local minimum of convex function is necessarily global

Theorem 1.

Let f:S 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 ξU. We prove f(x)f(y) for arbitrary yS.

Consider the convex combinationMathworldPlanetmath (1-t)x+ty for 0t1:

Since scalar multiplication and vector addition are, by definition, continuousMathworldPlanetmath in a topological vector space, the convex combination approaches x as t0. 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