Weak Duality Theorem

Definition

The weak duality theorem is a fundamental concept in linear programming that states the relationship between the solutions of a primal problem and its dual problem. Specifically, it asserts that the objective value of any feasible solution to the primal problem is always less than or equal to the objective value of any feasible solution to the dual problem. This theorem lays the groundwork for understanding optimality in linear programming, as it implies that if both problems have feasible solutions, the optimal solution of one cannot exceed that of the other.

5 Must Know Facts For Your Next Test

1. Weak duality holds for any feasible solution of both the primal and dual problems, regardless of whether they are optimal.
2. If the primal problem is feasible and bounded, the weak duality theorem ensures that there exists an optimal solution for the dual problem as well.
3. The theorem is essential for deriving strong duality, which states that if both problems have optimal solutions, their objective values are equal.
4. Weak duality can be used to determine infeasibility by showing that no feasible solution exists for one of the problems.
5. In practical applications, weak duality provides bounds on the optimal value of the objective function, which can guide decision-making in optimization.

Review Questions

• How does the weak duality theorem support decision-making in linear programming?
  • The weak duality theorem helps decision-makers understand the bounds on the optimal value of an objective function. By establishing that any feasible solution to the primal problem will have an objective value less than or equal to any feasible solution to its dual, it provides a way to assess possible outcomes and determine how close one is to an optimal solution. This information can guide adjustments in constraints or objectives to improve solutions effectively.
• What implications does weak duality have when analyzing the feasibility of a linear programming problem?
  • When analyzing feasibility in linear programming, weak duality indicates that if one of the problems—either primal or dual—has no feasible solution, then the other must also be infeasible. This relationship helps identify potential issues in model formulation and can highlight areas that need revision or re-evaluation. Therefore, checking weak duality can serve as a diagnostic tool for determining whether further investigation is needed into either formulation.
• Evaluate how weak duality relates to strong duality in terms of their roles within linear programming.
  • Weak duality serves as a foundational concept leading to strong duality in linear programming. While weak duality assures us that feasible solutions yield bounded relationships between primal and dual objectives, strong duality takes this further by asserting equality of optimal values when both problems have solutions. Understanding weak duality allows us to grasp why solving one problem effectively informs about the other, establishing a complete framework for optimization analysis and providing deeper insights into their interconnectedness.

Related terms

Primal Problem: The original linear programming problem, which seeks to maximize or minimize a linear objective function subject to certain constraints.

Dual Problem: A derived linear programming problem that corresponds to the primal problem, where the roles of the objective function and constraints are switched.

Feasibility: The property of a solution that satisfies all constraints of a linear programming problem, whether it be primal or dual.