This note summarizes the optimization formulations used in the study of Markov decision processes. We consider both the discounted and undiscounted processes under the standard and the entropy-regularized settings. For each setting, we first summarize the primal, dual, and primal-dual problems of the linear programming formulation. We then detail the connections between these problems and other formulations for Markov decision processes such as the Bellman equation and the policy gradient method.
We discuss solution algorithms for calibrating a tumor growth model using imaging data posed as a deterministic inverse problem. The forward model consists of a nonlinear and time-dependent reaction-diffusion partial differential equation (PDE) with unknown parameters (diffusivity and proliferation rate) being spatial fields. We use a dimension-independent globalized, inexact Newton Conjugate Gradient algorithm to solve the PDE-constrained optimization. The required gradient and Hessian actions are also presented using the adjoint method and Lagrangian formalism.
In this brief note, we find formulas for the distribution and the transition probability matrices of a stochastic process described as a time-reversion in a finite time window of a Markov chain, with cluster observation of the Markov state in the extremes of that window.
Weather forecast information will very likely find increasing application in the control of future energy systems. In this paper, we introduce an augmented state space model formulation with linear dynamics, within which one can incorporate forecast information that is dynamically revealed alongside the evolution of the underlying state variable. We use the martingale model for forecast evolution (MMFE) to enforce the necessary consistency properties that must govern the joint evolution of forecasts with the underlying state. The formulation also generates jointly Markovian dynamics that give …
In this short note we derive a relationship between the Bregman divergence from the current policy to the optimal policy and the suboptimality of the current value function in a regularized Markov decision process. This result has implications for multi-task reinforcement learning, offline reinforcement learning, and regret analysis under function approximation, among others.
