  The projected subgradient method for constrained minimization repeatedly
interlaces subgradient steps for the objective function with projections onto
the feasible region, which is the intersection of closed and convex constraints
sets, to regain feasibility. The latter poses a computational difficulty and,
therefore, the projected subgradient method is applicable only when the
feasible region is "simple to project onto". In contrast to this, in the
superiorization methodology a feasibility-seeking algorithm leads the overall
process and objective function steps are interlaced into it. This makes a
difference because the feasibility-seeking algorithm employs projections onto
the individual constraints sets and not onto the entire feasible region.
  We present the two approaches side-by-side and demonstrate their performance
on a problem of computerized tomography image reconstruction, posed as a
constrained minimization problem aiming at finding a constraint-compatible
solution that has a reduced value of the total variation of the reconstructed
image.
