  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.
