Superiorization Methodology: Feasibility & Optimization

• Superiorization is a framework that perturbs feasibility-seeking algorithms to steer iterates toward both feasibility and reduced objective function values.
• It leverages the perturbation resilience of projection-type iterative methods, making it applicable in optimization, medical physics, and signal processing.
• The approach systematically balances computational efficiency with enhanced solution quality by incorporating bounded, summable perturbations at each iteration.

Superiorization methodology is a framework for perturbing feasibility-seeking algorithms in order to obtain solutions that, in addition to being constraint-compatible (feasible), yield reduced values of an exogenous objective function, without incurring the full computational cost of constrained minimization. The approach leverages the perturbation resilience properties of many projection-type or fixed-point iterative algorithms. When applied, it systematically “steers” the iterates of a basic algorithm toward points that are not only feasible but are also, in a well-defined sense, superior with respect to a target function. Superiorization has been widely developed across mathematical programming, optimization, inverse problems, medical physics, signal processing, and large-scale computational engineering.

1. Conceptual Framework and Key Definitions

Superiorization sits between pure feasibility-seeking and full constrained minimization. Let the primary computational goal be to find $x^* \in C$ where $C = \bigcap_{i=1}^m C_i$ is a nonempty intersection of closed convex sets in a real Hilbert space $H$. A feasibility-seeking algorithm is an iterative scheme

$x^{k+1} = \mathcal{A}(x^k)$

that converges to $C$ for any starting $x^0$. Its performance can be measured by a proximity function $\mathsf{Pr}_C : H \to \mathbb{R}_+$, where $x$ is called $\varepsilon$-compatible if $\mathsf{Pr}_C(x)\leq \varepsilon$.

The superiorization methodology augments this basic algorithm by inserting bounded, summable perturbations at each iteration, chosen (often via subgradient or directional search) to reduce some real-valued target function $C = \bigcap_{i=1}^m C_i$0 (e