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From the book of bertesekas(1999),borwein(2006), we learned some constraint qualifications on R^n spaces,such as:

Linear independence constraint qualification(1951) Mangasarian–Fromovitz constraint qualification(1967) , Constant rank constraint qualification , Constant positive linear dependence constraint qualification , Quasi-normality constraint qualification

when it is in Infinite-dimensional optimization, Question: How to get constraint qualifications conditions for optimization on banach spaces

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You must look at the study on Lagrange multipliers in Banach spaces in the book of Clarke : Optimization and Nonsmooth Analysis where he defines the calmness constraint qualification. For example in convex optimization the Slater condition can be defined even in Banach spaces and we can derive the Karush-Kuhn-Tucker conditions. For example one can define even an Abadie type constraint qualification in Banach spaces. See the book titled : Techniques of variational analysis by Borwein and Zhu.

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