Convergent Design Sequences, for Sufficiently Regular Optimality Criteria

Abstract

For an optimality criterion function $\Phi$ and a design $\xi_n$, approximate $\Phi(\mathbf{M}(\xi))$ near $\xi_n$ by a quadratic Taylor expansion, let $\xi_n + \eta$ minimize this approximation, and let $\xi_{n+1} = \xi_n + \alpha\eta$, with $\alpha$ minimizing $\Phi(\mathbf{M}(\xi_{n+1}))$. If $\Phi$ satisfies regularity conditions, including strict convexity, possession of three continuous derivatives, and finiteness only for nonsingular $\mathbf{M}$, then $\mathbf{M}(\xi_n)$ converges to the optimal value for both the Federov steepest descent sequence and the above quadratic sequence, with the quadratic sequence having a faster asymptotic convergence rate. Methods are discussed for collapsing clusters of design points during the iterative process. In a simple example with $D$-optimality, the two methods are comparable. In a more complicated example the quadratic method is far superior.