Karush–Kuhn–Tucker conditions

In mathematical optimization, the Karush–Kuhn–Tucker (KKT) conditions, also known as the Kuhn–Tucker conditions, are first-order necessary conditions for a solution in nonlinear programming to be optimal, provided that some regularity conditions are satisfied. Allowing inequality constraints, the KKT approach to nonlinear programming generalizes the method of Lagrange multipliers, which allows only equality constraints. The system of equations corresponding to the KKT conditions is usually not solved directly, except in the few special cases where a closed-form solution can be derived analytically. In general, many optimization algorithms can be interpreted as methods for numerically solving the KKT system of equations.

The KKT conditions were originally named after Harold W. Kuhn, and Albert W. Tucker, who first published the conditions in 1951. Later scholars discovered that the necessary conditions for this problem had been stated by William Karush in his master's thesis in 1939.

Consider the following nonlinear optimization problem:

where x is the optimization variable, ${\displaystyle f}$ is the objective or utility function, ${\displaystyle g_{i}\ (i=1,\ldots ,m)}$ are the inequality constraint functions, and ${\displaystyle h_{j}\ (j=1,\ldots ,\ell )}$ are the equality constraint functions. The numbers of inequality and equality constraints are denoted m and ℓ, respectively.

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