The nonlinear programming (NLP) problem with one objective function and constraint functions , which are continuously differentiable, is defined as follows:

In the preceding notation, is the dimension of the function , and is the number of equality constraints. The linear combination of objective and constraint functions

is the Lagrange function, and the coefficients are the Lagrange multipliers.

If the functions and are twice differentiable, the point is an isolated local minimizer of the NLP problem, if there exists a vector that meets the following conditions:

• Kuhn-Tucker conditions

• second-order condition

  Each nonzero vector with

  satisfies

In practice, you cannot expect the constraint functions to vanish within machine precision, and determining the set of active constraints at the solution might not be simple.