Final answer:
The dual of the linear programming problem given can be obtained by formulating a maximization problem with the transpose of the constraint matrix and the inequality relations reversed, resulting in a problem that maximizes b^Ty subject to A^Ty >= c and y >= 0.
Step-by-step explanation:
To find the dual of the given linear programming problem (P): minimize cTx, subject to Ax ≤ b, we can follow a standard process. This process takes the primal problem, which is in this case a minimization problem, and formulates the corresponding dual problem.
The dual of a linear programming problem in standard form can be defined by introducing a vector of dual variables, which we'll denote by y. If we have a minimization problem with constraints of the form Ax ≤ b, the dual problem will be a maximization problem with constraints that make use of the transpose of matrix A, denoted by AT.
The dual problem (D) corresponding to the primal problem (P) will therefore have the following form:
- Maximize bTy
- Subject to ATy ≥ c
- And y ≥ 0 (since the primal constraints are inequalities)
The rationale behind duality in linear programming is based on the concept that each constraint in the primal space relates to a variable in the dual space, and likewise, each primal variable relates to a constraint in the dual space. The objective in the primal is to minimize costs or resources, represented here by cTx, while the objective in the dual problem is to maximize the value or benefit, indicated by bTy.
The relationship between the primal and dual problems is defined by the weak and strong duality theorems. The weak duality theorem states that the value of the objective function for any feasible solution to the dual problem provides a bound to the value of the objective function for any feasible solution to the primal problem. On the flip side, the strong duality theorem states that if both the primal and dual problems have feasible solutions, then they both have optimal solutions, and the values of their objective functions at the optimal solutions are equal.
Here's a step-by-step guide to formulating the dual:
- For each inequality constraint in the primal problem (Ax ≤ b), introduce a non-negative dual variable.
- Form the dual objective function by taking the product of the dual variables with the primal constraint constants (b), and set it to be maximized (since the primal is a minimization problem).
- Construct the dual constraints by relating the original primal variables to the newly introduced dual variables using the primal constraint coefficients (A).
- The inequality of the dual constraints will be the opposite of the primal constraints; since the primal has a ≤ form, the dual will have a ≥ form.
In summary, the dual problem will maximize bTy, subject to the constraints ATy ≥ c and y ≥ 0.