# On tangent cone to systems of inequalities and equations in Banach spaces under relaxed constant rank condition

## TL;DR

This paper proves that under the relaxed constant rank condition, the linearized cone is contained in the tangent cone for systems of inequalities and equations in Banach spaces, extending classical results.

## Contribution

It establishes the Abadie condition under the relaxed constant rank condition for Banach space systems, broadening the scope of tangent cone analysis.

## Key findings

- Linearized cone is contained in the tangent cone under the relaxed constant rank condition.
- Extension of classical tangent cone results to Banach spaces.
- Applicable to systems of inequalities and equations with differentiable functions.

## Abstract

Under the relaxed constant rank condition, introduced by Minchenko and Stakhovski, we prove that the linearized cone is contained in the tangent cone (Abadie condition) for sets represented as solution sets to systems of finite numbers of inequalities and equations given by continuously differentiable functions defined on Banach spaces.