Derivative-Free Superiorization: Principle and Algorithm

TL;DR

This paper introduces a derivative-free superiorization method that adapts feasibility-seeking algorithms to reduce target functions without derivatives, demonstrated on image reconstruction data.

Contribution

It develops a novel derivative-free superiorization algorithm that can handle target functions without derivatives, enhancing applicability in constrained optimization.

Findings

The derivative-free superiorization algorithm effectively reduces target function values.

Proximity-target curves help compare iterative methods for specific problems.

Numerical experiments show the proposed method's advantage over existing approaches.

Abstract

The superiorization methodology is intended to work with input data of constrained minimization problems, that is, a target function and a set of constraints. However, it is based on an antipodal way of thinking to what leads to constrained minimization methods. Instead of adapting unconstrained minimization algorithms to handling constraints, it adapts feasibility-seeking algorithms to reduce (not necessarily minimize) target function values. This is done by inserting target-function-reducing perturbations into a feasibility-seeking algorithm while retaining its feasibility-seeking ability and without paying a high computational price. A superiorized algorithm that employs component-wise target function reduction steps is presented. This enables derivative-free superiorization (DFS), meaning that superiorization can be applied to target functions that have no calculable partial derivatives or subgradients. The numerical behavior of our derivative-free superiorization algorithm is illustrated on a data set generated by simulating a problem of image reconstruction from projections. We present a tool (we call it a proximity-target curve) for deciding which of two iterative methods is "better" for solving a particular problem. The plots of proximity-target curves of our experiments demonstrate the advantage of the proposed derivative-free superiorization algorithm.