 Active-set clipping in cost-weighted baseline-tolerance allocation

Active-set clipping in cost-weighted baseline-tolerance allocation enforces the reference caps that an interior solution of cost-weighted allocation of baseline-response tolerances may violate. Keep the parent's stipulated logarithmic effort modelHere $B_j{\delta }_j$ is a physical bias contribution in the common units of $E$. If $E\ge {\sum }_j{B}_j{\delta }_j^{\mathrm{ref}}$, no tightening is needed. Otherwise the budget binds, and the Karush-Kuhn-Tucker conditions giveThe positive Lagrange multiplier $\lambda$ prices scarce bias budget. For a set $A$ of capped components and remaining free set $F$, stationarity givesprovided the denominator is positive and the resulting choices agree with the proposed capped/free status. The monotonicity of the multiplier equation for capped tolerances permits a scalar search; strict convexity of the objective gives a unique allocation.

For an original cap-activating version of the parent's apparatus example, let $E=10\mathrm{nV}$, $a_1=1$, $a_2=3$, $B_1=0.01\mathrm{V}$, and $B_2=100\mathrm{V}{\mathrm{s}}^{-1}$. Set reference bias contributions $B_1{\delta }_1^{\mathrm{ref}}=1\mathrm{nV}$ and $B_2{\delta }_2^{\mathrm{ref}}=20\mathrm{nV}$. The unconstrained shares $2.5$ and $7.5\mathrm{nV}$ violate the first cap. Capping it leaves $9\mathrm{nV}$ for the second component, soMerely clipping the first share and leaving the second at $7.5\mathrm{nV}$ wastes $1.5\mathrm{nV}$ of permitted bias and unnecessarily increases the chosen effort. Reallocation lowers that effort by $3\log (9/7.5)\approx 0.547$ units.

The physical motivation remains leakage from residual offset and drift in a pulsed-radiometer baseline, documented in NIST's calibration apparatus.^[ref] The logarithmic cost and the numerical caps here are illustrative assumptions, not reported apparatus constants.