Building the Dual

Every LP Has a Dual

A dual LP prices the resources of a primal LP. This lesson shows the transpose rule that turns primal rows and columns into dual variables and constraints. The coefficients are exact entries, not a visual analogy.

highlighted = computed this step

Dual shape

In general, a max problem with resource matrix A has a min problem with A transposed. Why: the dual prices resources, so each primal constraint gets a dual variable.

dual uses the transpose\text{dual uses the transpose}
Matrix and transposeIn general, the dual constraints use the transpose of the primal resource matrix.A2131A transpose2311

Rows become prices

There is one dual variable for each primal resource row and one dual constraint for each primal decision variable. Why: prices must cover every unit of objective value.

rows price resources; columns become constraints\text{rows price resources; columns become constraints}
Matrix and transposeIn general, the dual constraints use the transpose of the primal resource matrix.A2131A transpose2311

Diagram note

The example matrix is intentionally non-symmetric, so the row-column swap is visible. Pixel positions are rounded for layout; every number shown is exact.

the dual is built from the same coefficients\text{the dual is built from the same coefficients}
Matrix and transposeIn general, the dual constraints use the transpose of the primal resource matrix.A2131A transpose2311