---
title: 'Superiorization Methodology: Feasibility & Optimization'
url: https://www.emergentmind.com/topics/superiorization-methodology
type: topic
---

# Superiorization Methodology: Feasibility & Optimization

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 $\phi:H\to\mathbb{R}$ (e

Source: https://www.emergentmind.com/topics/superiorization-methodology