Multi-Stage Planning and the Extended Linear-Quadratic-Gaussian Control Problem

Multi-stage planning processes arise in many situations in operations research and economics. The purpose of the planning process is to adapt to a future described by certain state variables x(k). This paper considers the problem of finding optimal forecasts for the state variables which explicitly take account of a cost tradeoff between costs associated to errors in predicting the values of the state variables and costs associated with changes in forecasts for a particular time k made at successive time periods k₁, k₂. That is, there is a tradeoff between the accuracy of individual forecasts x̃(k ∣ k₂) for x(k) and the stability of successive forecasts x̃(k ∣ k₁) and x̃(k ∣ k₂). This paper gives an exact solution in the case where the state variables x̃(k) are generated by a Gaussian linear system and the cost function is quadratic plus linear in the state variables and forecasts. In this case the optimal forecasts are matrix weighted linear combinations of Kalman filter estimates, with weights depending only on the loss function. In the special case that all costs associated with a change in forecasts occur in the period the change is made, these weights have a simple recursive form. This problem is a special case of a stochastic control problem we call the extended linear-quadratic-Gaussian control problem.