local minimum of convex function is necessarily global
Theorem 1.
A local minimum (resp. local maximum) of a convex function
(resp. concave function) on a
convex subset of a topological vector space
, is always a global extremum.
Proof.
Let f:S→ℝ be a convex function on a convex set S in a topological vector space.
Suppose x is a local minimum for f; that is, there is an open neighborhood U of x where f(x)≤f(ξ) for all ξ∈U. We prove f(x)≤f(y) for arbitrary y∈S.
Consider the convex combination (1-t)x+ty for 0≤t≤1:
Since scalar multiplication and vector addition are, by definition,
continuous in a topological vector space, the convex combination approaches
x as t→0. Therefore for small enough t,
(1-t)x+ty is in the neighborhood U.
Then
f(x) | ≤f(ty+(1-t)x) | for small t>0 | ||
≤tf(y)+(1-t)f(x) | since f is convex. |
Rearranging , we have f(x)≤f(y).
To show the analogous situation for a concave function f, the above reasoning can be applied after replacing f with -f. ∎
Title | local minimum of convex function is necessarily global |
---|---|
Canonical name | LocalMinimumOfConvexFunctionIsNecessarilyGlobal |
Date of creation | 2013-03-22 13:33:37 |
Last modified on | 2013-03-22 13:33:37 |
Owner | stevecheng (10074) |
Last modified by | stevecheng (10074) |
Numerical id | 13 |
Author | stevecheng (10074) |
Entry type | Theorem |
Classification | msc 26B25 |
Synonym | extremal value of convex/concave functions |
Synonym | local maximum of concave function is necessarily global |