# Qualification Conditions in Semi-algebraic Programming

arXiv: https://arxiv.org/abs/1705.08219 · HTML: https://mlsanity.com/paper/1705.08219/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 May 2017.

## Built on by (most cited first, 1 of 1)

- [Accelerated First-Order Optimization under Nonlinear Constraints](https://mlsanity.com/paper/2302.00316/) (2023)

## Abstract (arXiv)

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.