Can the Dinkelbach method solve nonlinear fractional programming problems, where the functions in the numerator and denominator are not necessarily quadratic and not convex either?
If not is there a variant or a hybrid version of Dinkelbach's method that caters to these cases?
Denote the problem as $$\min_{x \in X} \frac{f(x)}{g(x)}$$ and define: $$h(\lambda) = \min_{x \in X} \{ f(x)-\lambda g(x) \}$$ When $g(x) \geq 0$ for all $x$ in $X$, it is still imminent that $h(\lambda) \leq 0$ if and only if $\min_{x \in X} \frac{f(x)}{g(x)} \leq \lambda$. The problem now is to find the smallest $\lambda$ such that $h(\lambda) \leq 0$.
Without concavity of $g$, the subproblem for computing $h(\lambda)$ is not convex, so it may be hard to compute $h(\lambda)$. However, the function $h$, being the minimum of functions that are affine in $\lambda$, is still concave. Therefore, it is still straightforward to find the optimal $\lambda$ with, e.g., bisection search.