Multiplicative Programming (MP) pertains to a spectrum of optimization problems that involve product term(s). As computational paradigms of communication systems continue to evolve, particularly concerning the offloading strategies of computationally intensive tasks simultaneously to centralized or decentralized servers, designing or optimizing effective communication systems with MP techniques becomes increasingly indispensable. Similarly, Fractional Programming (FP) is another significant branch in the optimization domain, addressing various essential scenarios in communication. For instance, in minimization optimization problems, transmission power and processing delay of communication systems are considered critical metrics. In a very recent JSAC paper by Zhao et al. [2], an innovative transform (Zhao's Optimization Transform) was proposed for solving the minimization of MP and FP problems. Nevertheless, the resolution of optimization problems in communication systems encounters several limitations when adopting Zhao's optimization transform, especially in MP problems. Primarily, objective functions proposed in these optimization problems typically involve sum-of-products terms and the optimization variables are always discrete leading to NP-hard problems. Furthermore, multiple functions mapping to the non-negative domain in these scenarios can result in auxiliary variables being zero values, while the same situation is avoidable in FP problems due to the presence of these functions in the denominator. In this paper, we introduce an updated transform, building on the foundations of Zhao's original method, designed to effectively overcome these challenges by reformulating the original problem into a series of convex or concave problems. This introduced problem reformulation provides a superior iteration algorithm with demonstrable convergence to a stationary point.

翻译：乘性规划涉及一类含乘积项的优化问题。随着通信系统计算范式持续演进，特别是将计算密集型任务同时卸载至集中式或分布式服务器时，利用乘性规划技术设计与优化高效通信系统变得日益重要。同样地，分式规划作为优化领域的另一重要分支，可解决通信中多种关键场景。例如在最小化优化问题中，通信系统的传输功率与处理延迟被视为关键指标。Zhao等人近期在JSAC期刊论文中提出一种创新变换（Zhao优化变换），用于求解乘性规划与分式规划的最小化问题。然而，在通信系统优化问题的求解过程中，采用Zhao优化变换存在若干局限，尤其体现在乘性规划问题中：首先，这些优化问题所提出的目标函数通常包含乘积求和项，且优化变量始终为离散型，导致问题具有NP难度；其次，此类场景中多个映射至非负域的函数可能导致辅助变量取零值，而同等情况下因这些函数出现在分母位置，该问题在分式规划中可避免。本文基于Zhao原始方法的基础提出一种改进变换，通过将原问题重构为一系列凸问题或凹问题，有效克服上述挑战。该重构方法可提供更优的迭代算法，并能证明其收敛至平稳点。