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?


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