Much of the continuing appeal and challenge of electromagnetism stems from the same root: given some desired objective (enhancing radiation from a quantum emitter, the field intensity in a photovoltaic cell, the radiative cross section of an antenna) subject to some physical constraints (material compatibility, fabrication tolerances, or system size) there is currently no method for finding or assessing uniquely best wave solutions. While improvements in nanofabrication and computational methods have driven dramatic progress in expanding the range of achievable optical characteristics, they have also greatly increased design complexity. These developments have led to heightened relevance for the study of fundamental limits on optical response. Here, we review present results pertaining to an emerging theoretical method based on Lagrange duality that combines techniques from convex optimization and conservation laws to frame the calculation of physical limits on classical and quantum wave problems as quadratically constrained quadratic programs. Results pertaining to canonical electromagnetic problems such as thermal emission, scattering cross sections, Purcell enhancement, and power routing are presented.
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