In some texts I have seen arguments for the fact that the Lagrange multipliers of a constrained optimization problem remain bounded. Are there general conditions for that fact?
In particular,
Let a mathematical program: \begin{align} &\mathrm{min}~ f(x)\\ &\mathrm{s.t.}~ g_i(x) \leq 0, ~\forall i\in I\\ &\mathrm{~~~~~~}~ h_j(x) \leq 0, ~\forall j\in J. \end{align}
The lagrangian function of the constrained optimization problem is the following: \begin{equation} \mathcal{L}(x, \lambda, \mu) = f(x) + \langle \lambda, g(x) \rangle + \langle \mu, h(x) \rangle. \end{equation}
Under which conditions do we know that $\| \lambda \|_2 < \infty$?