Difference between revisions of "Convex function"
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==Definition== | ==Definition== | ||
| − | Let | + | Let {{M|S\in\mathcal{P}(\mathbb{R}^n)}} be an arbitrary [[subset of]] [[Euclidean n-space|Euclidean {{n|space}}]], {{M|\mathbb{R}^n}}, and let {{M|f:S\rightarrow\mathbb{R} }} be a [[function]]. We say {{M|f}} is a ''convex function'' if both of the following hold{{rAFCIRAPM}}: |
# {{M|S}} is a [[convex set]] itself, {{ie}} the line connecting any two points in {{M|S}} is also entirely contained in {{M|S}} | # {{M|S}} is a [[convex set]] itself, {{ie}} the line connecting any two points in {{M|S}} is also entirely contained in {{M|S}} | ||
#* In symbols: {{M|\forall x,y\in S\forall t\in [0,1]\subset\mathbb{R}[x+t(y-x)\in S]}}, and | #* In symbols: {{M|\forall x,y\in S\forall t\in [0,1]\subset\mathbb{R}[x+t(y-x)\in S]}}, and | ||
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#* In symbols: {{M|\forall t\in [0,1]\subset\mathbb{R}[f(x+t(y-x))\le f(x)+t(f(y)-f(x))]}} | #* In symbols: {{M|\forall t\in [0,1]\subset\mathbb{R}[f(x+t(y-x))\le f(x)+t(f(y)-f(x))]}} | ||
{{Requires work|msg=A picture would be great|grade=C}} | {{Requires work|msg=A picture would be great|grade=C}} | ||
| + | ==Equivalent statements== | ||
| + | * "''[[a function is convex if and only if its domain is convex and its epigraph are convex sets]]''"{{rAFCIRAPM}} | ||
==References== | ==References== | ||
<references/> | <references/> | ||
{{Definition|Analysis|Functional Analysis|Combinatorial Optimisation|Convex Optimisation}} | {{Definition|Analysis|Functional Analysis|Combinatorial Optimisation|Convex Optimisation}} | ||
Latest revision as of 10:54, 10 February 2017
- See convex for other uses of the word (eg a convex set)
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Definition
Let [ilmath]S\in\mathcal{P}(\mathbb{R}^n)[/ilmath] be an arbitrary subset of Euclidean [ilmath]n[/ilmath]-space, [ilmath]\mathbb{R}^n[/ilmath], and let [ilmath]f:S\rightarrow\mathbb{R} [/ilmath] be a function. We say [ilmath]f[/ilmath] is a convex function if both of the following holdTemplate:RAFCIRAPM:
- [ilmath]S[/ilmath] is a convex set itself, i.e. the line connecting any two points in [ilmath]S[/ilmath] is also entirely contained in [ilmath]S[/ilmath]
- In symbols: [ilmath]\forall x,y\in S\forall t\in [0,1]\subset\mathbb{R}[x+t(y-x)\in S][/ilmath], and
- The image of a point [ilmath]t[/ilmath]-far along the line [ilmath][x,y][/ilmath] is [ilmath]\le[/ilmath] the point [ilmath]t[/ilmath]-far along the line [ilmath]f(x)[/ilmath] to [ilmath]f(y)[/ilmath]
- In symbols: [ilmath]\forall t\in [0,1]\subset\mathbb{R}[f(x+t(y-x))\le f(x)+t(f(y)-f(x))][/ilmath]
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Equivalent statements
- "a function is convex if and only if its domain is convex and its epigraph are convex sets"Template:RAFCIRAPM
