# Dual

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## Definition: Dual

Dual is an alternate formulation, of a linear programming problem which can be used to obtain the optimal solution.

Every linear programming problem, referred to as a primal problem can be converted into a dual problem.

Usefulness of dual:

The number of decision variables in the primal is equal to the number of constraints in the dual. The number of decision variables in the dual is equal to the number of constraints in the primal. Since it is computationally easier to solve problems with fewer constraints in comparison to solving problems with fewer variables, the dual gives us the flexibility to choose which problem to solve.

Mathematical representation

 In matrix form, we can express the primal problem as: 1)      Maximize CTx subject to Ax ≤ B, x ≥ 0;   And corresponding dual problem is, Minimize BTy subject to ATy ≥ C, y ≥ 0.   2)      An alternative primal formulation is Maximize CTx subject to Ax ≤ C;   with the corresponding asymmetric dual problem, Minimize BTy subject to ATy = C, y ≥ 0.

An example:

 The primal is minimizing   40x1 + 44 x2 +48x3 subject to   x1 + 2 x2 + 3x3 >= 20 4 x1+4 x2+4 x3 >=30 x1, x2 and x3 >=0 The dual of the problem is   Maximize 20y1 + 30y2 subject to y1+ 4 y2 <= 40 2y1 + 4y2 <= 44 3y1 + 4y2 <= 48   y1,y2 >= 0

Hence, this concludes the definition of Dual along with its overview.

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