Finite-budget multiplicative design updates preserve the simplex and nonsingularity

Fix an update at a current approximate design with positive-definite $M={\sum }_x{\xi }_x{f}_x{f}_x^T$. For each $x$, put $s_x=f_x^T{M}^{-1}{f}_x$. Since $M$ is symmetric positive definite, $s_x\ge 0$, and $s_x>0$ whenever $f_x\mathrm{\ne }0$: writing $u=M^{-1}{f}_x$, one has $f_x^T{M}^{-1}{f}_x=u^{T}Mu>0$ for $f_x\mathrm{\ne }0$. By [F4] and [F5], ${\sum }_x{\xi }_x{s}_x={\sum }_x{\xi }_x\mathrm{tr}(M^{-1}{f}_x{f}_x^T)=\mathrm{tr}(M^{-1}M)=\mathrm{tr}(I_p)=p$. Therefore ${\xi }_x^{+}={\xi }_x{s}_x\mathrm{/}p$ is nonnegative and ${\sum }_x{\xi }_x^{+}=1$, so it is again an approximate design.

For every feature $f_x$, a positive old weight remains positive after the update exactly when $f_x\mathrm{\ne }0$: step 1.1 gives $s_x>0$ for nonzero $f_x$, and $s_x=0$ for $f_x=0$; zero weights remain zero. Thus the stacked scalar sensitivity matrix loses at most zero rows, so its kernel is unchanged. The current matrix is positive definite, so [F2] gives this kernel is trivial; applying [F2] to the new weights gives positive definiteness of the updated matrix. Induction applies this argument at every update, so no later inverse is undefined.

The loop checks one finite candidate set at each index $k=0,\dots ,N$, returns immediately on zero violation, and otherwise makes an update only when $k<N$. Thus it makes at most $N$ updates and at most $N+1$ sensitivity scans, and at $k=N$ returns the current design and its finite violation. The invariant argument above ensures every scan and update is well-defined.