Consider following nonlinear optimization \begin{align} P(y):\min_{x}\ & g(x,y),\\ \text{s.t.}\ \ & f(x,y) = 0, \end{align} where $y$ is given. Suppose $P(y^*)$ satisfies second-order sufficient condition at $x^*$, let's denote its Lagrange multiplier by $\lambda^*$. Further, we assume the reduced Hessian is always positive definite at $(x,\lambda)$ such that $\|(x,\lambda) - (x^*,\lambda^*)\|\leq C$ for some $C$ (note $\lambda$ may not be the multiplier associated to $x$). Then it seems to me when we move $y^*$ a little bit, say to $y^0$, $P(y^0)$ still has local minimizer near $x^*$ and still satisfies SOSC at that minimizer.
Am I correct? If so, how to quantitatively bound how far we can move $y$. Any other reasonable assumption if needed is fine.