 Tolerance-budget residual bound for multiplier bisection

A tolerance-budget residual bound for multiplier bisection controls the physical bias left unused by a numerical allocation, rather than only the numerical error in its Lagrange multiplier. Use $Q(\lambda )={\sum }_j\min (c_j,a_j/\lambda )$ from monotonicity of the multiplier equation for capped tolerances. Suppose a positive bracket $[\ell ,u]$ satisfies $Q(\ell )\ge E\ge Q(u)$. Choosing the upper endpoint gives a feasible allocation, with residualChoosing the lower endpoint can violate the budget. The direction of monotonicity therefore matters even when both endpoints are numerically close.

For an a priori bound, put $A={\sum }_j{a}_j$. On the entire bracket, each summand obeys Lipschitz continuity with constant at most $a_j/{\ell }^2$, including across its breakpoint. HenceAfter $m$ bisections of an original bracket $[{\ell }_0,u_0]$ with ${\ell }_0>0$, the corresponding worst-case bound is $A(u_0-{\ell }_0)/(2^m{\ell }_0^2)$. This is conservative; evaluating the two endpoint budgets can give a much tighter residual certificate.

For an original stopping example, use $(a_j)=(1,2,1)$, $(c_j)=(1,1,4)$ and $E=3$ in common normalized bias units. The bracket $[0.99,1.03]$ straddles the root. Its feasible upper endpoint leaves $3-Q(1.03)\approx 0.05825$ unused. One bisection evaluates $Q(1.01)<3$ and replaces the upper endpoint, reducing that unused amount to about $0.01980$. A residual allowance of $0.03$ is now met, despite a breakpoint lying inside the bracket.

This gives a thresholded stopping of a capped-tolerance multiplier search in the same units as the permitted bias. Floating-point comparisons also need an allowance for rounding if they are to serve as hard certificates. The underlying bias coefficients and cost model remain assumptions of cost-weighted allocation of baseline-response tolerances; numerical precision cannot validate missing physical terms. NIST's pulse-radiometer account motivates the offset/leakage constraints rather than prescribing this numerical stopping rule.^[ref]