Complementary slackness

Definition

Complementary slackness is a condition in optimization problems, specifically in linear programming, that relates the primal and dual solutions. It states that for each pair of primal and dual constraints, if a primal constraint is not tight (slack), the corresponding dual variable must be zero, and vice versa. This concept is essential in determining optimal solutions and understanding the relationships between different constraints in constrained optimization problems.

5 Must Know Facts For Your Next Test

1. Complementary slackness plays a crucial role in verifying optimality for linear programming solutions, as it connects primal and dual solutions.
2. If a primal constraint has slack, then the corresponding dual variable must be zero, indicating no influence on the optimal value.
3. Conversely, if a dual constraint is not satisfied, its corresponding primal variable will be zero, indicating that it does not contribute to the solution.
4. This principle helps identify which constraints are binding (tight) at the optimal solution and which are non-binding (slack).
5. Complementary slackness can be used as a tool for sensitivity analysis, providing insights into how changes in constraints affect optimal solutions.

Review Questions

• How does complementary slackness help determine optimality in linear programming?
  • Complementary slackness helps determine optimality by establishing a relationship between the primal and dual solutions. For an optimal solution, if a primal constraint has slack (is not binding), the corresponding dual variable must equal zero. This relationship allows us to quickly verify whether both solutions are optimal since any deviation from this condition indicates that we may not have found the best solution yet.
• Discuss the implications of complementary slackness for interpreting primal and dual constraints in an optimization problem.
  • The implications of complementary slackness are significant when interpreting primal and dual constraints. It shows that each primal constraint's status (tight or slack) directly affects the value of its corresponding dual variable. If a constraint in the primal is not active at optimality (i.e., has slack), it suggests that the related dual variable contributes nothing to the objective function, emphasizing how different constraints interact with one another and influence overall optimality.
• Evaluate how understanding complementary slackness can aid in sensitivity analysis within linear programming.
  • Understanding complementary slackness enhances sensitivity analysis by clarifying how changes in constraints impact optimal solutions. When one knows which primal or dual variables are active or inactive at the optimal point, they can predict how adjustments in resource availability or constraint limits will affect the overall solution. This understanding allows for strategic decision-making by highlighting critical constraints that drive optimality and those that can be adjusted with minimal impact on outcomes.

Related terms

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

Dual Problem: A derived optimization problem that corresponds to the primal problem, focusing on the dual variables associated with the constraints of the primal.

Feasibility: A condition in optimization where a solution satisfies all constraints of the problem.