Fractional programming (FP) plays a crucial role in wireless network design because many relevant problems involve maximizing or minimizing ratio terms. Notice that the maximization case and the minimization case of FP cannot be converted to each other in general, so they have to be dealt with separately in most of the previous studies. Thus, an existing FP method for maximizing ratios typically does not work for the minimization case, and vice versa. However, the FP objective can be mixed max-and-min, e.g., one may wish to maximize the signal-to-interference-plus-noise ratio (SINR) of the legitimate receiver while minimizing that of the eavesdropper. We aim to fill the gap between max-FP and min-FP by devising a unified optimization framework. The main results are three-fold. First, we extend the existing max-FP technique called quadratic transform to the min-FP, and further develop a full generalization for the mixed case. Second. we provide a minorization-maximization (MM) interpretation of the proposed unified approach, thereby establishing its convergence and also obtaining a matrix extension; another result we obtain is a generalized Lagrangian dual transform which facilitates the solving of the logarithmic FP. Finally, we present three typical applications: the age-of-information (AoI) minimization, the Cramer-Rao bound minimization for sensing, and the secure data rate maximization, none of which can be efficiently addressed by the previous FP methods.

翻译：分式规划（FP）在无线网络设计中扮演着关键角色，因为许多相关问题涉及最大化或最小化分式项。注意到FP的最大化情形与最小化情形通常不能相互转化，因此在以往的大多数研究中必须分别处理。这意味着现有针对最大化分式的FP方法通常不适用于最小化情形，反之亦然。然而，FP的目标函数可以是混合最大-最小形式，例如，可能希望最大化合法接收端的信干噪比（SINR），同时最小化窃听端的信干噪比。我们旨在通过设计一个统一优化框架来填补最大-FP与最小-FP之间的空白。主要成果体现在三个方面。首先，我们将现有的最大-FP技术（称为二次变换）扩展到最小-FP，并进一步发展为混合情形的完全泛化形式。其次，我们为所提出的统一方法提供了最小化-最大化（MM）解释，从而建立其收敛性，并进一步获得矩阵扩展；另一项成果是推导出广义拉格朗日对偶变换，该变换有助于求解对数型FP问题。最后，我们展示三个典型应用：信息年龄（AoI）最小化、传感克拉美-罗界最小化以及安全数据速率最大化，这些应用此前均无法通过现有FP方法有效处理。