  Recent advances in fundamental performance limits for power quantities based
on Lagrange duality are proving to be a powerful theoretical tool for
understanding electromagnetic wave phenomena. To date, however, in any approach
seeking to enforce a high degree of physical reality, the linearity of the wave
equation plays a critical role. In this manuscript, we generalize the current
quadratically constrained quadratic program framework for evaluating linear
photonics limits to incorporate nonlinear processes under the undepleted pump
approximation. Via the exemplary objective of enhancing second harmonic
generation in a (free-form) wavelength-scale structure, we illustrate a model
constraint scheme that can be used in conjunction with standard convex
relaxations to bound performance in the presence of nonlinear dynamics.
Representative bounds are found to anticipate features observed in optimized
structures discovered via computational inverse design. The formulation can be
straightforwardly modified to treat other frequency-conversion processes,
including Raman scattering and four-wave mixing.
