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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 \u201cregular\u201d problems approximating an arbitrary semialgebraic\/definable problem. Applications to sequential quadratic programming methods and sum of squares relaxation are provided.<\/jats:p>","DOI":"10.1137\/16m1133889","type":"journal-article","created":{"date-parts":[[2018,6,28]],"date-time":"2018-06-28T12:25:53Z","timestamp":1530188753000},"page":"1867-1891","source":"Crossref","is-referenced-by-count":6,"title":["Qualification Conditions in Semialgebraic Programming"],"prefix":"10.1137","volume":"28","author":[{"given":"J\u00e9r\u00f4me","family":"Bolte","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Antoine","family":"Hochart","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Edouard","family":"Pauwels","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,6,28]]},"reference":[{"key":"atypb1","unstructured":"M. Abril Bucero and B. 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O'Shea,\n                      Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra\n                      , Undergrad. Texts Math., 4th ed., Springer, New York, 2015,https:\/\/doi.org\/10.1007\/978-3-319-16721-3.","DOI":"10.1007\/978-3-319-16721-3"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1112\/jlms\/jdq084"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1016\/j.jpaa.2006.05.028"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-005-0672-6"},{"key":"atypb21","first-page":"281","volume":"13","author":"Dontchev A. L.","year":"2006","journal-title":"J. Convex Anal."},{"key":"atypb22","doi-asserted-by":"crossref","unstructured":"A. L. Dontchev and R. T. Rockafellar,\n                      Implicit Functions and Solution Mappings\n                      , Springer Ser. Oper. Res. Financ. 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