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.

翻译：我们定义了一个基于图的速率优化问题并研究其计算方法，该方法为计算多种理论极限提供了统一框架，包括（条件）图熵、率失真函数以及带边信息的容量-成本函数。相较于经典对应问题，带边信息的理论极限计算更为困难，因为其优化问题表征具有更大且更复杂的可行域。遵循统一框架，我们开发了有效方法来解决这一难题。在理论层面，我们推导了带边信息的率失真函数与容量-成本函数的图表征，并通过减少决策变量数量简化了特殊情形下的表征形式。在计算层面，我们设计了一种高效的交替最小化算法来处理该基于图的问题，该算法通过灵活的乘子更新策略处理不等式约束。此外，我们利用简化的图表征并引入收缩技术，从而大幅减少了计算时间。理论分析表明该算法收敛至最优解。数值实验验证了算法的准确性与效率，并证明其相较于现有方法的显著优势。