ma213 x 2

Question 1

how to know if y* is optimal?

Answer 1

check against the variables of the dual solution!!

Question 2

ineffective and effective?

Answer 2

ineffective constraints (> or <)’s variables are = 0 in the dual (f)

Question 3

Does the problem have multiple optimal solutions?

Answer 3

Since the objective function is parallel to one of the binding constraints, it can slide along that boundary and still maintain the same objective value.

If two corner (extreme) points of the feasible region lie on this line, then all points on the segment connecting them are optimal.

Hence, the feasible region has a face (edge) where the objective function is constant ⇒ multiple optimal solutions.

Question 4

optimal solution criteria

Answer 4

1. where the objective function passes through
2. if it doesn’t pass through anything, then where its parallel constraint (the slope is the same) passes through

Question 5

effective vs basic

Answer 5

basic focuses on x*. if x3 is 0 it is nonbasic and must be eliminated from primal(f)

effective is when substituting x* is equal to its answer, ineffective constraints must be eliminated from the dual(f)

Question 6

complementary slackness

Answer 6

For each constraint and variable:

If a dual variable yj>0, then the corresponding primal constraint must be tight (holds as equality).

If a primal variable xj>0, then the corresponding dual constraint must be tight (equality).

Or flipped:

If a primal constraint is slack (not tight), then the dual variable must be 0.

If a dual constraint is slack, then the primal variable must be 0.

Question 7

slack and tight

Answer 7

tight = 0
slack doesn’t = 0

Question 8

where does minimum and maximum occur

Answer 8

minimum at f’(x) = 0
max at substituting the value of minimum in f(x)

Question 9

convex and concave

Answer 9

concave at f’‘(x)<0
- sad face, less than zero

convex at f’‘(x)>0