Math 171A Homework Assignment # 1
Instructor: Jiawang Nie
Assigned Date: January 6, 2010 Due Date: January 13, 2010
1. Find the optimal solution to the following LP
maximize x1+ 4x2+ 3x3
subject to 2x1+x2+x3≤4
x1−x3= 1
x2≥0, x3≥0.
2. Draw the feasible region defined by the following six constraints:
x1+ 2x2≤6, x1−x2≥2, x2≤1, x1−x2≤4, x1≥0, x2≥0.
Find all the corner points.
3. Convert the following LP into the standard form (minimize cTxsubject to Ax ≥b)
minimize x1+ 2x2+ 3x3
subject to 2 ≤x1+x2≤3
4≤x1+x3≤5
x1≥0, x2≥0, x3≥0.
4. Consider the inequality constraint aTx≤bwhere a6= 0. Show that starting from any point
on its boundary, the constraint normal points into the infeasible half-space.
5. Transform the following optimization equivalently into an LP of the standard form (minimize
cTxsubject to Ax ≥b)
minimize |x|+|y|+|z|
subject to x+y≤1
2x+z= 3.
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