Standard Form Worked Examples
Standard Form Worked Examples
Course convention
\[\min c^Tx\quad \text{s.t. } Ax=b,\;x\ge0.\]Example 1 — simple max to min
Original:
\[\max 2x_1+x_2\]subject to
\[x_1+2x_2\le2,\] \[-x_1+2x_2\le2,\] \[x_2\ge0,\]with $x_1$ free.
Set $x_1=x_1^+-x_1^-$.
Standard form:
\[\min -2x_1^+ +2x_1^- -x_2\]subject to
\[x_1^+-x_1^-+2x_2+s_1=2,\] \[-x_1^+ +x_1^-+2x_2+s_2=2,\] \[x_1^+,x_1^-,x_2,s_1,s_2\ge0.\]Example 2 — mixed signs and negative RHS
Original:
\[\min -x_1-x_2\]subject to
\[-x_1+x_2\le2,\] \[x_1+x_2\ge1,\] \[x_1\le1,\] \[x_1,x_2\ge0.\]Standard form:
\[\min -x_1-x_2\]subject to
\[-x_1+x_2+s_1=2,\] \[x_1+x_2-s_2=1,\] \[x_1+s_3=1,\] \[x_1,x_2,s_1,s_2,s_3\ge0.\]Example 3 — equality already standard
Original:
\[\min 2x_1+3x_2+4x_3+x_4\]subject to
\[2x_1+x_2+x_3=1,\] \[x_1+3x_2+0.5x_4=2,\] \[x_j\ge0.\]This is already in standard form.
Example 4 — $\ge$ constraint
Original:
\[\min 12x_1+20x_2\]subject to
\[3x_1-4x_2\ge2,\] \[6x_1+2x_2\le8,\] \[5x_1-7x_2=-2,\] \[x_1,x_2\ge0.\]Standard form:
\[\min 12x_1+20x_2\]subject to
\[3x_1-4x_2-s_1=2,\] \[6x_1+2x_2+s_2=8,\] \[5x_1-7x_2=-2,\] \[x_1,x_2,s_1,s_2\ge0.\]The equality row may have a negative RHS; standard form allows $b_i$ to be negative unless the simplex starting basis requires otherwise.
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.