= Monotonicity of the multiplier equation for capped tolerances
<Monotonicity of the multiplier equation for capped tolerances> makes the allocation in <active-set clipping in cost-weighted baseline-tolerance allocation> computable by a scalar search, even at a change of active constraints. Express each capped physical bias contribution as $c_j=B_j\delta_j^{\rm ref}>0$. The total allocated bias at positive <Lagrange multiplier> $\lambda$ is
$$
Q(\lambda)=\sum_j\min(c_j,a_j/\lambda),\qquad a_j>0.
$$
Each summand is continuous and nonincreasing. Its breakpoint is $\lambda=a_j/c_j$; below that value it is capped, and above it it is strictly decreasing. Between breakpoints,
$$
Q'(\lambda)=-\frac1{\lambda^2}\sum_{j\ {\rm free}}a_j.
$$
Thus $Q$ is initially constant at $\sum_jc_j$ and becomes strictly decreasing once the first component is free. Continuity persists when the derivative changes.
For a binding budget $0<E<\sum_jc_j$, there is exactly one solution of $Q(\lambda)=E$. It lies in the explicit positive bracket
$$
\lambda_{\rm lo}=\min_j\frac{a_j}{c_j},\qquad
\lambda_{\rm hi}=\frac{\sum_ja_j}{E},
$$
because $Q(\lambda_{\rm lo})=\sum_jc_j>E$ and $Q(\lambda_{\rm hi})\le E$. The <bisection method> preserves a bracket without requiring a derivative at the root. A <tolerance-budget residual bound for multiplier bisection> can then relate the stopping rule to the physical bias budget instead of only to the multiplier's units.
For an original dimensionless illustration, take $(a_1,a_2,a_3)=(1,2,1)$, $(c_1,c_2,c_3)=(1,1,4)$, and $E=3$, with bias expressed in one common reference unit. The bracket is $[1/4,4/3]$. At $\lambda=1$, the contributions are $(1,1,1)$ and sum to three. This root is itself a breakpoint: the first contribution changes between capped and free there. The derivative is $-1$ just below it and $-2$ just above it, while the function remains continuous and strictly decreasing.
The argument uses the positive coefficients and logarithmic effort model specified by the parent. NIST's pulse-radiometer discussion motivates controlling offset and leakage, but does not prescribe that mathematical cost model.<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}
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