We define a regularized variant of the dual dynamic programming algorithm called DDP-REG 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 DDP-REG and SDDP-REG. We assess the performance of DDP-REG 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 DDP-REG 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).

中文翻译：

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凸非线性优化问题的正则随机对偶动态规划

我们定义了称为DDP-REG的双重动态规划算法的规则化变体，以求解非线性动态规划方程。我们扩展算法来求解非线性随机动态规划方程。相应的算法称为SDDP-REG，可以看作是最近引入的随机双动态规划（SDDP）算法的正则化的扩展，该算法仅针对线性问题进行了研究，通用中心较少。我们展示了DDP-REG和SDDP-REG的融合。我们用直接交易和市场影响成本评估DDP-REG和SDDP-REG在投资组合模型上的表现。特别是，我们提出了一种风险中立的投资组合选择模型，该模型可以转换为多阶段随机二阶锥程序。该公式的制定是受市场影响成本对大型投资组合再平衡操作的影响。数值模拟表明，在所有考虑的问题实例上，DDP-REG都比DDP快得多（比DDP快184倍），而在经过测试的市场影响成本的投资组合选择问题上，SDDP-REG更快，而在考虑了市场影响成本的情况下，SDDP-REG更快。实施风险中性的多阶段随机线性程序的实例（速度提高了8.2倍）。