Methods and apparatus to solve for a minimum variance solution for a constrained weight problem in a scalable and memory efficient manner are disclosed. An example method of reducing memory usage of a computing device disclosed herein comprises storing a constraint matrix and a population matrix into a memory, the population matrix representing a population related to data of the constraint matrix, determining, via a processor, a linear system based on a set of constraints including the constraint matrix and the population matrix, determining, via a processor, an inverse of a matrix of the linear system, the determination of the inverse including breaking down the matrix into blocks, the dimensions of the blocks being no larger than dimensions of the constraint matrix, and outputting a set of weights based on the inverse of the linear system, the set of weights satisfying the set of constraints.