A nonlinear preconditioner for experimental design problems

TL;DR

This paper introduces a nonlinear preconditioner for experimental design problems that stabilizes and accelerates convergence of SQP methods by controlling the condition number, especially for large problems with poor initial designs.

Contribution

The paper develops a novel nonlinear preconditioner that bounds the condition number, improving stability and convergence of SQP methods in experimental design optimization.

Findings

Preconditioner keeps the condition number bounded.

Significant improvements in stability and convergence observed.

Effective for large, ill-conditioned experimental design problems.

Abstract

We address the slow convergence and poor stability of quasi-newton sequential quadratic programming (SQP) methods that is observed when solving experimental design problems, in particular when they are large. Our findings suggest that this behavior is due to the fact that these problems often have bad absolute condition numbers. To shed light onto the structure of the problem close to the solution, we formulate a model problem (based on the $A$-criterion), that is defined in terms of a given initial design that is to be improved. We prove that the absolute condition number of the model problem grows without bounds as the quality of the initial design improves. Additionally, we devise a preconditioner that ensures that the condition number will instead stay uniformly bounded. Using numerical experiments, we study the effect of this reformulation on relevant cases of the general problem, and find that it leads to significant improvements in stability and convergence behavior.