Complementary slackness explains how the primal and dual optima fit together constraint by constraint. A positive variable pairs with a tight constraint, while unused slack pairs with a zero price. This lesson shows the exact products that must vanish at the certificate.
highlighted = computed this step
Primal slack products
Each primal resource slack times its dual price is 0. Why: a positive resource price can only appear when the resource is tight.
primal slack⋅y∗=0
Reduced-cost products
Each reduced cost times its primal variable value is 0. Why: a positive primal variable can only appear when the corresponding dual constraint is tight.
reduced cost⋅x∗=0
What binds
At (4/3, 4/3), the binding constraints make all four displayed products 0. Why: complementary slackness is the local shape of the certificate.
all products =0
Diagram note
The first table uses constraint slacks and prices; the second uses reduced costs and primal values. Pixel positions are rounded for layout; every number shown is exact.