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Integer Programming ISE 418

Lecture 4

Dr. Ted Ralphs ISE 418 Lecture 4 1 Reading for This Lecture

• N&W Sections I.4.1-I.4.3

1 ISE 418 Lecture 4 2 Review: Definition 1. A finite collection of vectors x1, . . . , xk ∈ Rn is linearly Pk i independent if the unique solution to i=1 λix = 0 is λi = 0, i ∈ [1..k]. Otherwise, the vectors are linearly dependent.

Let A be a square . Then, the following statements are equivalent:

• The matrix A is invertible. • The matrix A> is invertible. • The determinant of A is nonzero. • The rows of A are linearly independent. • The columns of A are linearly independent. • For every vector b, the system Ax = b has a unique solution. • There exists some vector b for which the system Ax = b has a unique solution.

2 ISE 418 Lecture 4 3 Linear Algebra Review: Affine Independence Definition 2. A finite collection of vectors x1, . . . , xk ∈ Rn is affinely independent if the vectors x2 − x1, . . . , xk − x1 ∈ Rn are linearly independent.

• Linear independence implies affine independence, but not vice versa. • The property of linear independence is with respect to a given origin. • Affine independence is essentially a “coordinate-free” version of linear independence. Proposition 1. The following statements are equivalent:

n 1. x1, . . . , xk ∈ R are affinely independent.

2. x2 − x1, . . . , xk − x1 are linearly independent. n+1 3. (x1, 1),..., (xk, 1) ∈ R are linearly independent.

3 ISE 418 Lecture 4 4 Linear Algebra Review: Subspaces Definition 3. A nonempty H ⊆ Rn is called a subspace if αx+γy ∈ H ∀x, y ∈ H and ∀α, γ ∈ R. Definition 4. A of a collection of vectors x1, . . . xk ∈ Rn n Pk i k is any vector y ∈ R such that y = i=1 λix for some λ ∈ R . Definition 5. The span of a collection of vectors x1, . . . xk ∈ Rn is the set of all linear combinations of those vectors.

Definition 6. Given a subspace H ⊆ Rn, a collection of linearly independent vectors whose span is H is called a of H. The number of vectors in the basis is the of the subspace.

4 ISE 418 Lecture 4 5 Linear Algebra Review: Subspaces and Bases

• A given subspace has an infinite number of bases. • Each basis has the same number of vectors in it (the dimension).

• If S and T are subspaces such that S ⊆ T ⊆ Rn, then a basis of S can be extended to a basis of T . • The span of the columns of a matrix A is a subspace called the column or the range, denoted range(A). • The span of the rows of a matrix A is a subspace called the row space. • The of the column space and row space are always equal. We call this number rank(A). • Clearly, rank(A) ≤ min{m, n}. If rank(A) = min{m, n}, then A is said to have full rank.

• The set {x ∈ Rn | Ax = 0} is called the nullspace of A (denoted null(A)) and has dimension n − rank(A).

5 ISE 418 Lecture 4 6 Some Properties of Subspaces Proposition 2. The following are equivalent:

1. H ⊆ Rn is a subspace. 2. There is an m × n matrix A such that H = {x ∈ Rn | Ax = 0}. 3. There is a k×n matrix B such that H = {x ∈ Rn | x> = uB, u ∈ Rk}.

Proposition 3. If {x ∈ Rn | Ax = b}= 6 ∅, the maximum number of affinely independent solutions of Ax = b is n + 1 − rank(A).

Proposition 4. If H ⊆ Rn is a subspace, the subspace {x ∈ Rn | x>y = 0 ∀ y ∈ H} is a subspace called the orthogonal subspace or orthognal complement and denoted H⊥.

Proposition 5. If H = {x ∈ Rn | Ax = 0}, with A being an m×n matrix, then H⊥ = {x ∈ Rn | x = A>u, u ∈ Rm} and dim(H) + dim(H⊥) = n.

6 ISE 418 Lecture 4 7 Affine Spaces Definition 7. An affine combination of a collection of vectors x1, . . . xk ∈ n n Pk i k R is any vector y ∈ R such that y = i=1 λix for some λ ∈ R with Pk j=1 λj = 1. Definition 8. A nonempty subset A ⊆ Rn is called an affine space if A is closed with respect to affine combination.

Definition 9. A basis of an affine space A ⊆ Rn is a maximal set of affinely independent points of A. Definition 10. The inclusionwise minimal affine space containing a set S is called the affine hull of S, denoted aff(S). Definition 11. All bases of an affine space A have the same cardinality and this cardinality exceeds the dimension of the affine space by one.

7 ISE 418 Lecture 4 8 Projections Definition 12. If p ∈ Rn and H is a subspace, the of p onto H is the vector q ∈ H such that p − q ∈ H⊥.

• Note that this is a decomposition of a vector p into the sum of a vector in H and a vector in H⊥. • The projection of a set is the union of the projections of all its members. • Projections play a very important role in discrete optimization, as we will see later in the course.

8 ISE 418 Lecture 4 9 Polyhedra, , and Half-spaces Definition 13. A is a set of the form {x ∈ Rn | Ax ≤ b}, where A ∈ Rm×n and b ∈ Rm. Definition 14. A polyhedron P ⊂ Rn is bounded if there exists a constant K such that |xi| < K ∀x ∈ S, ∀i ∈ [1, n]. Definition 15. A bounded polyhedron is called a .

Definition 16. Let a ∈ Rn and b ∈ R be given.

• The set {x ∈ Rn | a>x = b} is called a . • The set {x ∈ Rn | a>x ≤ b} is called a half-space.

9 ISE 418 Lecture 4 10 Convex Sets

Definition 17. A set S ⊆ Rn is convex if ∀x, y ∈ S, λ ∈ [0, 1], we have λx + (1 − λ)y ∈ S.

1 k n k > Definition 18. Let x , . . . , x ∈ R and λ ∈ R+ be given such that λ 1 = 1. Then Pk i 1 k 1. The vector i=1 λix is said to be a of x , . . . , x . 2. The of x1, . . . , xk is the set of all convex combinations of these vectors.

1 j n j Definition 19. Let r , . . . , r ∈ R and µ ∈ R+ be given. Then Pj i 1 k 1. The vector i=1 µir is said to be a conic combination of r , . . . , r . 2. The conic hull of r1, . . . , rj is the set of all conic combinations of these vectors. • The convex hull of two points is a segment. • A set is convex if and only if for any two points in the set, the joining those two points lies entirely in the set. • All polyhedra are convex.

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