[Math] Pointwise infimum of affine functions is concave

Question

So I was just starting on convex optimization and was having a slightly hard time visualizing the lagrangian being always concave because it is the pointwise infimum of a family of affine functions.

Can anyone help explain this? I've googled extensively but most places just state this without elaboration or examples.

Thanks.

Answer 1

Daniel Fischer gave a transparent explanation in terms of epigraphs $\{(x,y): y\ge f(x)\}$:

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 $\{(x,y): y\le f(x)\}$: this set is convex if and only if $f$ is concave. Taking infimum of functions results in taking the intersection of such sets.