Cs 598csc: Combinatorial Optimization
نویسنده
چکیده
Throughout this lecture we will use affhull to denote the affine hull, linspace to be the linear space, charcone to denote the characteristic cone and convexhull to be the convex hull. Recall that P = {x | Ax ≤ b} is a polyhedron in Rn where A is a m× n matrix and b is a m× 1 matrix. An inequality aix ≤ bi in Ax ≤ b is an implicit equality if aix = bi ∀x ∈ P . Let I ⊆ {1, 2, . . . ,m} be the index set of all implicit equalities in Ax ≤ b. Then we can partition A into A=x ≤ b= and A+x ≤ b+. Here A= consists of the rows of A with indices in I and A+ are the remaining rows of A. Therefore, P = {x | A=x = b=, A+x ≤ b+}. In other words, P lies in an affine subspace defined by A=x = b=.
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