II.1.4.2: Univariate and Separable Functions

Many of our examples are majorizations of a real-valued function of a single real variable over the real line This is partly for mathematical convenience, because many results are simpler in the univariate case. And partly for didactic reasons, because plots and tables are more easy to visualize and interpret.

As pointed out by De Leeuw and Lange [2009] looking at the univariate case is obviously restrictive, but not as restrictive as it seems. Many of the functions in optimization and statistics are separable, which means they are of the form and majorizing each of the univariate gives a majorization of . Note that if , where the are positive, can be turned into a separable problem by taking logarithms.

In addition it is often possible to majorize a non-separable function by a separable one. Suppose, for example, that where is positive semi-definite. Suppose we can find a diagonal such that then for any two vectors and in where Thus with The majorizer we have constructed is separable.