High order Bellman equations and weakly chained diagonally dominant tensors
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. Mtensors. In this case, we give a policy iteration algorithm to compute this solution. We also prove that a weakly diagonally dominant Ztensor with nonnegative diagonals is a strong Mtensor 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. "Mtensors 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 Mtensors and optimal control has not been previously established.
 Publication:

arXiv eprints
 Pub Date:
 March 2018
 arXiv:
 arXiv:1803.08870
 Bibcode:
 2018arXiv180308870A
 Keywords:

 Mathematics  Rings and Algebras;
 Mathematics  Numerical Analysis;
 15A69;
 65H10 (Primary) 65N22 (Secondary)
 EPrint:
 27 pages, 3 figures, 2 tables