Complementary Slackness Theorem

Definition

The Complementary Slackness Theorem is a key principle in linear programming that establishes a relationship between the primal and dual solutions of an optimization problem. It states that for each pair of corresponding primal and dual constraints, either the primal constraint is tight (active) and the corresponding dual variable is non-zero, or the primal constraint is slack (inactive) and the corresponding dual variable is zero. This theorem helps in identifying optimal solutions by providing conditions under which the primal and dual solutions align.

5 Must Know Facts For Your Next Test

1. Complementary slackness provides necessary and sufficient conditions for optimality in linear programming problems, helping to determine whether a candidate solution is optimal.
2. If a primal constraint is not binding (i.e., it has slack), then the corresponding dual variable must be equal to zero, indicating that it does not influence the optimal value.
3. Conversely, if a dual variable is positive, then its associated primal constraint must be binding, meaning it actively constrains the solution.
4. The complementary slackness conditions can be used to simplify the process of solving both primal and dual problems by checking these relationships.
5. Understanding complementary slackness is crucial for interpreting sensitivity analysis, as it helps identify how changes in constraints can affect optimal solutions.

Review Questions

• How do the complementary slackness conditions help in determining whether a given solution to a linear programming problem is optimal?
  • The complementary slackness conditions provide a direct way to assess whether a candidate solution is optimal by examining the relationships between primal and dual variables. If all pairs of corresponding primal constraints and dual variables satisfy the conditions, it indicates that both solutions are optimal. If any condition fails, it signals that adjustments may be needed in either the primal or dual problem.
• Discuss how the concepts of binding and non-binding constraints relate to complementary slackness in linear programming.
  • Binding constraints are those that have no slack and directly impact the feasible solution, while non-binding constraints have slack. Complementary slackness states that for a non-binding primal constraint, its associated dual variable must be zero. Conversely, if a dual variable is positive, its corresponding primal constraint must be binding. This relationship illustrates how constraints interact within optimization problems.
• Evaluate the implications of complementary slackness on sensitivity analysis within linear programming models.
  • Complementary slackness has significant implications for sensitivity analysis by highlighting how changes in constraints can affect optimal solutions. When analyzing how perturbations in coefficients or right-hand sides impact solutions, understanding which constraints are binding helps predict shifts in feasibility and optimality. This knowledge allows decision-makers to identify critical constraints and make informed adjustments to maintain optimality.

Related terms

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

Dual Problem: The derived optimization problem associated with the primal problem, which provides bounds on the optimal value of the primal solution.

Feasibility: The condition in which a solution satisfies all constraints of an optimization problem, ensuring it lies within the feasible region.