Purpose: To describe and mathematically validate the superiorization methodology, which is a recently developed heuristic approach to optimization, and to discuss its applicability to medical physics problem formulations that specify the desired solution (of physically given or otherwise obtained constraints) by an optimization criterion. Methods: The superiorization methodology is presented as a heuristic solver for a large class of constrained optimization problems. The constraints come from the desire to produce a solution that is constraints-compatible, in the sense of meeting requirements provided by physically or otherwise obtained constraints. The underlying idea is that many iterative algorithms for finding such a solution are perturbation resilient in the sense that, even if certain kinds of changes are made at the end of each iterative step, the algorithm still produces a constraints- compatible solution. This property is exploited by using permitted changes to steer the algorithm to a solution that is not only constraints-compatible, but is also desirable according to a specified optimization criterion. The approach is very general, it is applicable to many iterative procedures and optimization criteria used in medical physics. Results: The main practical contribution is a procedure for automatically producing from any given iterative algorithm its superiorized version, which will supply solutions that are superior according to a given optimization criterion. It is shown that if the original iterative algorithm satisfies certain mathematical conditions, then the output of its superiorized version is guaranteed to be as constraints-compatible as the output of the original algorithm, but it is superior to the latter according to the optimization criterion. This intuitive description is made precise in the paper and the stated claims are rigorously proved. Superiorization is illustrated on simulated computerized tomography data of a head cross section and, in spite of its generality, superiorization is shown to be competitive to an optimization algorithm that is specifically designed to minimize total variation. Conclusions: The range of applicability of superiorization to constrained optimization problems is very large. Its major utility is in the automatic nature of producing a superiorization algorithm from an algorithm aimed at only constraints-compatibility; while nonheuristic (exact) approaches need to be redesigned for a new optimization criterion. Thus superiorization provides a quick route to algorithms for the practical solution of constrained optimization problems.
Bibliographical noteFunding Information:
The detailed and penetrating comments of three reviewers and the editors helped us to improve this paper in a significant way. The authors thank Professor Xiaochuan Pan and his co-workers from the University of Chicago for providing them with the reconstruction from their data using their implementation of their ASD-POCS algorithm. Our work is supported by the National Science Foundation Award No. DMS-1114901, the United States-Israel Binational Science Foundation (BSF) Grant No. 200912, the U.S. Department of Defense Prostate Cancer Research Program Award No. W81XWH-12-1-0122, and the U.S. Department of Army Award No. W81XWH-10-1-0170.
- constrained optimization
- heuristic optimization
- total variation
ASJC Scopus subject areas
- Radiology Nuclear Medicine and imaging