I'm confused about a theorem located at page 444 of "Linear and nonlinear programming" by Luenberger D. and Ye Y. I report it here.
Local Duality Theorem. Suppose that the problem $$\begin{matrix}\text{min }f(x)\\\text{s.t. }h(x)=0\end{matrix}$$ (where f, h are nonlinear) has a local solution at $x^\star$ with corresponding value $r^\star$ and Lagrange multiplier $\lambda^\star$. Suppose also that $x^\star$ is a regular point of the constraints and that the corresponding Hessian of the Lagrangian $\nabla_{xx}^2L(x^\star,\lambda^\star)$ is positive definite. Then the dual problm $$\text{max }\phi(\lambda)$$has a local solution at $\lambda^\star$ with corresponding value $r^\star$ and $x^\star$ as the point corresponding to $\lambda^\star$ in the definition of $\phi$.
How is it possible that $$\text{max } \phi(\lambda)=r^\star =\text{min } f(x)?$$ Wouldn't this statement imply strong duality and that $x^\star$, $\lambda^\star$ are global optima, which would be clearly false for nonlinear problems? Am I misenterpreting this theorem? Thank you!