CSc 545 Lecture topic: The Criss-Cross method of Linear Programming

Transcription

1 CSc 545 Lecture topic: The Criss-Cross method of Linear Programming Wanda B. Boyer University of Victoria November 21, 2012

2 Presentation Outline 1 Outline Please note: I would be extremely grateful if questions could be left until the end of the lecture! 1

3 Why so Cross? Given the prevalence of the Simplex method, what is so interesting about the Criss-Cross method? 2

4 Why so Cross? Given the prevalence of the Simplex method, what is so interesting about the Criss-Cross method? It s all well and good when you have nice problems with feasible origins; but we ve seen the trouble with negative b i values in the constraints... 2

5 Why so Cross? Given the prevalence of the Simplex method, what is so interesting about the Criss-Cross method? The Simplex method needs to find an initial feasible origin before it can solve such a problem. 2

6 Why so Cross? Given the prevalence of the Simplex method, what is so interesting about the Criss-Cross method? The various Criss-cross methods, on the other hand, do not need to solve an auxiliary problem, nor do they require artificial variables (i.e. the two-phase simplex method s x 0 ) to work towards a solution. 2

7 Why so Cross? Given the prevalence of the Simplex method, what is so interesting about the Criss-Cross method? But there is a more important motivation: the simplex method still took too long using the limited computing resources of Even today, we are still trying to find a guaranteed polytime algorithm for solving LP problems. 2

8 A general description of the method The variant of the criss-cross method first proposed by Zionts in 1969 partitions the primal and dual problems and then alternates between pivoting in the primal and dual problem until one is feasible, at which point the standard simplex method is used on that feasible problem to reach an optimal solution, if there is one. The problem is partitioned in the following way: Minimize c1 T u 1 + c2 T u 2 subject to A 1,1 u 1 + A 1,2 u 2 b 1 A 2,1 u 1 + A 2,2 u 2 b 2 u 1, u 2 0 3

9 A general description of the method (continued 1) Minimize c1 T u 1 + c2 T u 2 subject to A 1,1 u 1 + A 1,2 u 2 b 1 A 2,1 u 1 + A 2,2 u 2 b 2 u 1, u 2 0 Order the x i s in the objective function so that those with negative coefficients are listed first as components of the vector u 1 ; the latter x i s, which have only positive coefficients, are components of the vector u 2. Order the constraints such that those with negative b i values come first in vector b 1, followed by the constraints with positive b i s in b 2. Note that in the actual implementation, you don t actually have to re-order the problem; this partitioning simply makes it easier for Zionts to reason about the correctness of the method. 4

10 A worked example This example shows what happens when both primal and dual infeasibility strike! Maximize 2x 1 x 2 subject to 2x 1 2x 2 2 3x 1 + 3x 2 6 x 1, x 2 0 Recall from primal-dual theory that the correspondence between primal and dual variables is: Primal Dual x 1 y 4 x 2 y 5 x 3 y 1 x 4 y 2 5

11 A worked example (continued) Setup: Solving using the CRISS-CROSS method *************** Problem 1 *************** The initial primal dictionary: X3 = X X2 X4 = X X z = X X2 The initial dual dictionary: Y3 = Y Y2 Y4 = Y Y w = Y Y2 Because X4 corresponds to the most negative b i value, we choose Y2 to enter in the dual. But no entering row exists among those corresponding to the positive c i s (i.e. the row for Y4). 6

12 A worked example (continued) The first iteration: ======================================================= Iteration 0 of the criss-cross method: ======================================================= This is a primal iteration: Try to choose pivot row No options in feasible part as suggested by Zionts. Finding a pivot failed so try in dual instead. Try to choose pivot row The dual dictionary: Y3 = Y Y2 Y4 = Y Y w = Y Y2 The primal dictionary: X3 = X X2 X4 = X X z = X X2 X1 enters. X3 leaves. w =

13 A worked example (continued) The first iteration: After 1 pivot The dual dictionary: Y1 = Y Y3 Y4 = Y Y w = Y Y3 The primal dictionary: X1 = X X3 X4 = X X z = X X3 8

14 A worked example (continued) The second iteration: The primal dictionary: X2 = X X3 X4 = X X z = X X3 The dual dictionary: Y1 = Y Y3 Y4 = Y Y w = Y Y3 9

15 A worked example (continued) 10 The second iteration: ======================================================= Iteration 1 of the criss-cross method: ======================================================= This is a dual iteration: Try to choose pivot row No options in feasible part as suggested by Zionts. But finding a pivot failed so try in primal instead. Try to choose pivot row No options in feasible part as suggested by Zionts. Both infeasible, hence no solution exists.

16 11 Correctness Outline In discussing the conditions of convergence for the criss-cross method, Zionts used the theories of the dual simplex method and regularization to cover when a dual unbounded solution is reached, or where the regularization constraint rejects the possibility of a finite optimal feasible solution.

17 11 Correctness Outline In discussing the conditions of convergence for the criss-cross method, Zionts used the theories of the dual simplex method and regularization to cover when a dual unbounded solution is reached, or where the regularization constraint rejects the possibility of a finite optimal feasible solution. To clarify: regularization is how Zionts made sure his implementation didn t go off into la-la land, by guessing a sufficiently high number and made that a constraint of the problem. If that constraint was tight, then the problem was probably unbounded. He had to find a balance between choosing a large enough number so that his computer wouldn t run out of resources, but that he wouldn t be cutting off any corners of the polytope and excluding potentially optimal solutions.

18 Correctness (continued) 12 Recalling the partitioned problem: Minimize c1 T u 1 + c2 T u 2 subject to A 1,1 u 1 + A 1,2 u 2 b 1 A 2,1 u 1 + A 2,2 u 2 b 2 u 1, u 2 0 For the case in which the problem is neither primal nor dual feasible, Zionts goes to show that if the method cannot perform another step, it is indicative that there is no finite optimal feasible solution for the problem. In his proof, Zionts relies on the fact that when the problem is partitioned as above, if another iteration cannot be made, it is because the elements of A 1,2 are non-negative, and the elements of A 2,1 are non-positive.

19 Correctness (continued) 12 Recalling the partitioned problem: Minimize c1 T u 1 + c2 T u 2 subject to A 1,1 u 1 + A 1,2 u 2 b 1 A 2,1 u 1 + A 2,2 u 2 b 2 u 1, u 2 0 If you can t pivot in one problem, Zionts method then flips the roles of the primal and dual problems and tries again. Only when neither works do we conclude that the problem is infeasible.

20 13 Finiteness Outline Zionts stipulated that failure to halt indicates cycling; this results from the pidgeonhole principle, since there are only ( ) n+m m possible choices for the basis. He proposed that one must show that n + 1 iteration cycles are not possible given that n-teration cycles are not possible, since the basis cases of the impossibility of two- and three-iteration cycles had been proven.

21 14 Performance Outline Zionts performed empirical tests to determine the efficiency of his modified criss-cross method (which discovers unboundedness and infeasibility faster than his original criss-cross method) versus a baseline implementation of Dantzig s simplex method.

22 Performance (continued 1) 15 In order to ensure fairness, simplex method and criss-cross method implementations used many of the same subroutines (For example, the pivoting algorithm, once given a column and a row to work with, would be the same) and were run on problems generated using a uniform distribution; therefore, it most likely that this sampling of problems was diverse enough to ensure that there was no bias towards questions that were bad for the simplex method, favoring the use of a criss-cross method. However, these problems were very small due to constraints on computing resources; further empirical results hint that this criss-cross method fares no better than the simplex method.

23 Performance (continued 2) 16 Bonates and Maculan go further in their efficiency analysis, saying that although the combinatorial criss-cross variants encourage simpler proofs of finiteness, they hypothesize that non-combinatorial variants would fare far better in comparison to implementations of the simplex method.

24 Performance (continued 3) 17 Roos outlines a case in which Terlaky s pivoting rule for the criss-cross method takes exponential time. In dealing with the n-dimensional convex polytope P n, Roos finds a relationship between choosing the leaving variable and a mathematically equivalent Gray code; his result for the number of pivots required in this case asymptotically agree with Chvatal s earlier result, although there are differences in the estimation of the number of iterations, possibly due to there being something missing in the combinatorial equivalence of the convex polytopes discussed in either paper.

25 18 Questions? Outline Thank you for listening! Hopefully I can answer any questions you might have