For an arbitrary finite family of semialgebraic/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 the 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 semialgebraic/definable problem. Applications to sequential quadratic programming methods and sum of squares relaxation are provided.


  1. constraint qualification
  2. Mangasarian--Fromovitz
  3. Arrow--Hurwicz--Uzawa
  4. Lagrange multipliers
  5. optimality conditions
  6. tame programming

MSC codes

  1. Primary
  2. 26D10; Secondary
  3. 32B20
  4. 49K24
  5. 49J52
  6. 37B35
  7. 14P15

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Information & Authors


Published In

cover image SIAM Journal on Optimization
SIAM Journal on Optimization
Pages: 1867 - 1891
ISSN (online): 1095-7189


Submitted: 9 June 2017
Accepted: 26 February 2018
Published online: 28 June 2018


  1. constraint qualification
  2. Mangasarian--Fromovitz
  3. Arrow--Hurwicz--Uzawa
  4. Lagrange multipliers
  5. optimality conditions
  6. tame programming

MSC codes

  1. Primary
  2. 26D10; Secondary
  3. 32B20
  4. 49K24
  5. 49J52
  6. 37B35
  7. 14P15



Funding Information


Funding Information

Air Force Office of Scientific Research https://doi.org/10.13039/100000181 : FA9550-15-1-0500

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