In this paper we discuss the properties and optimization of the linear variance function. The linear variance function is a measure of the how variation in variables and parameters is transmitted to design functions. By minimizing this function, it is possible to develop robust designs that are more tolerant of variation. We discuss general mathematical properties of this function as well as properties of the variance of quadratic functions. We show how these properties can be used to more efficiently optimize the variance function, and report test results on seven problems. An example problem of a fluid flow check valve is presented showing how robustness can be achieved by minimizing the linear variance function.