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.

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! ^_^!

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) $.

1. 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.

2. 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! ^_^