The discovery of efficient and accurate descriptions for the macroscopic constitutive behavior of heterogeneous materials with complex microstructure remains an outstanding challenge in mechanics. On the one hand, great accuracy can be achieved by modeling small domains of a material including all the details in the microstructure, however, at the expense of a large computational cost. On the other hand, efficient descriptions of the macroscopic material behavior can be obtained by empirical constitutive laws, at the expense of a tedious calibration process and limited accuracy. The challenge is in finding an optimal balance between preserving enough small-scale detail and keeping the computational expense low, without the need for empirically calibrated models. Based on the Lippmann-Schwinger integral equation, two novel reduced-order homogenization methods have been developed in this dissertation.