The Fritz John conditions (abbr. FJ conditions), in mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, are a necessary condition

Necessary and sufficient conditions

In logic, the words necessity and sufficiency refer to the implicational relationships between statements. The assertion that one statement is a necessary and sufficient condition of another means that the former statement is true if and only if the latter is true.-Definitions:A necessary condition...

 for a solution in nonlinear programming

Nonlinear programming

In mathematics, nonlinear programming is the process of solving a system of equalities and inequalities, collectively termed constraints, over a set of unknown real variables, along with an objective function to be maximized or minimized, where some of the constraints or the objective function are...

 to be optimal

Optimization (mathematics)

In mathematics, computational science, or management science, mathematical optimization refers to the selection of a best element from some set of available alternatives....

. They are used as lemma in the proof of the Karush–Kuhn–Tucker conditions. We consider the following optimization problem

Optimization problem

In mathematics and computer science, an optimization problem is the problem of finding the best solution from all feasible solutions. Optimization problems can be divided into two categories depending on whether the variables are continuous or discrete. An optimization problem with discrete...

: where ƒ is the function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 to be minimized, $g\_i$ the inequality constraints

Constraint (mathematics)

In mathematics, a constraint is a condition that a solution to an optimization problem must satisfy. There are two types of constraints: equality constraints and inequality constraints...

 and $h\_j$ the equality constraints, and where, respectively, $\backslash mathcal\{I\}$, $\backslash mathcal\{I\text{'}\}$ and $\backslash mathcal\{E\}$ are the indices

Index (mathematics)

The word index is used in variety of senses in mathematics.- General :* In perhaps the most frequent sense, an index is a number or other symbol that indicates the location of a variable in a list or array of numbers or other mathematical objects. This type of index is usually written as a...

 set of inactive, active and equality constraints and $x^*$ is a optimal solution of $f$, then there exists a non-zero number $\backslash lambda\; \_0$ and a non-zero vector $\backslash lambda=[\backslash lambda\; \_1,\; \backslash lambda\; \_2,\backslash dots,\backslash lambda\; \_n]$ such that: NEWLINE

NEWLINE

$\backslash begin\{cases\}$

NEWLINE \lambda_0 \nabla f(x^*) = \sum\limits_{i\in \mathcal{I}'} \lambda_i \nabla g_i(x^*) + \sum\limits_{i\in \mathcal{E}} \lambda_i \nabla h_i (x^*) \\[10pt] \lambda_i \ge 0,\ i\in \mathcal{I}' \\[10pt] \exists i\in \left( \{0,1,\ldots ,n\} \backslash \mathcal{I} \right) \left( \lambda_i \ne 0 \right) \end{cases} $\backslash lambda\_0=0$ iff

IFF

IFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radio-based identification system using transponders...

 the $\backslash nabla\; g\_i\; (i\backslash in\backslash mathcal\{I\}\text{'})$ and $\backslash nabla\; h\_i\; (i\backslash in\backslash mathcal\{E\})$ are linearly dependent and $\backslash lambda\_i\backslash neq\; 0,\backslash ,\backslash forall\; i\backslash in\backslash mathcal\{I\}\text{'}\backslash cup\backslash mathcal\{E\}$, i.e. if the constraint qualifications do not hold. Named after Fritz John

Fritz John

Fritz John was a German-born mathematician specialising in partial differential equations and ill-posed problems. His early work was on the Radon transform and he is remembered for John's equation.-Biography:...

, these conditions are equivalent to the Karush–Kuhn–Tucker conditions in the case $\backslash lambda\_0\; =\; 1$.