We report using Inverse Electromagnetic Design to computationally optimize the geometric shapes of metallic optical antennas or near-field transducers (NFTs) and dielectric waveguide structures that comprise a sub-wavelength optical focusing system for practical use in Heat Assisted Magnetic Recording (HAMR). This magnetic data-recording scheme relies on focusing optical energy to locally heat the area of a single bit, several hundred square nanometers on a hard disk, to the Curie temperature of the magnetic storage layer. There are three specifications of the optical system that must be met to enable HAMR as a commercial technology. First, to heat the media at scan rates upward of 10 m/s, ~1mW of light (>1% of typical laser diode output power) must be focused to a 30nm×30nm spot on the media. Second, the required lifetime of many years necessitates that the nano-scale NFT must not over-heat from optical absorption. Third, to avoid undesired erasing or interference of adjacent tracks on the media, there must be minimal stray optical radiation away from the hotspot on the hard disk. One cannot design the light delivery system by tackling each of these challenges independently, because they are governed by coupled electromagnetic phenomena. Instead, we propose multiobjective optimization using Inverse Electromagnetic Design in conjunction with a commercial 3D FDTD Maxwell's equations solver. We computationally generated designs of a metallic NFT and a high-index waveguide grating that meet the HAMR specifications simultaneously. Compared to a mock industry design, our proposed design has a similar optical coupling efficiency, ~3x improved suppression of stray optical radiation, and a 60% (280°C) reduction in NFT temperature rise. We also distributed the Inverse Electromagnetic Design software online so that industry partners can use it as a repeatable design process.