Reliable communication over a discrete memoryless channel with the help of a relay has aroused interest due to its widespread applications in practical scenarios. By considering the system with a mismatched decoder, previous works have provided optimization models to evaluate the mismatch capacity in these scenarios. The proposed models, however, are difficult due to the complicated structure of the mismatched decoding problem with the information flows in hops given by the relay. Existing methods, such as the grid search, become impractical as they involve finding all roots of a nonlinear system, with the growing size of the alphabet. To address this problem, we reformulate the max-min optimization model as a consistent maximization form, by considering the dual form of the inner minimization problem and the Lagrangian with a fixed multiplier. Based on the proposed formulation, an alternating maximization framework is designed, which provides the closed-form solution with simple iterations in each step by introducing a suitable variable transformation. The effectiveness of the proposed approach is demonstrated by the simulations over practical scenarios, including Quaternary and Gaussian channels. Moreover, the simulation results of the transitional probability also shed light on the promising application attribute to the quantizer design in the relay node.

翻译：借助中继实现离散无记忆信道上的可靠通信因其在实际场景中的广泛应用而备受关注。通过考虑采用失配解码器的系统，先前的研究已提出优化模型来评估此类场景下的失配容量。然而，由于中继引入的跳数信息流使失配解码问题结构复杂，所提出的模型求解困难。现有方法（如网格搜索）需要求解非线性系统的全部根，随着字母表规模的增大，这些方法变得不切实际。为解决该问题，我们通过考虑内部最小化问题的对偶形式及固定乘子下的拉格朗日函数，将原最大-最小优化模型重构为一致的最大化形式。基于此重构模型，设计了交替最大化框架：通过引入合适的变量变换，该框架在每一步迭代中均可提供闭式解。通过在四进制信道和高斯信道等实际场景中的仿真验证了所提方法的有效性。此外，转移概率的仿真结果亦揭示了该方法在中继节点量化器设计方面的潜在应用价值。