The two special cases are multiple optimal solutions and degeneracy. Multiple optimal solutions mean that infinitely many solutions give the same optimal objective value, while degeneracy means that more constraints are binding than necessary at a solution. Neither implies the other because multiple optima depend on the objective function, while degeneracy depends on the constraints and solution structure.

The two special cases are multiple optimal solutions and degeneracy. Multiple optimal solutions occur when more than one feasible solution gives the same optimal objective value, while degeneracy occurs when an optimal basic feasible solution has at least one basic variable equal to zero. Neither implies the other because an LP can have multiple optimal solutions without a degenerate optimal solution, and it can have a degenerate optimal solution with only one optimal solution.

The two properties are multiple optimal solutions and degeneracy.

Multiple optimal solutions happen when the objective line runs parallel to a constraint boundary, making every point along an entire edge optimal. This depends on objective slope alignment.

Degeneracy happens when a basic variable equals zero, occurring when extra constraint lines intersect at a single corner point. This depends on constraint geometry.

Multiple optima do not cause degeneracy because an optimal edge can connect standard, non-degenerate corners. Degeneracy does not cause multiple optima because a unique optimal solution can sit directly on a degenerate corner point without parallel constraints.

The two special cases are multiple optimal solutions and degeneracy. Multiple optimal solutions mean different feasible solutions have the same optimal objective value. Degeneracy means an optimal basic solution has more active constraints than necessary. Neither implies the other because a problem can have multiple optima without degeneracy, and a degenerate solution can still have a unique optimal solution.

The two special-case properties are multiple optimal solutions and degeneracy. Multiple optimal solutions occur because infinitely many feasible solutions achieve the same optimal objective value, while degeneracy occurs when at least one basic variable is zero at an optimal solution. These properties are independent because having multiple optimal solutions does not require any of them to be degenerate, and a degenerate solution does not automatically mean that other optimal solutions exist. In this case, the LP simply has both properties at the same time: infinitely many optimal solutions and at least one degenerate optimal solution.

The two special-case properties are multiple (alternative) optimal solutions and degeneracy. Multiple optimal solutions mean that infinitely many feasible solutions produce the same optimal objective value, usually because the objective function is parallel to a binding edge of the feasible region. Degeneracy means that at least one optimal basic feasible solution has a basic variable equal to zero, often because more constraints are binding than necessary. Neither implies the other: an LP can have multiple optimal solutions without being degenerate, and it can have a degenerate optimal solution while still having a unique optimum.