Algorithms aimed at solving some problems are discussed. For other problems, algorithms tend to need more computer memory and longer execution time. A methodology is presented to produce automatically a "superiorized version" of an algorithm.
Iterative algorithms aimed at solving some problems are discussed. For
certain problems, such as finding a common point in the intersection of a
finite number of convex sets, there often exist iterative algorithms that
impose very little demand on computer resources. For other problems, such as
finding that point in the intersection at which the value of a given function
is optimal, algorithms tend to need more computer memory and longer execution
time. A methodology is presented whose aim is to produce automatically for an
iterative algorithm of the first kind a "superiorized version" of it that
retains its computational efficiency but nevertheless goes a long way towards
solving an optimization problem. This is possible to do if the original
algorithm is "perturbation resilient," which is shown to be the case for
various projection algorithms for solving the consistent convex feasibility
problem. The superiorized versions of such algorithms use perturbations that
drive the process in the direction of the optimizer of the given function.
After presenting these intuitive ideas in a precise mathematical form, they are
illustrated in image reconstruction from projections for two different
projection algorithms superiorized for the function whose value is the total
variation of the image.