Industrial Engineering & Operations Research, UC Berkeley IEOR269 Integer Programming and Combinatorial Optimization Summary: Industrial Engineering & Operations Research, UC Berkeley IEOR269 Integer Programming and Combinatorial Optimization Semester: Spring 2004 Instructor: Alper Atamt¨urk Midterm Exam ­ Due: March 17, 2004, 5PM (4175 Etcheverry) 1. Let A be a rational valued m × n matrix and b be a rational valued m­dimensional column vector. Let P = {x IRn : Ax b} be a nonempty polyhedron and x P. (a) Prove that for x, the violation of any Chv´atal-Gomory (CG) inequality = yT Ax - yT b , where y IRm + , yT A integral, is upper bounded by a constant. (b) Let s = b - Ax and call a CG inequality complementary (with respect to x) if yis i = 0 for all i = 1, . . . , m. Prove that there is a polynomial algorithm that determines whether there is a complementary CG inequality violated by x or not (and in the former case outputs such an inequality). 2. Consider the affine transformation T(x) = QA-1/2(x - a) defined in the ellipsoid method of Khachiyan. Prove that T( ^E(A, a)) = ^E(I, 0). 3. Prove that if H and G are two faces of a polyhedron P of dimension r and r + s (r 0, s > 0 and both integer), respectively, and H is a face of G, then there exists Collections: Engineering