AN INVERSE CONTROL PROCESS AND AN INVERSE ALLOCATION PROCESS
Seiichi Iwamoto · Journal of the Operations Research Society of Japan · 1981
An inverse theory of sequential decision processes, including the standard control process and allocation process, is developed. A finite-stage deterministic invertible (main) dynamic program (DP) whose reward functions depend not only on action but also on state is formulated as a sequential decision process. The main DP is transformed into an equivalent inverse DP by an algebraic inversion. The main DP maximizes a generalized total reward, while the inverse DP minimizes a generalized total state. An inverse theorem is established. It characterizes optimal solutions (optimal reward functions and optimal policy) of inverse DP by those of main DP through inverse and composition. The main DP includes a linear equation and quadratic criterion (main) control process on the half-line and a typical multi-stage (main) allocation process. Therefore, the inverse DP generates an inverse control process and an inverse allocation process, respectively. Not solving the recursive equation directly but applying the inverse theorem, optimal solutions of both inverse processes are easily calculated by use of those of the corresponding main processes.