Sign Restricted variables
Sign-Restricted Variables
Standard form requires every variable to be nonnegative.
If a variable has a different sign restriction, it must be transformed.
Standard-Form Requirement
In standard form:
\[x\ge 0\]This means:
\[x_j\ge 0\]for every variable $j$.
Nonpositive Variables
If a variable satisfies:
\[x_j\le 0\]replace it by:
\[x_j=-y_j\]where:
\[y_j\ge 0\]Now the new variable $y_j$ satisfies the standard-form sign rule.
Example
Suppose:
\[x_1\le 0\]and the constraint is:
\[2x_1+x_2\le 5\]Replace:
\[x_1=-y_1, \qquad y_1\ge 0\]Then:
\[2(-y_1)+x_2\le 5\]so:
\[-2y_1+x_2\le 5\]Objective Example
If the objective is:
\[\min 3x_1+4x_2\]and:
\[x_1\le 0\]then after $x_1=-y_1$:
\[\min -3y_1+4x_2\]with:
\[y_1,x_2\ge 0\]Variables With Lower Bounds
If:
\[x_j\ge L\]where $L$ is a constant, define:
\[y_j=x_j-L\]so:
\[y_j\ge 0\]and:
\[x_j=y_j+L\]Then substitute $y_j+L$ everywhere.
Variables With Upper Bounds
If:
\[x_j\le U\]one method is to define:
\[y_j=U-x_j\]so:
\[y_j\ge 0\]and:
\[x_j=U-y_j\]Then substitute $U-y_j$ everywhere.
Free Variables
If a variable has no sign restriction:
\[x_j\in\mathbb{R}\]replace it by:
\[x_j=x_j^+-x_j^-\]where:
\[x_j^+,x_j^-\ge 0\]Summary Table
| Original restriction | Replacement | New restriction |
|---|---|---|
| $x_j\ge 0$ | keep $x_j$ | $x_j\ge 0$ |
| $x_j\le 0$ | $x_j=-y_j$ | $y_j\ge 0$ |
| $x_j$ free | $x_j=x_j^+-x_j^-$ | $x_j^+,x_j^-\ge 0$ |
| $x_j\ge L$ | $x_j=y_j+L$ | $y_j\ge 0$ |
| $x_j\le U$ | $x_j=U-y_j$ | $y_j\ge 0$ |
Common Mistake
When substituting a transformed variable, update every occurrence:
- objective function
- every constraint
- sign restrictions
Changing it in only one place creates a different problem.
Checklist
You understand sign-restricted variables if you can:
- identify variables not already nonnegative
- transform $x_j\le 0$ using $x_j=-y_j$
- transform free variables using two nonnegative variables
- handle lower and upper bounds by shifting variables
- substitute consistently throughout the whole model
See Also
Exam checkpoint
For standard-form questions, use the course convention $\min c^Tx$ subject to $Ax=b$, $x\ge0$. Convert max objectives, add slack to $\le$, subtract surplus from $\ge$, and split free variables.