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：分数规划(FP)在无线网络设计中具有关键作用，因为许多相关问题涉及最大化或最小化比率项。值得注意的是，FP的最大化情形与最小化情形通常无法相互转换，因此在以往大多数研究中必须分别处理这两类问题。现有的分数比率最大化方法通常无法直接适用于最小化情形，反之亦然。然而，FP目标函数可能呈现混合最大-最小形式，例如，我们可能希望最大化合法接收器的信干噪比(SINR)，同时最小化窃听者的SINR。本文旨在通过构建统一优化框架来填补最大分数规划(max-FP)与最小分数规划(min-FP)之间的空白。主要研究成果包含三方面：首先，将现有的max-FP技术（即二次变换）扩展至min-FP，并进一步发展为混合情形的完全泛化形式；其次，提出所提统一方法的极小化-最大化(MM)解释，从而确立其收敛性并实现矩阵扩展，同时推导出广义拉格朗日对偶变换以辅助求解对数型分数规划；最后，给出三个典型应用实例：信息年龄(AoI)最小化、感知克拉美罗界最小化以及安全数据速率最大化，这些应用均无法通过先前的FP方法高效求解。