I want to add $f(\vec{x})=\frac{ \vec{x}^{\top}\vec{b}} {\vec{x}^{\top}\vec{c}}$ to an optimization problem as an additional objective, such that: $$\min_x g(\vec{x})+f(\vec{x}) \\ s.t ~~x_i \ge 0$$
in which $g(x)$ is convex, and $\vec{b}$ and $\vec{c}$ are positive vectors. I tried to checked the hessian of $f(x)$ for its convexity, and apparently depending on $\vec{x}$ it may be convex or concave (generally non-convex).
So when i use a convex solver (matlab fmincon), the result varies depending on the initial point. However when i use another constraint such as $\sum{x_i}=1$ or $\vec{x}^{\top}\vec{c}=1$ in combination with $x_i \geq 0$ it always converges to a certain answer $\vec{x}^*$ regardless of initial $x$.
So my question is, do those constraints make the optimization problem convex, although $f(x)$ is not convex? And if yes, can we optimize it using typical constrained convex optimization methods, for instance based on gradient-descent?