Finite-budget multiplicative update for approximate D-optimal design

Let $p\ge 1$, let $\mathcal{X}$ be a finite nonempty candidate set, and let $f_x\in {\mathbb{R}}^p$ be the feature vector for each $x\in \mathcal{X}$. An approximate design $\xi$ and information matrix $M(\xi )={\sum }_x{\xi }_x{f}_x{f}_x^T$ use the convention of Approximate designs and linear-model optimality criteria. The same vectors may be scalar-observation sensitivities at a fixed parameter, as in Finite nonlinear observation designs, local identifiability, and weighted information, with a scalar precision absorbed into $f_x$. The procedure uses exact arithmetic.

Input: A finite nonempty $\mathcal{X}$, $p\ge 1$, its feature vectors $(f_x{)}_{x\in \mathcal{X}}$, an approximate initial design ${\xi }^{(0)}$, and a finite nonnegative integer budget $N$ counting weight updates.

Procedure: Form $M_0=M({\xi }^{(0)})$. If $M_0$ is not positive definite, return $({\xi }^{(0)},\text{undefined violation},\text{invalid initial design})$. Otherwise, for $k=0,1,\dots ,N$, with current ${\xi }^{(k)}$ and $M_k=M({\xi }^{(k)})$, compute $s_x^{(k)}=f_x^T{M}_k^{-1}{f}_x(x\in \mathcal{X}),v_k={\max }_{x\in \mathcal{X}}{s}_x^{(k)}-p\mathrm{.}$ If $v_k=0$, return $({\xi }^{(k)},v_k,\text{equivalence-certified})$. If $k=N$, return $({\xi }^{(k)},v_k,\text{budget-exhausted})$. Otherwise set ${\xi }_x^{(k+1)}={\xi }_x^{(k)}\frac{s_x^{(k)}}{p}(x\in \mathcal{X}),$ form $M_{k+1}={\sum }_x{\xi }_x^{(k+1)}{f}_x{f}_x^T$, and continue.

Finite-budget multiplicative design updates preserve the simplex and nonsingularity ↗ proves preservation of positive definiteness and the bounds of at most $N$ updates and $N+1$ sensitivity scans. A nonzero returned violation is reported as-is; it is not an exact D-optimality certificate. The exact integer allocation problem remains separate from this approximate-weight procedure.