On $\mathcalVU$-theory for Functions with Primal-Dual Gradient Structure

We consider a general class of convex functions having what we call primal-dual gradient structure. It includes finitely determined max-functions and maximum eigenvalue functions as well as other infinitely defined max-functions. For a function in this class, we discuss a space decomposition that allows us to identify a subspace on which the function appears to be smooth. Moreover, using the special structure of such a function, we compute smooth trajectories along which certain second-order expansions can be obtained. We also give an explicit expression for the Hessian of a related Lagrangian.