Fractional programming (FP) is a branch of mathematical optimization that deals with the optimization of ratios. It is an invaluable tool for signal processing and machine learning, because many key metrics in these fields are fractionally structured, e.g., the signal-to-interference-plus-noise ratio (SINR) in wireless communications, the Cramér-Rao bound (CRB) in radar sensing, the normalized cut in graph clustering, and the margin in support vector machine (SVM). This article provides a comprehensive review of both the theory and applications of a recently developed FP technique known as the quadratic transform, which can be applied to a wide variety of FP problems, including both the minimization and the maximization of the sum of functions of ratios as well as matrix-ratio problems.
