Consider the following optimization problem
$$\begin{array}{ll} \text{minimize} & f(x)\\ \text{subject to} & g(x,y) \leq 0\end{array}$$
where the objective function $f$ is linear. When I fix $y=y_0$, the region $\{(x,y_0)\in(\mathbb{R^n,R}) \mid g(x,y_0)\leq 0\}$ is convex, but $\{(x,y)\in(\mathbb{R^n,R}) \mid g(x,y) \leq 0\}$ is not convex.
Is there any algorithm to find the global minimum?
PS: Using a local minimum optimization algorithm(SQP), I find the outcome is always equal to the global minimum. I guess that the objective function $f(x)$ only has one local minimum, but I dont't know how to prove it.