  We present an optimization-based framework to construct confidence intervals
for functionals in constrained inverse problems, ensuring valid one-at-a-time
frequentist coverage guarantees. Our approach builds upon the now-called strict
bounds intervals, originally pioneered by Burrus (1965) and Rust and Burrus
(1972), which offer ways to directly incorporate any side information about the
parameters during inference without introducing external biases. This family of
methods allows for uncertainty quantification in ill-posed inverse problems
without needing to select a regularizing prior. By tying optimization-based
intervals to an inversion of a constrained likelihood ratio test, we translate
interval coverage guarantees into type I error control and characterize the
resulting interval via solutions to optimization problems. Along the way, we
refute the Burrus conjecture, which posited that, for possibly rank-deficient
linear Gaussian models with positivity constraints, a correction based on the
quantile of the chi-squared distribution with one degree of freedom suffices to
shorten intervals while maintaining frequentist coverage guarantees. Our
framework provides a novel approach to analyzing the conjecture, and we
construct a counterexample employing a stochastic dominance argument, which we
also use to disprove a general form of the conjecture. We illustrate our
framework with several numerical examples and provide directions for extensions
beyond the Rust-Burrus method for nonlinear, non-Gaussian settings with general
constraints.
