Homework Assigned on: Nov 30, 2001 Due on: Dec 7, 2001 by 4 p.m. Note: Please return the homework to Linda Weybright at MGL 1318, 494-5409. 1. In this problem, you'll compute two iterations of Karmarkar's algorithm. The problem will give you a feel for the numerical computation involved in the algorithm. Consider the following LP (which is in Karmarkar Normal Form). M in -x1 + 1 S.T. x2 - x3 = 0 x1 + x2 + x3 = 1 x1 , x 2 , x 3 0 (a) Draw the feasible region of the LP. (b) Verify that the LP satisfies all the assumptions that Karmarkar's algorithm makes about an LP. e (c) Let X (0) = n . Compute the points X (1) and X (2) that Karmarkar's algorithm would generate. (d) Draw the (zig-zag) trajectory of the algorithm in the above (x1 , x2 , x3 ) coordinates. 2. This problem will develop the details needed to justify the assumption that the optimal <a href="/keyword/objective-function-value/" >objective function value</a> in Karmarkar's algorithm is known to be zero. Consider the problem M in cT x S.T. Ax = 0; A: m n matrix, rank(A) = m eT x = 1 x0 and its dual M ax z S.T. AT w + ze c; eT = (1 1 . . . 1) Assume that the optimal <a href="/keyword/objective-function-value/" >objective function value</a> of the primal is zero. (a) Show that the dual LP is always feasible. (b) Let x = (x , . . . , x ) be the optimal solution of the primal and (w , z ), the optimal 1 n solution of the dual. Further, define a diagonal matrix x 1 X = x 2 ... x n (the off-diagonal entries are all zeroes). Using complementary slackness conditions show that X AT w = X c. IE 535, Fall 2001 1 Instructor: Prof. Prabhu Homework Assigned on: Nov 30, 2001 Due on: Dec 7, 2001 by 4 p.m. (c) If x(k) n is an interior point of n , and = (k) x1 (k) X (k) x2 ... x(k) n then show that A(X (k) )2 AT is invertible. (d) Karmarkar's algorithm generates a sequence of interior points x(1) , x(2) , . . . n-1 which converge to the optimal solution x (that lies on the boundary of n-1 ). Show that if x(k) is an interior primal solution, then the following is a corresponding dual solution w(k) = (A(X (k) )2 AT )-1 A(X (k) )2 c z (k) = min(c - AT w(k) )j j (1) (e) Show that, as x(1) , x(2) , . . . converge to the primal optimal solution x , the sequence of dual solutions shown in (1) converge to optimal dual solution. Using the above information, I'll show you in the lecture, how one can justify the assumption that the optimal <a href="/keyword/objective-function-value/" >objective function value</a> is zero. IE 535, Fall 2001 2 Instructor: Prof. Prabhu