  We define a regularized variant of the Dual Dynamic Programming algorithm
called REDDP (REgularized Dual Dynamic Programming) to solve nonlinear dynamic
programming equations. We extend the algorithm to solve nonlinear stochastic
dynamic programming equations. The corresponding algorithm, called SDDP-REG,
can be seen as an extension of a regularization of the Stochastic Dual Dynamic
Programming (SDDP) algorithm recently introduced which was studied for linear
problems only and with less general prox-centers. We show the convergence of
REDDP and SDDP-REG. We assess the performance of REDDP and SDDP-REG on
portfolio models with direct transaction and market impact costs. In
particular, we propose a risk-neutral portfolio selection model which can be
cast as a multistage stochastic second-order cone program. The formulation is
motivated by the impact of market impact costs on large portfolio rebalancing
operations. Numerical simulations show that REDDP is much quicker than DDP on
all problem instances considered (up to 184 times quicker than DDP) and that
SDDP-REG is quicker on the instances of portfolio selection problems with
market impact costs tested and much faster on the instance of risk-neutral
multistage stochastic linear program implemented (8.2 times faster).
