  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.

Computational Complexity;Discrete Mathematics;Numerical Analysis;Performance;Numerical Analysis