local minimum of convex function is necessarily global
Let be a convex function on a convex set in a topological vector space.
Suppose is a local minimum for ; that is, there is an open neighborhood of where for all . We prove for arbitrary .
Consider the convex combination for :
Since scalar multiplication and vector addition are, by definition, continuous in a topological vector space, the convex combination approaches as . Therefore for small enough , is in the neighborhood . Then
|since is convex.|
Rearranging , we have .
To show the analogous situation for a concave function , the above reasoning can be applied after replacing with . ∎
|Title||local minimum of convex function is necessarily global|
|Date of creation||2013-03-22 13:33:37|
|Last modified on||2013-03-22 13:33:37|
|Last modified by||stevecheng (10074)|
|Synonym||extremal value of convex/concave functions|
|Synonym||local maximum of concave function is necessarily global|