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Multiplicative Programming (MP) pertains to a spectrum of optimizationproblems that involve product term(s). As computational paradigms ofcommunication systems continue to evolve, particularly concerning theoffloading strategies of computationally intensive tasks simultaneously tocentralized or decentralized servers, designing or optimizing effectivecommunication systems with MP techniques becomes increasingly indispensable.Similarly, Fractional Programming (FP) is another significant branch in theoptimization domain, addressing various essential scenarios in communication.For instance, in minimization optimization problems, transmission power andprocessing delay of communication systems are considered critical metrics. In avery recent JSAC paper by Zhao et al. [2], an innovative transform (Zhao'sOptimization Transform) was proposed for solving the minimization of MP and FPproblems. Nevertheless, the resolution of optimization problems incommunication systems encounters several limitations when adopting Zhao'soptimization transform, especially in MP problems. Primarily, objectivefunctions proposed in these optimization problems typically involvesum-of-products terms and the optimization variables are always discreteleading to NP-hard problems. Furthermore, multiple functions mapping to thenon-negative domain in these scenarios can result in auxiliary variables beingzero values, while the same situation is avoidable in FP problems due to thepresence of these functions in the denominator. In this paper, we introduce anupdated transform, building on the foundations of Zhao's original method,designed to effectively overcome these challenges by reformulating the originalproblem into a series of convex or concave problems. This introduced problemreformulation provides a superior iteration algorithm with demonstrableconvergence to a stationary point.

Transforms for Multiplicative and Fractional Programming with Broad Applications in Edge Computing and Communication Networks