I have a constrained optimization problem,
$$
\min_x f(x) \quad\mathrm{s.t.}\quad g(x)\leq0.
$$
The feasible region of this optimization problem is a convex set. I can prove that the optimal solution must lie on the boundary of the feasible region, i.e., $g(x)=0$. Therefore, we can reformulate the problem equivalently as
$$
\min_x f(x) \quad\mathrm{s.t.}\quad g(x)=0.
$$
However, its feasible region is not convex.
My question is which formulation I should use in practice. If it matters, I intend to perform the optimization using first-order methods.
Best Answer
Generally speaking, you want to exploit convexity whenever you have the opportunity. I expect that an interior-point method applied to your inequality-constrained problem would work quite well. Or if by "first-order method" you mean that you want to use a variant of gradient descent, step along the gradient until you reach the boundary, and then flow along the boundary (using projected gradient descent, e.g.).
By contrast, trying to solve the non-convex equality-constrained problem can result in getting stuck at some local minimum instead, etc.