structural:summarization, information:highlighting, information:classification, stylistic:tone, structural:scholar_edition
Necessary conditions

Suppose that the objective function f: R n → R {\displaystyle f\colon \mathbb {R} ^{n}\rightarrow \mathbb {R} } and the constraint functions g i: R n → R {\displaystyle g_{i}\colon \mathbb {R} ^{n}\rightarrow \mathbb {R} } and h j: R n → R {\displaystyle h_{j}\colon \mathbb {R} ^{n}\rightarrow \mathbb {R} } have subderivatives at a point x ∗ ∈ R n {\displaystyle x^{*}\in \mathbb {R} ^{n}}. If x ∗ {\displaystyle x^{*}} is a local optimum and the optimization problem satisfies some regularity conditions (see below), then there exist constants μ i (i = 1, …, m) {\displaystyle \mu _{i}\ (i=1,\ldots,m)} and λ j (j = 1, …, ℓ) {\displaystyle \lambda _{j}\ (j=1,\ldots,\ell)}, called KKT multipliers, such that the following four groups of conditions hold:

The last condition is sometimes written in the equivalent form: μ i g i (x ∗) = 0, for i = 1, …, m. {\displaystyle \mu _{i}g_{i}(x^{*})=0,{\text{ for }}i=1,\ldots,m.}

In the particular case m = 0 {\displaystyle m=0}, i.e., when there are no inequality constraints, the KKT conditions turn into the Lagrange conditions, and the KKT multipliers are called Lagrange multipliers.