# Qualification Conditions in Semi-algebraic Programming

## TL;DR

This paper investigates qualification conditions in semi-algebraic programming, showing that most positive diagonal perturbations lead to regularity, and provides bounds on singular perturbations using Milnor-Thom theorem, with applications to optimization methods.

## Contribution

It introduces a perturbation approach ensuring regularity in semi-algebraic problems and bounds the number of singular perturbations using algebraic topology techniques.

## Key findings

- Most positive diagonal perturbations satisfy constraint qualification.
- Bound on singular perturbations is exponential, relevant for polynomial constraints.
- Perturbation method aids in approximating complex semi-algebraic problems.

## Abstract

For an arbitrary finite family of semi-algebraic/definable functions, we consider the corresponding inequality constraint set and we study qualification conditions for perturbations of this set. In particular we prove that all positive diagonal perturbations, save perhaps a finite number of them, ensure that any point within the feasible set satisfies Mangasarian-Fromovitz constraint qualification. Using the Milnor-Thom theorem, we provide a bound for the number of singular perturbations when the constraints are polynomial functions. Examples show that the order of magnitude of our exponential bound is relevant. Our perturbation approach provides a simple protocol to build sequences of "regular" problems approximating an arbitrary semi-algebraic/definable problem. Applications to sequential quadratic programming methods and sum of squares relaxation are provided.