Projected Subgradient Minimization versus Superiorization

TL;DR

This paper compares projected subgradient minimization and superiorization methods for constrained optimization, demonstrating their differences and performance in CT image reconstruction with total variation minimization.

Contribution

It provides a side-by-side comparison of the two approaches and evaluates their effectiveness in a practical CT reconstruction problem.

Findings

Superiorization uses simpler projections onto individual constraints.

Projected subgradient method requires projections onto the entire feasible region.

Both methods effectively reduce total variation in CT images.

Abstract

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.