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.

Answer 2

See picture below for some intuition on why that may be (bold line is the pointwise infimum of the four affine functions). Similarly, you can conclude that the

Pointwise supremum of the set of affine functions is convex