Convergent Design Sequences, for Sufficiently Regular Optimality Criteria, II: Singular Case

Abstract

Let an optimality criterion $\Phi$ satisfy regularity conditions, including convexity and possession of two continuous derivatives at nonsingular $\mathbf{M}$. $\Phi\{\mathbf{M})$ may be minimized at a singular $\mathbf{M}$. A class of design sequences $\{\xi_n\}$ is shown to make $\Phi\lbrack \mathbf{M}(\xi_n) \rbrack$ converge monotonically to the minimum value. An equivalence theorem for $\Phi$-optimality follows. Techniques which are applicable include vertex direction, conjugate gradient projection, quadratic and "diagonalized quadratic" methods for changing the design weights, and gradient-based methods for making small changes in the support points. Methods are also considered which approximate $\Phi$ by a criterion which is infinite for singular $\mathbf{M}$. The results are applied to an example with $D_s$-optimality.