Singular Value Homogenization: a simple preconditioning technique for linearly constrained optimization and its potential applications in medical therapy

• Published in: Journal of mathematics in industry Vol. 6; no. 1; p. 1
• Main Authors: Erdmann-Pham, Dan-Daniel, Gibali, Aviv, Küfer, Karl-Heinz, Süss, Philipp
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 22-01-2016; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Applied mathematics; Drug therapy; Health care industry; Homogenization; Linear equations; Math. Appl. in Environmental Science; Mathematical and Computational Biology; Mathematical and Computational Engineering; Mathematical Methods in Physics; Mathematical Modeling and Industrial Mathematics; Mathematical models; Mathematics; Mathematics and Statistics; Optimization; Studies; spectral regularization; projection methods; 65F10; 65F22; 65F20; 90C05; linear least squares; singular value decomposition; ill-conditioned matrix
• ISSN: 2190-5983; 2190-5983
• DOI: 10.1186/s13362-016-0017-5

A wealth of problems occurring naturally in the applied sciences can be reformulated as optimization tasks whose argument is constrained to the solution set of a system of linear equations. Solving these efficiently typically requires computation of feasible descent directions and proper step sizes - the quality of which depends largely on conditioning of the linear equality constraint. In this paper we present a way of transforming such ill-conditioned problems into easily solvable, equivalent formulations by means of directly changing the singular values of the system’s associated matrix. This transformation allows us to solve problems for which corresponding routines in the LAPACK library as well as widely used projection methods converged either very slowly or not at all.