Differences of Convex Functions

For d.c. functions (differences of convex functions) such as we can write with This gives a convex majorizer. Interesting, because basically all twice differentiable functions are d.c.

Add: convexification 02/21/15

Suppose . If we look for a majorization then our first thought is to majorize the first term, because the second is already nicely quadratic. But in this case we proceed the other way around.

In fact, let's consider the more general case , where is convex and differentiable. Note is indeed the difference of two convex functions. We see that and .

Using tangential majorization for gives Clearly is convex in for every , and since we have The iteration radius at a fixed point turns out to be

For we have . Convergence is to , and thus, using l'Hôpital's rule, For we have if , and thus The algorithm finishes in a single step, with the correct solution.