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 Nonlinear Optimization Examples

## Kuhn-Tucker Conditions

The nonlinear programming (NLP) problem with one objective function f and m constraint functions ci, which are continuously differentiable, is defined as follows:
In the preceding notation, n is the dimension of the function f(x), and me 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 f and ci are twice differentiable, the point x* 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 that the constraint functions ci(x*) will vanish within machine precision, and determining the set of active constraints at the solution x* may not be simple.

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