Semilinear Dynamic Programming: Analysis, Algorithms, and Certainty Equivalence Properties

We consider a broad class of dynamic programming (DP) problems that involve a partially linear structure and some positivity properties in their system equation and cost function. We address deterministic and stochastic problems, possibly with Markov jump parameters. We focus primarily on infinite horizon problems and prove that under our assumptions, the optimal cost function is linear, and that an optimal policy can be computed efficiently with standard DP algorithms. Moreover, we show that forms of certainty equivalence hold for our stochastic problems, in analogy with the classical linear quadratic optimal control problems.