1. Let's first define the concepts in Linear Programming (LP). 2. Alternative optimal solutions occur when multiple optimal points (solutions) yield the same optimal objective value. This happens when the objective function is parallel to a binding constraint, so the optimal solution lies on a line segment or face of the feasible region. 3. Mathematically, if the objective function's gradient vector is a linear combination of the gradients of the active constraints at the optimum, and these constraints form a face (not just a vertex), then multiple optimal solutions exist. 4. Halfline alternative optimal solutions arise when the feasible region extends infinitely in some direction, and the objective function remains constant along that direction, creating infinitely many optimal solutions forming a halfline (ray). 5. Degeneracy in LP happens when a basic feasible solution has one or more basic variables equal to zero, meaning the solution lies at the intersection of more constraints than the dimension requires. 6. In geometric terms, degeneracy occurs when more than $n$ constraints intersect at a vertex in an $n$-dimensional space, causing issues like cycling in algorithms. Summary: - Alternative optimal solutions appear when the objective function is parallel to a constraint boundary, leading to multiple optima. - Halfline alternative solutions occur when the objective function is constant along an infinite edge of the feasible region. - Degeneracy occurs when the number of binding constraints exceeds the dimension, causing basic variables to be zero at the solution.