Wikipedia reads, "The dual function g is concave, even when the initial problem is not convex, because it is a point-wise infimum of affine functions." Can someone explain this? Maybe provide a basic example? What do they mean by "point-wise" and how does the point-wise infimum of affine functions imply concavity?
I've read similar posts about this on here but they use the same argument without an example or much of a proof.
Best Answer
I don't know why I couldn't see it initially but for any given $\lambda$ we have an affine function, x being constant so the basic idea is (just looking at the graph):
wherein the "x-axis" is lambda and the "y-axis" is the value of the dual function g($\lambda$)