Standard Form Common Exam Conversions
Standard Form Common Exam Conversions
Conversion table
| Situation | Do this |
|---|---|
| Maximization | multiply objective by $-1$ and minimize |
| $\le$ constraint | add slack variable |
| $\ge$ constraint | subtract surplus variable |
| equality constraint | keep as equality |
| nonpositive variable $x_j\le0$ | set $x_j=-y_j$, $y_j\ge0$ |
| free variable | set $x_j=x_j^+-x_j^-$ |
| negative RHS in inequality | often multiply row by $-1$ before adding slack/surplus |
Signs to memorize
Slack:
\[a^Tx\le b \iff a^Tx+s=b, \qquad s\ge0.\]Surplus:
\[a^Tx\ge b \iff a^Tx-s=b, \qquad s\ge0.\]Free variable:
\[x=x^+-x^-.\]Nonpositive variable:
\[x\le0 \iff x=-y,\; y\ge0.\]Exam correction
Never write only “add slack variables” generically. The sign depends on the inequality direction.
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.