Convex Functions: Domain
Function:
Def
Extended-value function:
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Def
Optimization for ML
1
Convex Functions: Epigraph
Epigraph:
Def
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Optimization for ML
2
Convex Functions: Definition
Convex function:
Def
Convex function:
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Optimization for ML
3
Convex Functions: Definition
Convex function:
Def
Concave function:
Def
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Optimization for ML
4
Convex Functions: Definition
Concave function:
Def
Epigraph of a concave function is:
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1) concave
2) convex
3) neither concave nor
convex in general
4) none is correct
Optimization for ML
5
Convex Functions: Definition
Concave function:
Def
Epigraph of a concave function is:
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1) concave
2) convex
3) neither concave nor
convex in general
4) none is correct
Optimization for ML
6
Convex Functions: Definition
Convex function:
Def
Concave function:
Def
Linear function:
Def
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Optimization for ML
7
Convex Functions: Examples
Convex function on a convex set of constraints = convex:
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Optimization for ML
8
Convex Functions: Examples
Proposition:
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Optimization for ML
9
Convex Functions: Examples
Intersection of hyperplanes is convex:
Here could be your illustration!
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Optimization for ML
10
Convex Functions: Examples
Intersection of hyperplanes is convex:
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Optimization for ML
11
Convex Functions: Examples
Supremum of linear functions is convex:
Proof. Intersection of convex epigraphs is a convex set.
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Optimization for ML
12
Convex Functions: Examples
Supremum of convex functions:
?
1) Yes
2) No
3) I do not know
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Draw your picture
Optimization for ML
13
Convex Functions: Examples
Infinum of linear functions is concave:
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Optimization for ML
14
Convex Functions: Examples
Infinum of linear functions is concave:
?
1) Yes
2) No
3) I do not know
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Optimization for ML
15
Convex Functions: Examples
Infinum of linear functions is concave:
?
1) Yes
2) No
3) I do not know
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Optimization for ML
16