Finding Redundant Constraints in Sets of Linear Inequalities

Abstract

Consider the polyhedral set S = {x|x ≥ 0, Ax≤ b}, where A is m × n, x is n × 1, and b is m × 1. After we add slack variables, x_n+1,..., x_n+m, to the structural constraints, Ax ≤ b, we can also consider the polyhedral set S^H = {x|Hx = b, x ≥ 0}, where H = (A, I). S^H is an embedding of S in R^n+m. Each variable in the definition of S^H can be viewed as a slack variable for one of the constraints defining S, since the “structural variables” (x₁,...,x_n) can be viewed as the slacks of the nonnegativity constraints (x ≥ 0) defining S. Thus there is a one-to-one correspondence between all the constraints defining S and the nonnegativity constraints in the definition of S^H. Hence, the statements, “The kth constraint of S is redundant (nonredundant)” and “The constraint x_k ≥ 0 is redundant (nonredundant) in defining S^H” are equivalent. We let S^k be obtained from S by deleting the kth constraint.

This research was supported in part by a grant from the Business Foundation of North Carolina, Inc.