Qualification Conditions in Semialgebraic Programming
研究半代数/可定义函数不等式约束系统的扰动，证明除有限个正对角扰动外，可行集内所有点满足Mangasarian-Fromovitz约束规范，并利用Milnor-Thom定理给出多项式约束下奇异扰动数量的指数级上界，应用于序列二次规划和平方和松弛方法。
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.