= Cost-weighted allocation of baseline-response tolerances
{title2=$\delta_j^*=a_jE/(B_j\sum_ka_k)$}
\b[Cost-weighted allocation of baseline-response tolerances] chooses how tightly to enforce each baseline constraint when different tolerances require different effort. The <allocation of a deterministic bias budget among baseline constraints> gives the physical restriction $\sum_jB_j\delta_j\leq E$. An optimization also needs an explicit cost model; the bias bound alone does not choose a unique allocation.
For one illustrative model, let $0<\delta_j\leq\delta_j^{\rm ref}$ and minimize
$$
C(\boldsymbol\delta)=\sum_ja_j\log\frac{\delta_j^{\rm ref}}{\delta_j},\qquad a_j>0.
$$
Tighter tolerances cost more, and the logarithm compares quantities in matching units. This is a stipulated convex effort model, not an empirical law for every numerical solver or measurement procedure.
When the budget binds and no reference upper bound is active, a constraint multiplier gives $-a_j/\delta_j+\lambda B_j=0$. Therefore
$$
\delta_j^*=\frac{a_jE}{B_j\sum_ka_k},\qquad
B_j\delta_j^*=\frac{a_j}{\sum_ka_k}E.
$$
The positive curvature $a_j/\delta_j^2$ establishes the unique interior minimum. A more expensive-to-tighten condition receives more of the allowed physical bias, while its coefficient bound converts that share to response units.
For an original two-constraint example, take $E=10\,\mathrm{nV}$, offset bound $B_1=10\,\mathrm{mV}$, drift-rate bound $B_2=100\,\mathrm{V/s}$ and costs $a_1=1$, $a_2=3$. The optimum allocates $2.5$ and $7.5\,\mathrm{nV}$, giving $\delta_1=2.5\times10^{-7}$ and $\delta_2=7.5\times10^{-11}\,\mathrm s$. If their reference limits are $10^{-5}$ and $10^{-8}\,\mathrm s$, both are interior. Compared with equal physical bias shares, this optimum reduces the chosen cost by $\log(1/2)+3\log(3/2)\approx0.523$ cost units.
<Active-set clipping in cost-weighted baseline-tolerance allocation> is needed if an interior solution exceeds a reference limit. If all reference limits already satisfy the bias budget, no tightening is needed in this cost model. Otherwise cap active components and redistribute the remaining budget among the others.
NIST's pulse-energy baselines motivate controlling residual offset and drift responses.<Calibration Service for Low-level Pulsed-Laser Radiometers at 1.06 µm: Pulse Energy and Peak Power by Rodney W. Leonhardt>[§3.5, printed p. 8; §7.2, Leakage Effect]{ref} The selected tolerances control deterministic leakage within the stated coefficient bounds. Noise <variance> and unmodelled background shapes remain separate parts of the measurement analysis.
Back to article page
