High order Bellman equations and weakly chained diagonally dominant tensors

TL;DR

This paper extends classical Bellman equations to tensors, introduces weakly chained diagonally dominant tensors, and provides conditions for solutions and algorithms for high order Bellman equations, with applications to optimal control.

Contribution

It introduces high order Bellman equations and weakly chained diagonally dominant tensors, establishing existence, uniqueness, and algorithms for solutions in tensor-based optimal control.

Findings

W.c.d.d. tensors ensure unique positive solutions.

Policy iteration algorithm converges for high order Bellman equations.

Application to optimal control shows improved computational performance.

Abstract

We introduce high order Bellman equations, extending classical Bellman equations to the tensor setting. We introduce weakly chained diagonally dominant (w.c.d.d.) tensors and show that a sufficient condition for the existence and uniqueness of a positive solution to a high order Bellman equation is that the tensors appearing in the equation are w.c.d.d. M-tensors. In this case, we give a policy iteration algorithm to compute this solution. We also prove that a weakly diagonally dominant Z-tensor with nonnegative diagonals is a strong M-tensor if and only if it is w.c.d.d. This last point is analogous to a corresponding result in the matrix setting and tightens a result from [L. Zhang, L. Qi, and G. Zhou. "M-tensors and some applications." SIAM Journal on Matrix Analysis and Applications (2014)]. We apply our results to obtain a provably convergent numerical scheme for an optimal control problem using an "optimize then discretize" approach which outperforms (in both computation time and accuracy) a classical "discretize then optimize" approach. To the best of our knowledge, a link between M-tensors and optimal control has not been previously established.