Optimize concave function divided by a linear function

Question

If I divide a monoton concave upper and lower bounded function by a linear function, whould the result be concave? Or even easy to optimize?

Answer 1

In general, the result would not necessarily be concave. For instance, $x^{1/2}$ is concave, but $x^{1/2}/x$ is convex.

However, if the numerator is a concave quadratic and the denominator is a nonnegative linear (affine) function over its domain, the result is an easy to maximize concave function. If the numerator is a concave quadratic and the denominator is a nonpositive linear (affine) function over its domain, the result is an easy to minimize convex function. See the last paragraph of Example 3.4 on p. 76 of "Convex Optimization" by Boyd and Vandenberghe. If using CVX, the function quad_over_lin can be used to implement such cases.

Boundedness of the numerator is irrelevant to whether the quotient is concave.