Computation of a Unified Graph-Based Rate Optimization Problem

We define a graph-based rate optimization problem and consider its computation, which provides a unified approach to the computation of various theoretical limits, including the (conditional) graph entropy, rate-distortion functions and capacity-cost functions with side information. Compared with their classical counterparts, theoretical limits with side information are much more difficult to compute since their characterizations as optimization problems have larger and more complex feasible regions. Following the unified approach, we develop effective methods to resolve the difficulty. On the theoretical side, we derive graph characterizations for rate-distortion and capacity-cost functions with side information and simplify the characterizations in special cases by reducing the number of decision variables. On the computational side, we design an efficient alternating minimization algorithm for the graph-based problem, which deals with the inequality constraint by a flexible multiplier update strategy. Moreover, simplified graph characterizations are exploited and deflation techniques are introduced, so that the computing time is greatly reduced. Theoretical analysis shows that the algorithm converges to an optimal solution. By numerical experiments, the accuracy and efficiency of the algorithm are illustrated and its significant advantage over existing methods is demonstrated.