# Local minima for the quadratic penalty method

Let $$x\in\mathbb{R^n}$$ be a strict local minima for the problem

\begin{equation*} \begin{aligned} & \underset{x}{\text{minimize}} && f(x)\\ & \text{subject to} && h(x)=0 \end{aligned} \end{equation*}

Consider now the quadratic penality method

\begin{equation*} \begin{aligned} & \underset{x}{\text{minimize}} && P(x,C)=f(x)+\frac{C}{2}h(x)^Th(x)\\ \end{aligned} \end{equation*}

Is there a $$C^*\geq 0$$ such that x* is a local minima of $$P(x,C)$$ $$\forall C\geq C^*$$?

Now, since $$x^*$$ is an strict minima of the first problem,

1. $$\nabla f(x^*)+\lambda^Th(x^*)=0$$, and
2. $$h(x^*)=0$$

but I can't seem to figure out how to use this information to prove that there is a $$C^*$$ such that $$x^*$$ is a local minima of $$P(x,C)$$, $$C\geq C^*$$. I know that if $$x^*$$ is a local minima, then the first order conditions must be satisfied.

Is the statement even true?