  In this paper we present a new algorithmic realization of a projection-based
scheme for general convex constrained optimization problem. The general idea is
to transform the original optimization problem to a sequence of feasibility
problems by iteratively constraining the objective function from above until
the feasibility problem is inconsistent. For each of the feasibility problems
one may apply any of the existing projection methods for solving it. In
particular, the scheme allows the use of subgradient projections and does not
require exact projections onto the constraints sets as in existing similar
methods.
  We also apply the newly introduced concept of superiorization to optimization
formulation and compare its performance to our scheme. We provide some
numerical results for convex quadratic test problems as well as for real-life
optimization problems coming from medical treatment planning.
