Strong duality turns a bound into a proof. This lesson places the flagship primal optimum beside the dual optimum and checks that their exact objective values match. The same numbers connect the geometric vertex and the simplex shadow prices.
highlighted = computed this step
Primal side
The primal optimum is (4/3, 4/3) with z=8/3. Why: this is the same vertex found by geometry and simplex.
zP=8/3
Dual side
The dual optimum is (1/3, 1/3) with z=8/3. Why: these prices produce the smallest valid upper bound.
zD=8/3
Equal values
The two objective values are equal at 8/3. Why: a primal feasible point and a matching dual feasible bound certify optimality.
zP=zD=8/3
Diagram note
The paired tables are recomputed from the primal LP, its dual, and their exact optima. Pixel positions are rounded for layout; every number shown is exact.
matching primal and dual values certify optimality