Constraint qualification

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Conditions on the constraint functions (LaTeX: g and LaTeX: h) that are sufficient to make the Lagrange Multiplier Rule valid. Here is an example to illustrate what can go wrong:

LaTeX: \max x : x^2 \le 0.

Since LaTeX: x=0 is the only feasible solution, it is optimal. The Lagrange Multiplier Rule requires that there exist LaTeX: u for which LaTeX: f' - u^T g' = 0, but LaTeX: f' = 1 and LaTeX: g'= 0, so no such LaTeX: u exists. Slater used this example in illustrating his interiority condition. The classical qualification, given by Lagrange's multiplier theorem without inequality constraints, is that LaTeX: \nabla h(x) have full row rank, which stems from the Implicit Function Theorem. Another constraint qualification is that all constraint functions be affine (even with redundant constraints). Each constraint qualification gives a sufficient condition for the Lagrange Multiplier Theorem to be valid. A constraint qualification is necessary if it must hold in order to guarantee that the Lagrange Multiplier Rule is valid for all smooth LaTeX: f having optimal value at LaTeX: x.

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