Suppose I have the following optimization problem:
\begin{equation}
\min\limits_{\mathbf{x},\mathbf{y}} f(\mathbf{x},\mathbf{y})    \qquad \qquad \qquad (1)
\end{equation}

It is already known that the target function $f(\mathbf{x},\mathbf{y})$ is continuous and differentiable, which has a **unique** stationary point, and this stationary point is also a global minimum. 
But $f(\mathbf{x},\mathbf{y})$ is **not convex.** 

**Here I guess that $f(\mathbf{x},\mathbf{y})$ is a strict quasi-convex function, and my question is based on this.**

From the KKT condition, I find that the necessary condition for the optimal point is:
\begin{equation}
\mathbf x=g(\mathbf y)
\end{equation}


**The first question is:** 

Is my guess about this function correctly (just based on the properties discussed above) ?

If it is, then, we can use convex optimization algorithms to find a global optimal point of $f(\mathbf{x},\mathbf{y})$ since $f(\mathbf{x},\mathbf{y})$ is strict quasi-convex, [reference](https://www.sciencedirect.com/science/article/pii/0022247X67900959).

**Another question is:**

Can we substitute $\mathbf x=g(\mathbf y)$ into the target function and solve the consequent optimization problem. 

In words, is the solution to the following optimization problem equivalent to the original optimization problem in $(1)$? 

\begin{equation}
\min\limits_{\mathbf{y}} f(g(\mathbf y),\mathbf{y})
\end{equation}

Any helpful comments are appreciated! ^_^!