Complementary Slackness

Definition

Complementary slackness is a condition in optimization theory that relates the primal and dual solutions of a linear programming problem. It indicates that for each constraint in the primal problem, either the constraint is tight (active) and its corresponding dual variable is positive, or the constraint is slack (inactive) and its corresponding dual variable is zero. This principle connects the primal and dual formulations, illustrating the duality in optimization.

5 Must Know Facts For Your Next Test

1. Complementary slackness conditions are crucial for verifying optimality in both primal and dual linear programming problems.
2. If a primal constraint is satisfied with strict inequality, then the corresponding dual variable must equal zero.
3. Conversely, if a dual variable is positive, the corresponding primal constraint must hold as an equality.
4. The concept of complementary slackness extends beyond linear programming to other optimization contexts, such as convex optimization.
5. In semidefinite programming, complementary slackness plays a key role in understanding the relationships between feasible solutions of primal and dual formulations.

Review Questions

• How does complementary slackness help in determining the optimality of solutions in linear programming?
  • Complementary slackness serves as a diagnostic tool to assess whether a given pair of primal and dual solutions is optimal. For a solution to be optimal, it must satisfy the complementary slackness conditions: if a primal constraint is tight (active), then its corresponding dual variable should be positive; if it’s not tight (slack), then the dual variable must be zero. This relationship allows one to check quickly if both solutions are at their best without needing to recalculate the entire solution space.
• Discuss how complementary slackness relates to Farkas' lemma and its role in optimization problems.
  • Complementary slackness is closely linked to Farkas' lemma, which provides conditions for the solvability of systems of linear inequalities. Farkas' lemma helps establish the foundation for understanding when optimal solutions exist for primal and dual problems. By using complementary slackness alongside Farkas' lemma, one can more effectively navigate the relationships between constraints in primal problems and their counterparts in dual formulations, thus solidifying the concept of duality in optimization.
• Evaluate the significance of complementary slackness in semidefinite programming and how it impacts optimization strategies.
  • In semidefinite programming, complementary slackness provides critical insights into the structure of optimal solutions. It highlights how constraints interact within the context of matrix variables. By applying this principle, one can identify whether specific constraints are binding or not, which informs strategic decisions on tightening or relaxing constraints in future iterations. Understanding these relationships through complementary slackness enhances our ability to tackle complex optimization scenarios effectively.

Related terms

Primal Problem: The original optimization problem that seeks to maximize or minimize an objective function subject to constraints.

Dual Problem: An associated optimization problem derived from the primal problem, which typically seeks to maximize the lower bound of the primal objective.

Farkas' Lemma: A fundamental result in linear inequalities that provides necessary and sufficient conditions for the solvability of systems of linear inequalities, which underlies the duality in linear programming.