Optimization. Lecture08 12/06/11

Transcription

1 Optimization Lecture08 12/06/11

2 The RevisedSimplex Algorithm Assume againthatwearegivenan LP in canonicalform, whichwepad with slackvariables. Letusplacethe corresponding columns on the left of the initial tableau: As wehavearguedbefore, after literationsofthesimplex algorithm, we get: 2

3 The RevisedSimplex Algorithm We callthispartofthetableauin thel-th iterationcarry l Itisenoughtostorethispartofthetableauandtheordered set ofbasiscolumnstoexecutethesimplex algorithm: Simplex with Column Generation (1) [Pricing] Computerelative costs oneata time untilwe find a positive one, say, orconcludethatthesolutionisoptimal. (2) [ColumnGeneration] GeneratecolumnX ofthel-th tableauby. Determinethepivotelement, say, bytheususalratiotest min or discover that the optimal value is unbounded. (3) [Pivot]Update CARRY l to CARRY l byperformingtherowoperations determinedbycolumn whenpivotingon. (4) [Update Basis] Replacethe -thelement of by, the index of the new basis column. 3

4 The RevisedSimplex Algorithm Is the revised Simplex algorithm faster than the standard implementation that updates the full tableau in each step? In theworstcase, theadvantageissmall. However, in practice two reasons speak for the revised method: 1. The pricing operation(1) only needs to be performed untilwefind thefirstpositive value. Iftheaverage numberofcolumnsweneedtoexamineissmall, thisis fast. 2. Pricingusestherows oftheoriginal tableau. Ifthe original matrix issparse(whichitfrequentlyisin practice), these operations are fast. In some cases, we don tevenhavetogenerate(andstore) explicitely (e.g., adjacency matrix of some graph). 4

5 In most practical applications, Simplex turns out to be a very efficient algorithm. But what about provable runtime guarantees? Let us first recall some geometric facts about the canonical form LP max. s.t., where isofdimension. Here, non-negativity constraints(ifpresent) arepartofthesystem. If the systemhasfullrank, thenno satisfies 1constraints with equality. 5

6 In thecorrespondingpolytope, eachvertexisthenthe intersectionofexactly hyperplanes definingthehalfspaces givenbytheconstraints. Wesaythat issimple. We ve discussed(somewhat informally), that a pivot of the Simplex algorithm corresponds to moving from one vertex of toa neighboringonevia an edge. Recall thatan edgeisthe intersectionof 1defininghyperplanes. Algebraically, in a simple polytopea vertexisdefinedby constraintswhichwerequiretobetight. A pivotstep corresponds to loosening one constraint and selecting another one to become tight. 6

7 In a simple polytopewedon thavetoworryabouttheissue of degeneracy and so the Simplex pivot step is well-defined bytheselectionofa columntoenterthebasis. In our geometric view, this is equivalent to selecting the tight constraint to loosen. Ifconstraintsareindexed1,,, Bland spivotrule corresponds to loosening the tight constraint with smallest index which will lead to an improvement of the objective function value. Wearenowreadytoconstructa polytopeon whichthe Simplex algorithm with Bland s pivot rule needs an exponential number of pivot steps. 7

8 We definepolytope by: with 1/3. isa Klee-Mintycubeofdimension. Constraintsareindexed1,,2 asshownabove. Klee-Minty cubesareperturbationsof -dimensional hypercubes, which aredefinedby0 1for 1,,. 8

9 Theorem 11 The Simplex algorithm with Bland s pivot rule visits every vertexoftheklee-mintycube whensolving (1) max. s.t., ifstartedat 0,0,,0 or (2) min. s.t., ifstartedat 0,,0,1. Thus, the Simplex algorithm with Bland s rule will perform 2 1pivotstepson an LP ofdimension in theworst case. Similar lower bounds can be constructed for all the other pivot rules we have discussed. 9

10 Onemightsuspectthatthelowerboundin Theorem 11 can be extended to hold independently of any specific pivoting rule. However, this does not work. Wesaythedistancebetweena pair ofverticesofpolytope istheminimumnumberofpivots(oredges) weneedtoget fromonetotheother. The diameter ofthe polytope is the maximum distance between any pair of vertices. Theorem 12 For any -dimensional polytope with defininghalfspaces,. 10

11 For almost50 years, itwas believedthatin factitwas true that. This famoushirsch conjecturewas finallydisprovedin Notightboundsareknownatthis point So, ifwecan tprovethatall pivotrulesarebad, canweprove thatsomepivotruleisgood? The answertothisis no. To date, we don t know of any deterministic pivot rule that performs a polynomial(in the size of the input representation) numberofpivotstepsin theworstcase. However, there are theoretical results explaining the efficiency of Simplex in practice. 11

12 ThereareseveralresultsprovingthatSimplex performsan expected polynomial number of pivots when the input comes from some suitable probability distribution. It is generally difficult to judge whether a given probability distribution is in any way realistic. This is particularly so as we usually don t know all possible application scenarios of our algorithmatthetime weanalyzeit. A relatively recent approach to combine the benefits of worst-case and average-case analysis is what we call smoothed analysis. 12