Fenchel's duality theorem

In mathematics, Fenchel's duality theorem is a result in the theory of convex functions named after Werner Fenchel.

Let ƒ be a proper convex function on Rn and let g be a proper concave function on Rn.

Then, if regularity conditions are satisfied, where ƒ * is the convex conjugate of ƒ (also referred to as the Fenchel–Legendre transform) and g * is the concave conjugate of g. That is, Let X and Y be Banach spaces,

Then the Fenchel problems: satisfy weak duality, i.e.

Suppose that f,g, and A satisfy either Then strong duality holds, i.e.

[1] In the following figure, the minimization problem on the left side of the equation is illustrated.

One seeks to vary x such that the vertical distance between the convex and concave curves at x is as small as possible.

The position of the vertical line in the figure is the (approximate) optimum.

The next figure illustrates the maximization problem on the right hand side of the above equation.

Imagine the two tangents as metal bars with vertical springs between them that push them apart and against the two parabolas that are fixed in place.

Fenchel's theorem states that the two problems have the same solution.

The points having the minimum vertical separation are also the tangency points for the maximally separated parallel tangents.