The same geometry also detects cases with no finite optimum. This lesson separates infeasible systems, objectives that run forever, and unbounded regions that still have finite optima. The renderer labels each case from exact recomputation.
highlighted = computed this step
Unbounded
For max z=x+y with minus x plus y<=1 and nonnegative axes, feasible points keep moving toward larger z. The recomputed status is unbounded. Why: the feasible region has a direction that improves the objective forever.
−x+y≤1⇒unbounded
Other opening
The constraint x-y<=1 opens in a different direction, but the recomputed status is still unbounded. Why: the label unbounded is about the objective improving without a final touch, not about a particular slant.
x−y≤1⇒unbounded
Infeasible
The pair x+y<=1 and x+y>=3 cannot both hold. The recomputed status is infeasible. Why: no point can pass every constraint at once.
x+y≤1x+y≥3⇒infeasible
Unbounded region, finite max
For max z=x with x<=2 and nonnegative axes, the region keeps going upward but the recomputed max is z=2 at (2, 0). Why: an unbounded feasible region does not automatically mean an unbounded objective.
maxx=2
Geometry decides the case
A linear program may be optimal, unbounded, or infeasible. The diagram is not a proof by picture; it is a deterministic render of the exact recomputation. Why: the status comes from exact feasibility and recession checks.
status∈{optimal,unbounded,infeasible}
Diagram note
The cases differ by exact geometry: empty set, improving ray, or finite last touch. Pixel positions are rounded for layout; every number shown is exact.