----- - Let $X$ be a [[convex set|convex subset]] of a [[metric space]]; - Let $f: X \to \rr$ be a [[convex function]]. > [!proposition] Proposition. ([[local mins of convex functions are global mins]]) > If $\v u^* \in X$ is a [[local extrema|local minimizer]] of $f$, then $\v u^*$ is a [[global extrema|global minimizer]] of $f$. > [!proof]- Proof. ([[local mins of convex functions are global mins]]) > Suppose $\v u^*$ is a [[local extrema|local min]], but not a [[global extrema|global min]]. Then there exists $\v v^* \in X$ such that $f(\v v^*) < f(\v u ^*)$. Since $f$ is [[convex function|convex]], we have for all $t \in [0,1)$ that $\begin{align}f(t \v u^* + (1-t)\v v^*) \leq & tf(\v u ^*) + (1-t)f(\v v^{*}) \\ < & tf(\v u ^{*}) +(1-t)f(\v u ^{*} ) \\ & = tf(\v u ^{*}). \end{align}$ Letting $t$ approach $1$, the point $t \v u ^{*}$ becomes arbitrarily close to $\v u ^{*}$. That's a contradiction— hence $\v u^{*}$ must be a [[global extrema|global min]]. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```