Moreau’s Proximal Mappings and Convexity in Hamilton-Jacobi Theory

Abstract

Proximal mappings, which generalize projection mappings, were introduced by Moreau and shown to be valuable in understanding the subgradient properties of convex functions. Proximal mappings subsequently turned out to be important also in numerical methods of optimization and the solution of nonlinear partial differential equations and variational inequalities. Here it is shown that, when a convex function is propagated through time by a generalized Hamilton-Jacobi partial differential equation with a Hamiltonian that is concave in the state and convex in the co-state, the associated proximal mapping exhibits locally Lipschitz dependence on time. Furthermore, the subgradient mapping associated of the value function associated with this mapping is graphically Lipschitzian.