Explain this The lagrangian being always concave because it is the pointwise infimum of a family of affine functions.

dannigurl21ck2

dannigurl21ck2

Answered question

2022-11-05

Explain this
The lagrangian being always concave because it is the pointwise infimum of a family of affine functions.

Answer & Explanation

Biardiask3zd

Biardiask3zd

Beginner2022-11-06Added 16 answers

Daniel Fischer gave a transparent explanation in terms of epigraphs { ( x , y ) : y f ( x ) }:
A function is convex if and only if its epigraph is convex, and the epigraph of a pointwise supremum is the intersection of the epigraphs. [Hence,] the pointwise supremum of convex functions is convex.
One can similarly argue from concavity, using the sets { ( x , y ) : y f ( x ) }: this set is convex if and only if f is concave. Taking infimum of functions results in taking the intersection of such sets.

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