Nanophotonics is developing at a rapid pace, with ever more materials, form factors, and structural degrees of freedom now available. These large design spaces offer transformative technological possibility, but optimization subject to the nonconvex Maxwell-equation constraint is difficult. And incorporation of extraneous constraints such as fabrication tolerance further stymie various analytical approaches. I will discuss optimization concepts and physical insights that lead to global bounds, and structures that nearly achieve them, for three applications: (1) spontaneous emission, where causality and convexity can be united to identify power-bandwidth limits, (2) superresolution, where various physical relaxations lead to an analytically tractable quadratically constrained quadratic program, and (3) minimal mode volume, where we leverage Lagrangian duality for computational bounds that seamlessly incorporate fabrication-tolerance and multi-frequency-design constraints.