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arXiv · 1705.08219

Qualification Conditions in Semi-algebraic Programming

Jérôme Bolte·Antoine Hochart·Edouard Pauwels
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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.

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Jérôme Bolte, Antoine Hochart, Edouard Pauwels. 2017-05-23. Qualification Conditions in Semi-algebraic Programming. https://arxiv.org/abs/1705.08219

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