II.7.3.1: Existence

As we have seen in section II.5.5.2 quadratic majorizers exist if Although is independent of , the numerical value of does depend on , and of course on .

We say that a sharp quadratic majorizer exists if , i.e. if the supremum is attained. We the define the sharp quadratic majorizer of at as with The corresponding majorization algorithm is

If is convex we can generalize the tangential majorization algorithm of De Leeuw and Lange [2009, section 6] to compute For convex and thus with If the derivative of the minorizer on the right hand side of vanishes we have proportional to say It remains to maximize over . The maximum exists because convexity implies and it is attained for Thus tangential majorization gives the algorithm


[Insert exist.R Here](../code/exist.R)


We can compute the generalization with basically the same tangential majorization algorithm.

Quadratic majorization with updates with which has an iteration matrix at a solution equal to and an iteration radius where are the two largest eigenvalues of

As an example, consider defined by The function is convex, and as we have shown