Final answer:
The dual of the given primal linear programming problem is structured as maximizing bTy, with constraints ATy ≥ c and y ≥ 0. The dual variables y correspond to primal constraints, while the original primal coefficients c and constants b become part of the dual's constraints and objective function, respectively.
Step-by-step explanation:
The dual of a linear programming problem plays a critical role in understanding the structure of the original problem and can provide insights on how to solve it more efficiently. In the given problem, the primal linear programming problem is to minimize x cTx under the constraints Ax ≤ b and x ≥ 0. To find the dual problem, we need to apply the rules that guide the transition from primal to dual problems in linear programming.
For the given primal problem (P):
- Objective Function: Minimize cTx
- Subject to: Ax ≤ b
- And: x ≥ 0
The dual problem (D) will then have the following structure:
- Objective Function: Maximize bTy
- Subject to: ATy ≥ c
- And: y ≥ 0 (if original constraints are ≤ type)
The dual variables in y correspond to the constraints in the primal problem, and the constraints in the dual problem are derived from the original decision variables x. The relationship between c, A, and b in the primal flips in the dual, with A's transpose and inequality direction changed accordingly. The coefficients of the primal problem's objective function become the right-hand side constants in the dual's constraints, and the right-hand side constants in the primal constraints become the coefficients of the dual's objective function.