Completion-of-squares: Revisited and extended

Due to its simplicity, the completion-of-squares technique is quite popular in linear optimal control. However, this simple technique is limited to linear quadratic Gaussian systems. In this note, by interpreting the completion-of-squares from a new angle, we extend this technique to performance optimization of general Markov systems with the long run average criterion, leading to a new approach to performance optimization based on direct comparisons of the performance of two policies.