Degenerate problems of nonlinear analysis and extremum theory

Consider a system of nonlinear equations $\mathbf F(x)=y$, where $\mathbf F$ is a smooth map of some Banach space $X$ into another Banach space (for simplicity, these spaces can be regarded as finite-dimensional). If the point $x_0$ is degenerate, that is, the linear operator $\mathbf F'(x_0)$ is not surjective (e.g., $\mathbf F'(x_0)=0$), then, at the point $x_0$, the classical inverse function theorem can not be applied. Herein, we present theorems on inverse and implicit functions which can be applied to degenerate points as well.
Consider the classical extremal problem with constraints:
$$ \varphi(x)\to\min,\qquad f_i(x)=0,\quad i=1,2,\dots,k,\qquad x\in X. $$

Here, the smooth functions fi define the constraints, and $\varphi$ is the minimizing functional. Let $x_0$ be a local minimum. If the point $x_0$ is degenerate (abnormal), that is, the gradients $f_i'(x_0)$ are linearly dependent, then the Lagrange principle degenerates (i.e., it becomes non-informative), and the classical second order necessary optimality conditions do not hold true. We present a theory of necessary conditions of the first and second order which is equally meaningful for degenerate and non-degenerate problems. These results are a further development of the Lagrange principle.
A classic example of such abnormal problem: would a given quadratic form be non-negative (or, would it be zero) at the intersection of quadrics? The present theory allows us to give answers to these questions.
All the outlined in the report results are meaningful in the finite-dimensional case as well (even when $X$ is 3-dimensional).

References

1. A. V. Arutyunov, “Smooth abnormal problems in extremum theory and analysis”, Russian Math. Surveys, 67:3 (2012), 403–457