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Degeneracy in Linear Programming

The word "degenerate" comes from the Latin degenerare, which means "to fall from a kind" or "to become inferior." It is composed of de- (meaning "down" or "away") and generare (meaning "to generate" or "to produce"). The term originally referred to something that had declined from a normal or expected state.

Degeneracy in linear programming (LP) is a situation that occurs when there are more active constraints at a particular vertex (corner point) of the feasible region than necessary to define that point uniquely. This can lead to computational challenges with algorithms such as the Simplex method, potentially causing stalling or cycling behavior. In this article, we will explore the concept of degeneracy in detail, its causes, and its implications for solving linear programming problems.

What is Degeneracy?

Degeneracy in linear programming occurs when a basic feasible solution has one or more basic variables equal to zero. In geometric terms, this means that a vertex of the feasible region is defined by more constraints than strictly necessary. This excess of active constraints can create ambiguity in the pivot selection process of the Simplex method.

Geometric Interpretation of Degeneracy

From a geometric perspective, degeneracy occurs when more than n constraints intersect at a vertex in an n-dimensional problem. In a two-dimensional LP problem, the feasible region is a polygon, and normally exactly two constraints define each vertex. When three or more constraints intersect at the same point, that vertex is degenerate. While this doesn't affect the existence or uniqueness of optimal solutions, it can complicate the path to finding them.

Mathematical Definition of Degeneracy

Consider a standard-form LP problem:

    Minimize cTx
    Subject to:
    Ax = b
    x ≥ 0

Where x ∈ ℝⁿ, A ∈ ℝᵐˣⁿ, b ∈ ℝᵐ, and c ∈ ℝⁿ. A basic feasible solution is degenerate if one or more of its basic variables equals zero. Equivalently, degeneracy occurs when the rank of the active constraint matrix is less than the number of active constraints at a vertex.

Example of Degeneracy

Consider this two-variable linear programming problem:

    Maximize: z = x₁ + x₂
    Subject to:
    x₁ + x₂ ≤ 2
    x₁ ≥ 1
    x₂ ≥ 1
    2x₁ + 2x₂ = 4

In this example:

  • The feasible region is bounded by these constraints, forming a polygon
  • At the point (1, 1), three constraints are simultaneously active:
    • x₁ = 1 (from x₁ ≥ 1)
    • x₂ = 1 (from x₂ ≥ 1)
    • 2x₁ + 2x₂ = 4 (from the equality constraint)
  • Since only two constraints are needed to define a vertex in two dimensions, but we have three active constraints at (1, 1), this point is degenerate.

Impact on the Simplex Method

Degeneracy affects the Simplex method in several ways:

  • Zero Steps: When degeneracy occurs, the Simplex method may make a pivot that results in no improvement in the objective value (a degenerate pivot).
  • Cycling: In rare cases, the algorithm might cycle through a sequence of degenerate vertices without improving the objective value.
  • Multiple Optimal Solutions: Degeneracy can indicate the presence of multiple optimal solutions, though this isn't always the case.

Detection of Degeneracy

Degeneracy can be detected through several methods:

  • Basic Variable Analysis: Check if any basic variables in the current basic feasible solution equal zero.
  • Constraint Analysis: Count the number of binding constraints at a vertex - if it exceeds the dimension of the problem, the vertex is degenerate.
  • Rank Analysis: Check if the rank of the active constraint matrix is less than the number of active constraints.

Anti-Cycling Strategies

Several strategies exist to prevent cycling in degenerate problems:

  • Bland's Rule: Always select the entering and leaving variables with the smallest indices among those eligible.
  • Perturbation Methods: Slightly modify the right-hand side values (b vector) to eliminate degeneracy.
  • Lexicographic Method: Use a hierarchical system of objective functions to break ties in pivot selection.

Practical Implications

While degeneracy is common in practice, modern solvers handle it efficiently through:

  • Sophisticated pivot selection rules that reduce the likelihood of cycling.
  • Implementation of multiple anti-cycling strategies.
  • Advanced numerical methods to handle near-degenerate cases.
  • Preprocessing techniques that can identify and eliminate some forms of degeneracy.

Conclusion

Degeneracy is a fundamental concept in linear programming that arises naturally in many practical applications. While it can complicate the simplex solution process, modern algorithms and software implementations handle degeneracy effectively through various theoretical and practical techniques. Understanding degeneracy is crucial for practitioners working with optimization problems, as it helps in interpreting results and troubleshooting solution processes.







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