Additive manufacturing (AM) allows for the inclusion of complicated geometric features that are impractical or impossible to manufacture by other means. Among such features is the collection of intricate and periodic strut-like geometries known as lattice structures. Lattice structures are desirable for their ability to provide stiffness through a large number of supporting members while employing void space within the geometry as a means to reduce part material volume. Strut thicknesses of every lattice in a part are generally not well optimized in order to maximize part stiffness, and often every lattice unit cell is identical throughout the part. This work presents a lattice density optimization methodology that is able to find the optimal graded lattice density distribution for maximizing the part stiffness and also improving the additive manufacturability of the part. The material property interpolation scheme used in SIMP optimization is replaced by a representative volume element (RVE)-based interpolation scheme that more accurately captures the material properties of the prescribed lattice structure at an arbitrary density. A filter has been developed that allows for trimming of unnecessary lattices while simultaneously ensuring that the geometry remains self-supporting during the AM build process. This filter is incorporated seamlessly within the topology optimization routine. This increases the optimality of the resulting design when compared with full-domain lattice filling and increases the viability of the design from a manufacturing standpoint when compared with unconstrained lattice trimming.