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[parent] local minimum of convex function is necessarily global (Theorem)
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\to \mathbb{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) \leq f(\xi)$ for all $ \xi\in U$. We prove $ f(x) \leq f(y)$ for arbitrary $ y \in S$.

Consider the convex combination $ (1-t)x + ty$ for $ 0 \leq t \leq 1$:

\includegraphics{convex-minimum.eps}
Since scalar multiplication and vector addition are, by definition, continuous in a topological vector space, the convex combination approaches $ x$ as $ t \to 0$. Therefore for small enough $ t$, $ (1-t)x + ty$ is in the neighborhood $ U$. Then
$\displaystyle f(x)$ $\displaystyle \leq f\bigl(ty + (1-t) x\bigr)$ for small $ t > 0$    
  $\displaystyle \leq t f(y) + (1-t) f(x)$ since $ f$ is convex.    

Rearranging terms, we have $ f(x) \leq f(y)$.

To show the analogous situation for a concave function $ f$, the above reasoning can be applied after replacing $ f$ with $ -f$. $ \qedsymbol$



"local minimum of convex function is necessarily global" is owned by stevecheng. [ full author list (2) | owner history (3) ]
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Other names:  extremal value of convex/concave functions, local maximum of concave function is necessarily global

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Cross-references: continuous, vector addition, multiplication, scalar, convex combination, neighborhood, open, convex set, extremum, topological vector space, convex subset, concave function, convex function, local maximum, local minimum
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This is version 10 of local minimum of convex function is necessarily global, born on 2003-04-07, modified 2006-07-05.
Object id is 4165, canonical name is ExtremalValueOfConvexconcaveFunctions.
Accessed 7684 times total.

Classification:
AMS MSC: 26B25 (Real functions :: Functions of several variables :: Convexity, generalizations)
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