Complementary Slackness

Definition

Complementary slackness is a principle in linear programming that relates the optimal solutions of primal and dual problems. It states that for each pair of primal and dual constraints, at least one of the corresponding variables must be zero at the optimal solution, linking the two formulations in a meaningful way. This principle not only helps to determine feasibility but also aids in finding the optimal values by indicating when a constraint is binding or non-binding.

5 Must Know Facts For Your Next Test

1. Complementary slackness provides necessary and sufficient conditions for optimality between primal and dual problems.
2. If a primal variable is positive, then its corresponding dual constraint must be binding (i.e., equal to its upper bound).
3. Conversely, if a dual variable is positive, then its corresponding primal constraint must also be binding.
4. This principle is useful for sensitivity analysis as it helps identify which variables or constraints can change without affecting the optimal solution.
5. The concept of complementary slackness is essential when using algorithms like the Simplex method to solve linear programming problems.

Review Questions

• How does complementary slackness help in understanding the relationship between primal and dual problems?
  • Complementary slackness establishes a direct link between primal and dual problems by indicating that for each pair of primal and dual variables, at least one must be zero at the optimal solution. This means that if a primal variable has a positive value, its corresponding dual constraint must be tight or binding, while if a dual variable is positive, its related primal constraint must also be binding. This relationship not only helps confirm optimality but also offers insights into how changes in one formulation can affect the other.
• Discuss how complementary slackness can aid in performing sensitivity analysis in linear programming.
  • Complementary slackness aids in sensitivity analysis by clarifying which constraints are crucial to maintaining the current optimal solution. When analyzing potential changes in the constraints or objective function coefficients, this principle allows us to determine if certain variables can remain positive or must become zero. Understanding which constraints are binding versus non-binding provides valuable insights into how robust the solution is and what adjustments can be made without altering optimality.
• Evaluate the implications of complementary slackness when utilizing the Simplex method for solving linear programming problems.
  • When using the Simplex method, complementary slackness plays a critical role in identifying optimal solutions efficiently. The method iteratively moves along the edges of the feasible region towards an optimal vertex by pivoting on basic and non-basic variables. Complementary slackness ensures that during this process, we can quickly assess which constraints are active and which variables are non-binding. By applying this principle, we can streamline calculations and focus on maintaining optimality throughout the iterations, ultimately leading to a faster convergence to the solution.

Related terms

Primal Problem:

The original linear programming problem that seeks to maximize or minimize a linear objective function subject to linear constraints.

Dual Problem: A linear programming problem derived from the primal problem, which seeks to maximize or minimize a related objective function with its own set of constraints.

Binding Constraint: A constraint in a linear programming problem that holds as an equality at the optimal solution, meaning it directly affects the feasible region.