Inexact cuts in Stochastic Dual Dynamic Programming

We introduce an extension of Stochastic Dual Dynamic Programming (SDDP) to solve stochastic convex dynamic programming equations. This extension applies when some or all primal and dual subproblems to be solved along the forward and backward passes of the method are solved with bounded errors (inexactly). This inexact variant of SDDP is described both for linear problems (the corresponding variant being denoted by ISDDP-LP) and nonlinear problems (the corresponding variant being denoted by ISDDP-NLP). We prove convergence theorems for ISDDP-LP and ISDDP-NLP both for bounded and asymptotically vanishing errors. Finally, we present the results of numerical experiments comparing SDDP and ISDDP-LP on portfolio problem with direct transaction costs modelled as a multistage stochastic linear optimization problem. On these experiments, ISDDP-LP allows us to obtain a good policy faster than SDDP.