# Does the Lagrange multiplier method always give a saddle point problem?

I am considering the following problem: $$\min_u J(u)\text{, s.t. } \, H(u)=0.$$ Use Lagrange multiplier method, then $L(u, \lambda)=J + \lambda H(u)$. Does the critical point of the $L$ always correspond to a saddle point of $L$? If it does, how to show it? If it is not always true, how about a convex, or even quadratic $J(u)$? How to show it?

Thanks.

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