  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.
