Curse-of-complexity attenuation in the curse-of-dimensionality-free method for HJB PDEs

Recently, a curse-of-dimensionality-free method was developed for solution of Hamilton-Jacobi-Bellman partial differential equations (HJB PDEs) for nonlinear control problems, using semiconvex duality and max-plus analysis. The curse-of-dimensionality-free method may be applied to HJB PDEs where the Hamiltonian is given as (or well-approximated by) a pointwise maximum of quadratic forms. Such HJB PDEs also arise in certain switched linear systems. The method constructs the correct solution of an HJB PDE from a max-plus linear combination of quadratics. The method completely avoids the curse-of-dimensionality, and is subject to cubic computational growth as a function of space dimension. However, it is subject to a curse-of-complexity. In particular, the number of quadratics in the approximation grows exponentially with the number of iterations. Efficacy of such a method depends on the pruning of quadratics to keep the complexity growth at a reasonable level. Here we apply a pruning algorithm based on semidefinite programming. Computational speeds are exceptional, with an example HJB PDE in six-dimensional Euclidean space solved to the indicated quality in approximately 30 minutes on a typical desktop machine.