Polyhedron is unbounded iff it has direction

Dependencies:

  1. Polyhedral set and polyhedral cone
  2. Direction of a convex set and Recession cone
  3. Basic feasible solutions
  4. Point in polytope is convex combination of BFS
  5. Convex hull of a finite number of points in Euclidean space is bounded

Let $P \defeq \{x \in \mathbb{R}^n: (a_i^Tx \ge b_i, \forall i \in I) \textrm{ and } (a_i^Tx = b_i, \forall i \in E)\}$. Then $P$ is unbounded iff it has a non-zero direction $d$.

Proof

Suppose $P$ has a non-zero direction $d$. Since $d \neq 0$, $\exists k \in [n]$ such that $d_k \neq 0$. Let $\xhat \in P$. Then $\xhat + \lambda d \in P$ for all $\lambda \ge 0$. By making $\lambda$ arbitrarily large, $(\xhat + \lambda d)_k$ can be made arbitrarily large, and so $\|\xhat + \lambda d\|$ can be made arbitrarily large. Hence, $P$ is unbounded.

Suppose $P$ doesn't have a non-zero direction. Then $P$ doesn't contain a ray. Hence, any point in $P$ can be represented as a convex combination of at most $n+1$ BFSes of $P$. Let $X$ be the set of BFSes of $P$. Then $P$ is the convex hull of $X$. Hence, $P$ is bounded.

Dependency for:

  1. Bounded section of pointed cone

Info:

Transitive dependencies:

  1. /linear-algebra/vector-spaces/condition-for-subspace
  2. /linear-algebra/matrices/gauss-jordan-algo
  3. /sets-and-relations/equivalence-relation
  4. Group
  5. Ring
  6. Polynomial
  7. Integral Domain
  8. Comparing coefficients of a polynomial with disjoint variables
  9. Field
  10. Vector Space
  11. Linear independence
  12. Span
  13. Semiring
  14. Matrix
  15. Stacking
  16. System of linear equations
  17. Product of stacked matrices
  18. Matrix multiplication is associative
  19. Reduced Row Echelon Form (RREF)
  20. Matrices over a field form a vector space
  21. Row space
  22. Elementary row operation
  23. Every elementary row operation has a unique inverse
  24. Row equivalence of matrices
  25. Row equivalent matrices have the same row space
  26. RREF is unique
  27. Identity matrix
  28. Inverse of a matrix
  29. Inverse of product
  30. Elementary row operation is matrix pre-multiplication
  31. Row equivalence matrix
  32. Equations with row equivalent matrices have the same solution set
  33. Rank of a matrix
  34. Basis of a vector space
  35. Linearly independent set is not bigger than a span
  36. Homogeneous linear equations with more variables than equations
  37. Rank of a homogenous system of linear equations
  38. Cone
  39. Convex combination and convex hull
  40. Transitivity of convexity
  41. Convex set
  42. Polyhedral set and polyhedral cone
  43. Basic feasible solutions
  44. Condition for existence of BFS in a polyhedron
  45. Point in polytope is convex combination of BFS
  46. Convex hull of a finite number of points in Euclidean space is bounded
  47. Direction of a convex set and Recession cone