  We propose two novel numerical schemes for approximate implementation of the
dynamic programming~(DP) operation concerned with finite-horizon, optimal
control of discrete-time systems with input-affine dynamics. The proposed
algorithms involve discretization of the state and input spaces and are based
on an alternative path that solves the dual problem corresponding to the DP
operation. We provide error bounds for the proposed algorithms, along with a
detailed analysis of their computational complexity. In particular, for a
specific class of problems with separable data in the state and input
variables, the proposed approach can reduce the typical time complexity of the
DP operation from $O(XU)$ to $O (X+U)$, where $X$ and $U$ denote the size of
the discrete state and input spaces, respectively. This reduction is achieved
by an algorithmic transformation of the minimization in the DP operation to an
addition via discrete conjugation.
