Local versus global minimizers Theorem 2.4 shows that a convex problem has at most one local minimum. Moreover, if we find a local minimum for a convex problem, it is in fact the global minimum. If we have a strictly convex function and find a local minimizer, then the minimizer is unique. We emphasize: we will see that the nature of our iterative algorithms is that we can only guarantee, at best, that the sequence of iterates converges to a local optimizer. If the problem is convex, however, the local optimizer is global. Convexity is therefore a very important property that we should seek in problem formulation.
Choice of step directions Convexity enables us to relate the two goals of:
(i) moving from the current iterate ...