Convex Analysis/Strong convexity

From Wikibooks, open books for an open world
Jump to navigation Jump to search

Definition (strongly convex function):

Let be a Banach space over . A function is called strongly convex with parameter iff the following equation holds for all and :

Proposition (existence and uniqueness of minimizers of strongly convex functions):

Let be a Banach space over and let be a strongly convex function with parameter , which additionally is bounded below (say by ) and continuous. Then admits a unique minimizer (ie. an element which realizes the infimum of , where ranges over ).

Proof: Since , the value exists. Choose a sequence in such that

.[Note 1]

is a Cauchy sequence because if is such that and is such that , then




in particular, if we show that a minimizer exists, then it will be unique, for if we set and call any other minimizer , the above estimate holds for arbitrary. Since is Banach, is convergent, say to . If we show that

for all , then . By the continuity of , choose such that implies . By convergence of pick sufficiently large so that for all . Then choose such that . Then the triangle inequality implies


  1. If is separable, so that arbitrary products of nonempty open sets are nonempty, the continuity of implies that the axiom of choice is not required for this construction.