Question

Given the linear programming problem

maximizex,y c^Tx+d^Ty
subject to Ax+By<=b,
Dy=e
x>=0^(n1), y>=0^(n2)
Where A∈R^m1xn1, B∈R^m1xn2, D∈R^m2xn2,
b∈R^m1, r∈R^m2, c∈R^n1, and d∈R^n2
find its dual

Answer 1

The student's task involves identifying the dual of a given linear programming problem, which can be achieved using duality theory by introducing dual variables and restructuring the primal constraints and objective function accordingly.

Step-by-step explanation:

The question asks for the dual of a given linear programming problem. The primal linear programming problem is formulated as:

Maximize cTx + dTy

Subject to:

Using duality theory, we can derive its dual problem. Introducing dual variables u for the inequality constraints and v for the equality constraints, the dual is:

Minimize uTb + vTr

Subject to:

To find the dual, we used the fundamental duality relationships: The transpose of the coefficient matrix in the inequality constraints (A, B) is paired with the primal's objective function coefficients (c, d), with the inequality directions reversed. The dual variables associated with inequalities are non-negative, and those for equalities are free to vary.