Suppose I have the following optimization problem

$$ \min\limits_{\mathbf{x},\mathbf{y}} f(\mathbf{x},\mathbf{y}) \tag{1} $$

It is already known that the target function $f$ is continuous and differentiable, which has a **unique** stationary point, and this stationary point is also a global minimum. However, $f$ is **not** convex. Here, I guess that $f$ 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 $ \mathbf x=g(\mathbf y) $.

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 $f$ since $f$ is strictly quasi-convex.

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 $(1)$?

$$ \min\limits_{\mathbf{y}} f(g(\mathbf y),\mathbf{y}) $$

Any helpful comments are appreciated! ^_^