Degeneracy

Definition

Degeneracy refers to a situation in linear programming where a solution is not unique because multiple solutions yield the same optimal value. This can occur at vertices of the feasible region, where two or more constraints intersect, leading to an infinite number of optimal solutions. Recognizing degeneracy is important as it can affect the efficiency of the solution process and the interpretation of results.

5 Must Know Facts For Your Next Test

1. In a degenerate solution, at least one constraint is binding, but it does not change the optimal value due to multiple solutions being available.
2. Degeneracy can lead to cycling in the Simplex method, where the algorithm may revisit the same basic feasible solutions without making progress.
3. When encountering degeneracy, special techniques like perturbation methods or lexicographic ordering can be employed to ensure convergence in optimization algorithms.
4. Degeneracy does not imply a problem with the formulation; instead, it reflects the nature of the solution space.
5. In practical applications, degeneracy can complicate sensitivity analysis since small changes in coefficients might not affect the optimal solution.

Review Questions

• How does degeneracy affect the uniqueness of solutions in linear programming?
  • Degeneracy affects uniqueness by allowing multiple solutions to yield the same optimal objective value. This occurs when constraints intersect at a vertex of the feasible region, leading to several combinations of variable values that can achieve the same outcome. Recognizing this helps understand that while there may be a single optimal value, there could be infinitely many ways to reach it.
• Discuss how the Simplex method handles cases of degeneracy and what strategies might be implemented to avoid cycling.
  • The Simplex method can face challenges with degeneracy because it may cycle through basic feasible solutions without improving the objective value. To address this, strategies such as perturbation methods or lexicographic ordering can be applied. These techniques help ensure that the algorithm progresses toward an optimal solution by modifying constraint equations slightly or establishing a consistent tie-breaking rule for choosing among degenerate vertices.
• Evaluate the implications of degeneracy on sensitivity analysis in linear programming problems.
  • Degeneracy complicates sensitivity analysis because it indicates that small changes in the coefficients of the objective function or constraints may not lead to different optimal solutions. As a result, understanding how sensitive an optimal solution is to changes becomes more complex, since multiple degenerate solutions could respond differently to perturbations. This requires careful examination to determine which variables are truly pivotal in affecting outcomes, especially in real-world applications where decision-making relies on precise analyses.

Related terms

Feasible Region: The set of all possible points that satisfy the constraints of a linear programming problem.

Optimal Solution:

The best solution that maximizes or minimizes the objective function in a linear programming problem.

Basic Feasible Solution: A solution to a linear programming problem that is obtained by setting non-basic variables to zero and solving for basic variables, often associated with vertex points in the feasible region.