  Based on quasinormal-mode theory, we propose a novel approach enabling a deep
analytical insight into the multi-parameter design and optimization of
nonlinear photonic structures at subwavelength scale. A key distinction of our
method from previous formulations relying on multipolar Mie-scattering
expansions is that it directly exploits the natural resonant modes of the
nanostructures, which provide the field enhancement to achieve significant
nonlinear efficiency. Thanks to closed-form expression for the nonlinear
overlap integral between the interacting modes, we illustrate the potential of
our method with a two-order-of-magnitude boost of second harmonic generation in
a semiconductor nanostructure, by engineering both the sign of $\chi^{(2)}$ at
subwavelength scale and the structure of the pump beam.
