local minimum of convex function is
necessarily global∗
stevecheng†
2013-03-21 15:52:33
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
≤ t f (y) + (1 − t)f (x)
since f is convex.
Rearranging terms, 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 .
∗ hLocalMinimumOfConvexFunctionIsNecessarilyGlobali
created: h2013-03-21i
by:
hstevechengi version: h34165i Privacy setting: h1i hTheoremi h26B25i
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