Slip variable

Slack variables ( Engl. Slack variable ), and surplus variables mentioned are mathematical variables that are introduced to a problem to solve, but their value is not of interest. The additional slack variables are intended to reduce a problem to a simpler problem.

Applications

In linear optimization , slip variables are introduced to convert inequality constraints into equation constraints. This is based on the idea that the inequality is satisfied when the equation holds for any number . ${\displaystyle x<b}$ ${\displaystyle x=b-\chi }$ ${\displaystyle \chi >0}$

Lagrange multipliers are used to convert optimization problems with constraints into optimization problems without constraints.

In support vector machines , slip variables form so-called error terms , that is, they allow wrong decisions, but punish them at the same time.

example

Consider the system of inequalities

${\displaystyle {\begin{aligned}2x_{1}&+x_{2}&\leq 6\\7x_{1}&-3x_{2}&\leq 8\end{aligned}}}$

We introduce the slack variables to convert the inequalities into equations. Then follows ${\displaystyle x_{1}^{S},x_{2}^{S}\geq 0}$

${\displaystyle {\begin{aligned}2x_{1}&+x_{2}&+x_{1}^{S}&&=6\\7x_{1}&-3x_{2}&&+x_{2}^{S}&=8\end{aligned}}}$

The following matrix notation is often found in linear optimization:

${\displaystyle {\begin{pmatrix}2&1&1&0\\7&-3&0&1\end{pmatrix}}\cdot {\begin{pmatrix}x_{1}\\x_{2}\\x_{1}^{S}\\x_{2}^{S}\end{pmatrix}}={\begin{pmatrix}6\\8\end{pmatrix}}}$

The introduction of the slack variable leads to the addition of an identity matrix of suitable dimensions behind the coefficient matrix .

literature

• Hochstättler Winfried: Algorithmische Mathematik, Springer Berlin Heidelberg, 2010. ISBN 978-3-642-05421-1 , 202-203
• Peter Knabner , Wolf Barth : Lineare Algebra . Basics and applications (=  Springer textbook ). Springer, Berlin 2012, ISBN 978-3-642-32185-6 .