Fractional optimization

This notebook shows how to solve a simple concave fractional problem, in which the objective is to maximize the ratio of a nonnegative concave function and a positive convex function. Concave fractional problems are quasiconvex programs (QCPs). They can be specified using disciplined quasiconvex programming (DQCP), and hence can be solved using CVXPY.

!pip install --upgrade cvxpy
import cvxpy as cp
import numpy as np
import matplotlib.pyplot as plt

Our goal is to minimize the function


This function is not concave, but it is quasiconcave, as can be seen by inspecting its graph.

plt.plot([np.sqrt(y) / np.exp(y) for y in np.linspace(0, 10)])

The below code specifies and solves the QCP, using DQCP. The concave fraction function is DQCP-compliant, because the ratio atom is quasiconcave (actually, quasilinear), increasing in the numerator when the denominator is positive, and decreasing in the denominator when the numerator is nonnegative.

x = cp.Variable()
concave_fractional_fn = cp.sqrt(x) / cp.exp(x)
problem = cp.Problem(cp.Maximize(concave_fractional_fn))
assert problem.is_dqcp()