A Framework for Fractional Matrix Programming Problems with Applications in FBL MU-MIMO

An efficient framework is conceived for fractional matrix programming (FMP) optimization problems (OPs) namely for minimization and maximization. In each generic OP, either the objective or the constraints are functions of multiple arbitrary continuous-domain fractional functions (FFs). This ensures the framework's versatility, enabling it to solve a broader range of OPs than classical FMP solvers, like Dinkelbach-based algorithms. Specifically, the generalized Dinkelbach algorithm can only solve multiple-ratio FMP problems. By contrast, our framework solves OPs associated with a sum or product of multiple FFs as the objective or constraint functions. Additionally, our framework provides a single-loop solution, while most FMP solvers require twin-loop algorithms. Many popular performance metrics of wireless communications are FFs. For instance, latency has a fractional structure, and minimizing the sum delay leads to an FMP problem. Moreover, the mean square error (MSE) and energy efficiency (EE) metrics have fractional structures. Thus, optimizing EE-related metrics such as the sum or geometric mean of EEs and enhancing the metrics related to spectral-versus-energy-efficiency tradeoff yield FMP problems. Furthermore, both the signal-to-interference-plus-noise ratio and the channel dispersion are FFs. In this paper, we also develop resource allocation schemes for multi-user multiple-input multiple-output (MU-MIMO) systems, using finite block length (FBL) coding, demonstrating attractive practical applications of FMP by optimizing the aforementioned metrics.