generated
by earlier projection operations. By looking at a structured alternating
minimization problem, we show the convergence rate of Dykstra's algorithm when
reasonable conditions are imposed to guarantee a dual minimizer. We also
establish convergence of using a warmstart iterate for Dykstra's algorithm,
show how all the results for the Dykstra's algorithm can be carried over to the
simultaneous Dykstra's algorithm, and discuss a different way of incorporating
the SHQP strategy. Lastly, we show that the dual of the best approximation
problem can have an O(1/k^2) accelerated algorithm that also incorporates the
SHQP strategy.
