Consider the function which maps a vector to its maximum entry, ↦ maxᵢ ᵢ. While this function is non-smooth, a common trick in machine learning is to use a smooth approximation, LogSumExp, defined as follows.
LSE : R" → R, LSE(x) = ln » R, LSE(P) = ln [ (i=1) Σ eˣᶦ]
One of the nice properties of this function is that it is convex, which can be proved by showing its Hessian matrix is positive semidefinite.
To that end, compute its gradient and Hessian.