  Physical design problems, such as photonic inverse design, are typically
solved using local optimization methods. These methods often produce what
appear to be good or very good designs when compared to classical design
methods, but it is not known how far from optimal such designs really are. We
address this issue by developing methods for computing a bound on the true
optimal value of a physical design problem; physical designs with objective
smaller than our bound are impossible to achieve. Our bound is based on
Lagrange duality and exploits the special mathematical structure of these
physical design problems. For a multi-mode 2D Helmholtz resonator, numerical
examples show that the bounds we compute are often close to the objective
values obtained using local optimization methods, which reveals that the
designs are not only good, but in fact nearly optimal. Our computational
bounding method also produces, as a by-product, a reasonable starting point for
local optimization methods.
