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

In this paper, a kind of non regular constraints and a principle for seeking critical point under the constraint are presented, where no Lagrange multiplier is involved. Let $E, F$ be two Banach spaces, $g: E\rightarrow F$ a $c^1$ map defined 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 constraint and 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 a concerned focus in optimization theory. The principle also suits the case of regular constraint. Coordinately, the generalized regular constraint is introduced, and the critical point principle on generalized regular constraint is established. Let $f: U \rightarrow \mathbb{R}$ be a nonlinear functional. While the Lagrange multiplier $L$ in classical critical point principle is considered, 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 a regular constraint, then the Lagrange multiplier $L$ is unique; otherwise, $L$ is ill-posed. Hence, in case of the non regular constraint, it is very difficult to solve Euler equations, however, it is often the case in optimization theory. So the principle here seems to be new and applicable. By the 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 smooth diffeomorphic to certain Banach space. This is a new and interesting result in generalized inverse analysis.