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In this paper, a kind of non regular constraints and a principle for seekingcritical point under the constraint are presented, where no Lagrange multiplieris involved. Let $E, F$ be two Banach spaces, $g: E\rightarrow F$ a $c^1$ mapdefined on an open set $U$ in $E,$ and the constraint $S=$ the preimage$g^{-1}(y_0), y_0\in F.$ A main deference between the non regular constraintand regular constraint is that $g'(x)$ at any $x\in S$ is not surjective.Recently, the critical point theory under the non regular constraint is aconcerned focus in optimization theory. The principle also suits the case ofregular constraint. Coordinately, the generalized regular constraint isintroduced, and the critical point principle on generalized regular constraintis established. Let $f: U \rightarrow \mathbb{R}$ be a nonlinear functional.While the Lagrange multiplier $L$ in classical critical point principle isconsidered, and its expression is given by using generalized inverse${g'}^+(x)$ of $g'(x)$ as follows : if $x\in S$ is a critical point of $f|_S,$then $L=f'(x)\circ {g'}^+(x) \in F^*.$ Moreover, it is proved that if $S$ is aregular constraint, then the Lagrange multiplier $L$ is unique; otherwise, $L$is ill-posed. Hence, in case of the non regular constraint, it is verydifficult to solve Euler equations, however, it is often the case inoptimization theory. So the principle here seems to be new and applicable. Bythe way, the following theorem is proved; if $A\in B(E,F)$ is double split,then the set of all generalized inverses of $A,$ $GI(A)$ is smoothdiffeomorphic to certain Banach space. This is a new and interesting result ingeneralized inverse analysis.

A Principle for Critical Point under Generalized Regular Constraint and Ill- Posed Lagrange Multipliers under Non-Regular Constraints