Network Design Foundation Fall 2015 Iterative Methods in Combinatorial Optimization

1 Preliminaries 1.1 Linear Algebra In this section, we review some definitions and concepts related to linear algebra which will be useful in describing the iterative methods later. Definition 1 The row rank of a matrix A is the maximum number of linearly independent rows in A. The column rank of a matrix A is the maximum number of linearly independent columns in A. In other words, row rank of A = dim(span(row vectors of A)) column rank of A = dim(span(column vectors of A)) Theorem 1 The row rank and column rank of any given matrix are equal and therefore unambiguously referred to as the rank of the matrix. The general form of a linear program is as follows (this will be referenced in the definitions and lemmas given ahead): min cx s.t. Ax ≥ b . . . P x ≥ 0 Here, P corresponds to the set of feasible solutions (the ones that satisfy both given constraints) i.e P = {x : Ax ≥ b, x ≥ 0}. Definition 2 x ∈ R is an extreme point solution to P if there is no nonzero y ∈ R s.t x+y and x−y belong to P . Alternately, x cannot be written as a linear combination of any two y, z ∈ P (i.e x cannot be written as ay+(1−a)z).