15.081J/6.251J Introduction to Mathematical Programming. Lecture 6: The Simplex Method II

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1 15081J/6251J Intoduction to Mathematical Pogamming ectue 6: The Simplex Method II

2 1 Outline Revised Simplex method Slide 1 The full tableau implementation Anticycling 2 Revised Simplex Initial data: A, b, c Slide 2 1 Stat with basis B = [A B(1),, A B(m) ] and B 1 2 Compute p = c B B 1 c j = c j p A j If c j 0; x optimal; stop Else select j : c j < 0 3 Compute u = B 1 A j Slide 3 4 θ If u 0 cost unbounded; stop Else x B(i) u B(l) = min = 1 i m,u i >0 u i u l 5 Fom a new basis B by eplacing A B(l) with A j 6 y j = θ, y B(i) = x B(i) θ u i Slide 4 7 Fom [B 1 u] 8 Add to each one of its ows a multiple of the lth ow in ode to make the last column equal to the unit vecto e l 1 The fist m columns is B 21 Example min x 1 + 5x 2 2x 3 st x 1 + x 2 + x 3 4 x 1 2 x 3 3 3x 2 + x 3 6 x 1, x 2, x 3 0 Slide 5 Slide 6 1

3 B = {A 1, A 3, A 6, A 7 }, BFS: x = (2, 0, 2, 0, 0, 1, 4) c = (0, 7, 0, 2, 3, 0, 0) B B =, = (u 1, u 3, u 6 (, u 7 ) = ) B 1 A 5 = (1, 1, 1, 1) θ 2 = min 1, 1 1, 4 1 = 1, l = 6 l = 6 (A 6 exits the basis) Slide [B 1 u] = B = Pactical issues Numeical Stability B 1 needs to be computed fom scatch once in a while, as eos accumulate Slide 8 Spasity B 1 is epesented in tems of spase tiangula matices 3 Full tableau implementation c B 1 b B c c B 1 A B B 1 b B 1 A Slide 9 o, in moe detail, c B x B x B(1) x B(m) c 1 c n B 1 A 1 B 1 A n 2

4 31 Example min 10x 1 12x 2 12x 3 st x 1 + 2x 2 + 2x x 1 + x 2 + 2x x 1 + 2x 2 + x 3 20 x 1, x 2, x 3 0 Slide 10 min 10x 1 12x 2 12x 3 st x 1 + 2x 2 + 2x 3 + x 4 = 20 2x 1 + x 2 + 2x 3 + x 5 = 20 2x 1 + 2x 2 + x 3 + x 6 = 20 x 1,,x 6 0 BFS: x = (0, 0, 0, 20, 20, 20) B=[A 4, A 5, A 6 ] Slide 11 x 1 x 2 x 3 x 4 x 5 x x 4 = x 5 = 20 2* x 6 = c = c c B B 1 A = c = ( 10, 12, 12, 0, 0, 0) Slide x 4 = 10 x 1 = 10 x 6 = 0 x 1 x 2 x 3 x 4 x 5 x * Slide 13 x 1 x 2 x 3 x 4 x 5 x x 3 = x 1 = x 6 = * Slide 14 3

5 x 1 x 2 x 3 x 4 x 5 x x 3 = x 1 = x 2 = Slide 15 x 3 B = (0,0,10) A = (0,0,0) E = (4,4,4) C = (0,10,0) D = (10,0,0) x 1 x 2 4 Compaison of implementations Full tableau Revised simplex Memoy O(mn) O(m 2 ) Wost-case time O(mn) O(mn) Best-case time O(mn) O(m 2 ) Slide 16 5 Anticycling 51 Degeneacy in Pactice Does degeneacy eally happen in pactice? n x ij = 1 j=1 n x ij = 1 i=1 x ij 0 Slide 17 4

6 n! vetices: Fo each vetex 2 n 1 n n 2 diffeent bases (n = 8) fo each vetex 33, 554, 432 bases 52 Petubations 521 Theoem (P)min c x (P ǫ )min c x ǫ st Ax = b st ǫ 2 Ax = b + ǫ m x 0 x 0 Slide 18 Slide 19 ǫ 1 > 0: fo all 0 < ǫ < ǫ 1 ǫ Ax = b + ǫ m is non-degeneate x Poof et B 1,, B be all the bases ǫ b 1 + B 11ǫ + + B 1mǫ m B 1 b + = ǫ m b m + B m1 ǫ + + B mm ǫ m Slide 20 whee: B 11 B 1m b1 B 1, B 1 = b = B B m1 mm b m Slide 21 b + B + + B i i1 θ im θ m is a polynomial in θ Roots θ i, 1, θ i, 2,, θ i,m If ǫ = θ i, 1,, θ i,m b + B i 1ǫ + + B ǫ m i im = 0 et ǫ 1 the smallest positive oot 0 < ǫ < ǫ 1 all RHS ae = non-degeneacy 0 5

7 53 exicogaphy u is lexicogaphically lage than v, u > v, if u = v and the fist nonzeo component of u v is positive Slide 22 Example: (0, 2, 3, 0) > (0, 2, 1, 4), (0, 4, 5, 0) < (1, 2, 1, 2) 54 exicogaphy-petubation 541 Theoem et B be a basis of Ax = b, x 0 Then B is feasible fo Ax = b + (ǫ,,ǫ m ), x 0 fo sufficiently small ǫ if and only if Slide 23 u i = (b i, B i1,,b im ) > 0, i B 1 = (B ij ) (B 1 b) i = (b i ) 542 Poof B is feasible fo petubed poblem B 1 (b + (ǫ,, ǫ m ) ) 0 b i + B i1 ǫ + + B im ǫ m 0 i Fist non-zeo component of u i = (b i, B i1,,b im ) is positive i 55 Summay 1 We stat with: (P) : Ax = b, x 0 2 We intoduce (P ǫ ): Ax = b + (ǫ,,ǫ m ), x 0 Slide 24 Slide 25 3 A basis is feasible + non-degeneate in (P ǫ ) u i > 0 in (P) 4 If we maintain u i > 0 in (P) (P ǫ ) is non-degeneate Simplex is finite in (P ǫ ) fo sufficiently small ǫ 56 exicogaphic pivoting ule 1 Choose an enteing column A j abitaily, as long as c j < 0; u = B 1 A j Slide 26 2 Fo each i with u i > 0, divide the ith ow of the tableau (including the enty in the zeoth column) by u i and choose the lexicogaphically smallest ow If ow l is lexicogaphically smallest, then the lth basic vaiable x B(l) exits the basis 6

8 561 Example j = 3 Slide x B(1) /u 1 = 1/3 and x B(3) /u 3 = 3/9 = 1/3 We divide the fist and thid ows of the tableau by u 1 = 3 and u 3 = 9, espectively, to obtain: 1/3 0 5/3 1 1/3 0 7/9 1 Since 7/9 < 5/3, the thid ow is chosen to be the pivot ow, and the vaiable x B(3) exits the basis 562 Uniqueness Why lexicogaphic pivoting ule always leads to a unique choice fo the exiting vaiable? Slide 28 Othewise, two ows in tableau popotional ank(b 1 A) < m ank(a) < m 57 Theoem If simplex stats with all the ows in the simplex tableau, othe than the zeoth ow, lexicogaphically positive and the lexicogaphic pivoting ule is followed, then (a) Evey ow of the simplex tableau, othe than the zeoth ow, emains lexicogaphically positive thoughout the algoithm (b) The zeoth ow stictly inceases lexicogaphically at each iteation (c) The simplex method teminates afte a finite numbe of iteations 58 Smallest subscipt pivoting ule 1 Find the smallest j fo which the educed cost c j is negative and have the column A j ente the basis 2 Out of all vaiables x i that ae tied in the test fo choosing an exiting vaiable, select the one with the smallest value of i Slide 29 Slide 30 7

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