A bounded linear objective reaches its last feasible contour at a corner. This lesson compares the flagship objective, a tilted objective with a different winner, and an honest edge tie. The diagram shows how the same vertex rule handles all three.
highlighted = computed this step
Last touch
Slide the contour outward until it last touches the feasible region. The touching corner is (4/3, 4/3). Why: the best value is the final contour that still has any feasible contact.
last touch at (4/3,4/3)
Objective value
At that corner, z=x+y=4/3 plus 4/3=8/3. Why: the objective value is recomputed from the same exact point the dot marks.
z=x+y=4/3+4/3=8/3
Different winner
With z=3x+y on the same polygon, the recomputed winner moves to (2, 0) with z=6. Why: changing the objective direction can change which corner is touched last.
max(3x+y)=6 at (2,0)
Whole edge tie
For the tie example, a whole edge is optimal: the endpoints (0, 2) and (2, 0) both have z=2. Why: when the final contour lies on a boundary edge, the optimum is not a single point.
maxz=2 on an edge
Vertex rule
For a bounded linear program, a maximum occurs at a vertex, so checking corners is enough. The recomputed optimum is (4/3, 4/3) with z=8/3. Why: even an edge tie is detected from its endpoint vertices.
maxz=8/3 at (4/3,4/3)
Diagram note
The last-touch contour may name one corner, a different corner, or a whole edge. Pixel positions are rounded for layout; every number shown is exact.