Tut5
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Degeneracy in linear programming occurs when a basic feasible solution has a right-hand side coefficient of zero, allowing the objective value and solution to remain unchanged after a pivot. This introduces the possibility of cycling and an infinite simplex method. However, researchers have developed approaches like perturbation to guarantee the simplex method terminates by preventing degenerate bases or ensuring alternate optima are still feasible for the original problem.
![Cycling
• If a sequence of pivots starting from some basic feasible solution
ends up at the exact same basic feasible solution, then we refer to this
as “cycling.” If the simplex method cycles, it can cycle forever.
• Klee and Minty [1972] gave an example in which the simplex algorithm
really does cycle. Here is their example, with the pivot elements
outlined.
z x1 x2 x3 x4 x5 x6 x7 RHS
1 - 3/4 20 - 1/2 6 0 0 0 3
Initial tableau
0 1/4 -8 -1 9 1 0 0 0
0 1/2 -12 - 1/2 3 0 1 0 0
0 0 0 1 0 0 0 1 1
z x1 x2 x3 x4 x5 x6 x7 RHS
1 0 -4 -3 1/2 33 3 0 0 3
0 1 -32 -4 36 4 0 0 0 After 1 pivot
0 0 4 1 1/2 -15 -2 1 0 0
0 0 0 1 0 0 0 1 1 7](https://image.slidesharecdn.com/tut5-120528101359-phpapp02/85/Tut5-7-320.jpg)
Tut5
- 1. Degeneracy in Linear Programming Sorry, Tim. But Degeneracy? Students the topic is just as at MIT shouldn’t learn interesting. It’s about degeneracy. And about degeneracy in I heard that 15.053 Linear Programming. students have already I heard that studied convicts’ sex. today’s tutorial is all about Ellen DeGeneres Reverend Jerry Falwell Ellen DeGeneres Actually, they studied convex Tim, the turkey sets. 1
- 2. What is degeneracy? • As you know, the simplex algorithm starts at a corner point and moves to an adjacent corner point by increasing the value of a non-basic variable xs with a negative value in the z-row (objective function). • Typically, the entering variable xs does increase in value, and the objective value z improves. But it is possible that that xs does not increase at all. It will happen when one of the RHS coefficients is 0. • In this case, the objective value and solution does not change, but there is an exiting variable. This situation is called degeneracy. A basic feasible solution is z x1 x2 x3 x4 called degenerate if one of its RHS coefficients 1 3 -2 0 0 = 2 (excluding the objective value) is 0. 0 -3 3 1 0 = 6 This bfs is degenerate. 0 -4 2 0 1 = 0 2 2
- 3. No, Nooz**. This tutorial has many Great. I now more slides. Degeneracy adds know what complications to the simplex degeneracy is. algorithm. And if you understand Now we can move what occurs under degeneracy, you on to other really understand what is going on matters. with the simplex algorithm. ** As you know, “No, Nooz” is good news.” Ollie, the computationally wise owl. Nooz, the most Professor Orlin trusted name in fox. apologizes for this bad pun, but feels that he could not resist the temptation. Cleaver 3
- 4. Degeneracy and Basic Feasible Solutions • We may think that every two distinct bases lead to two different solutions. This would be true if there was no degeneracy. But with degeneracy, we can have two different bases, and the same feasible solution. z x1 x2 x3 x4 z x1 x2 x3 x4 1 3 -2 0 0 = 2 1 2 3 -2 0 0 1 0 = 2 0 -3 3 1 0 = 6 0 -3 -3/2 3 0 1 0 -3/2 = 6 0 -4 2 0 1 = 0 2 0 -1 -1/2 2 1 0 1 1/2 = 0 2 Both tableus correspond to the We now pivot on the “2” in same feasible solution with z = 2, Constraint 2 and obtain a second x1 = x2 = x4 = 0; x3 = 6. But the tableau. basic variables and the coefficients of the two tableaus are different. 4
- 5. z x1 x2 x3 x4 z x1 x2 x3 x4 1 3 -2 0 0 = 2 1 2 3 -2 0 0 1 0 = 2 0 -3 3 1 0 = 6 0 -3 -3/2 3 0 1 0 -3/2 = 6 0 -4 2 0 1 = 0 2 0 -1 -1/2 2 1 0 1 1/2 = 0 2 Not only are the two tableaus different, but the second tableau satisfies the optimality conditions. This means that the bfs is optimal, even though we were not aware that the bfs was optimal when we looked at the first tableau. But this all seems very technical. Is it Tim, it turns out to be very really important? important. We can show that the simplex algorithm is finite and guaranteed to be valid when there is no degeneracy. When there is degeneracy, we have to modify the algorithm to guarantee finiteness. Tim, the turkey Ollie 5
- 6. The Finiteness of the Simplex Algorithm when there is no degeneracy Recall that the simplex algorithm tries to increase a non-basic variable xs. If there is no degeneracy, then xs will be positive after the pivot, and the objective value will improve. Recall also that each solution produced by the simplex algorithm is a basic feasible solution with m basic variables, where m is the number of constraints. There are a finite number of ways of choosing the basic variables. (An upper bound is n! / (n-m)! m! , which is the number of ways of selecting m basic variables out of n.) So, the simplex algorithm moves from bfs to bfs. And it never repeats a bfs because the objective is constantly improving. This shows that the simplex method is finite, so long as there is no degeneracy. 6
- 7. Cycling • If a sequence of pivots starting from some basic feasible solution ends up at the exact same basic feasible solution, then we refer to this as “cycling.” If the simplex method cycles, it can cycle forever. • Klee and Minty [1972] gave an example in which the simplex algorithm really does cycle. Here is their example, with the pivot elements outlined. z x1 x2 x3 x4 x5 x6 x7 RHS 1 - 3/4 20 - 1/2 6 0 0 0 3 Initial tableau 0 1/4 -8 -1 9 1 0 0 0 0 1/2 -12 - 1/2 3 0 1 0 0 0 0 0 1 0 0 0 1 1 z x1 x2 x3 x4 x5 x6 x7 RHS 1 0 -4 -3 1/2 33 3 0 0 3 0 1 -32 -4 36 4 0 0 0 After 1 pivot 0 0 4 1 1/2 -15 -2 1 0 0 0 0 0 1 0 0 0 1 1 7
- 8. Cycling Example Continued z x1 x2 x3 x4 x5 x6 x7 RHS 1 0 0 -2 18 1 1 0 3 0 1 0 8 -84 -12 8 0 0 After 2 pivots 0 0 1 3/8 -3 3/4 - 1/2 1/4 0 0 0 0 0 1 0 0 0 1 1 z x1 x2 x3 x4 x5 x6 x7 RHS 1 1/4 0 0 -3 -2 3 0 3 After 3 pivots 0 1/8 0 1 -10 1/2 -1 1/2 1 0 0 0 - 3/64 1 0 3/16 1/16 - 1/8 0 0 0 - 1/8 0 0 10 1/2 1 1/2 -1 1 1 z x1 x2 x3 x4 x5 x6 x7 RHS 1 - 1/2 16 0 0 -1 1 0 3 0 -2 1/2 56 1 0 2 -6 0 0 After 4 pivots 0 - 1/4 5 1/3 0 1 1/3 - 2/3 0 0 0 2 1/2 -56 0 0 -2 6 1 1 8
- 9. Cycling Example Continued z x1 x2 x3 x4 x5 x6 x7 RHS 1 -1 3/4 44 1/2 0 0 -2 0 3 0 -1 1/4 28 1/2 0 1 -3 0 0 After 5 pivots 0 1/6 -4 - 1/6 1 0 1/3 0 0 0 0 0 1 0 0 0 1 1 z x1 x2 x3 x4 x5 x6 x7 RHS 1 - 3/4 20 - 1/2 6 0 0 0 3 0 1/4 -8 -1 9 1 0 0 0 After 6 pivots 0 1/2 -12 - 1/2 3 0 1 0 0 0 0 0 1 0 0 0 1 1 And Klee and Minty said “The first shall be last and the last shall be first”, and they saw that their example of cycling was good. 9 Cleaver
- 10. Is the simplex method finite? Tim, there are several We just proved that approaches to guaranteeing that the simplex method is the simplex method will be finite, finite if there is no including one developed by degeneracy. But how Professors Magnanti and Orlin. do we know that the We’ll show the standard simplex method will approach to guaranteeing terminate if there is finiteness, called the degeneracy? perturbation approach. Tim, the turkey Ollie 10
- 11. The Perturbation Approach • In the perturbation approach, we change the RHS by just a little. If the vector of right hand sides is b1, …, bm, we replace it by d1, …, dm, where dj is very close to bj. We want two properties to hold: 1. No basis is degenerate for the perturbed problem. 2. Any basis that is feasible for the perturbed problem is also feasible for the original problem. It turns out that if (1) and (2) are true, than an optimal basis for the perturbed problem will be optimal for the original problem. Before the perturbation. After the perturbation. z x1 x2 x3 x4 z x1 x2 x3 x4 1 3 -2 0 0 = 2 1 3 -2 0 0 = 2 0 -3 3 1 0 = 6 0 -3 3 1 0 = 6+ε1 0 -4 2 0 1 = 0 2 0 -4 2 0 1 = ε2 0 2 11
- 12. More on Perturbations I get it. So, we may let ε = .001. Actually, the perturbations are Nobody does these absurdly small. For perturbations in example, for a 100 practice; it would be variable problem, very impractical. It we may replace, bj was developed to by bj + 2-10,000j. guarantee that the simplex method would be finite. And for this purpose they work great! Nooz, the most trusted name in fox. Ollie 12
- 13. Degeneracy and the Simplex Algorithm The simplex method The simplex method without degeneracy with degeneracy The solution changes after each The solution may stay the same pivot. The objective value strictly after a pivot. The objective value improves after a pivot. may stay the same. The simplex method is guaranteed The simplex method may cycle and to be finite. be infinite. But it becomes finite if we use the perturbation approach or several other approaches. Two different tableaus in canonical It is possible that there are two form give two different solutions. different sets of basic variables that give the same solution. 13
- 14. Degeneracy vs. Alternative Optima Finally, degeneracy is similar to but different from the condition for alternate optima. In degeneracy, one of the z x1 x2 x3 x4 RHS values is 0. For alternate optima, in an optimal tableau 1 0 4 0 0 = 8 one of the non-basic z-row coefficients is 0. 0 1 3 1 0 = 6 0 -1 2 0 1 = 2 The optimal solution is: z = 8, x1 = x2 = 0, x3 = 6, and x4 = 8. x1 is nonbasic, and its z-row coefficient is 0. Increasing x1 (and adjusting x3 and x4) does not change z, and so the solution value remains optimal. Ollie 14
- 15. Summary • Degeneracy is important because we want the simplex method to be finite, and the generic simplex method is not finite if bases are permitted to be degenerate. • In principle, cycling can occur if there is degeneracy. In practice, cycling does not arise, but no one really knows why not. Perhaps it does occur, but people assume that the simplex algorithm is just taking too long for some other reason, and they never discover the cycling. • Researchers have developed several different approaches to ensure the finiteness of the simplex method, even if the bases can be degenerate. One such method is called the perturbation approach. The perturbation approach (in the form described here) is not practical, but it serves its purpose. It does give a way of doing simplex pivoting that is guaranteed to be finite. 15
- 16. Last Slide And that is all for our tutorial on degeneracy. We hope to see you again for our next tutorial. And I want to thank Tim, Ollie, and Nooz for sharing all of their insights on degeneracy. I also want to thank Ellen DeGeneres for making a cameo appearance. Cleaver 16