1 Outline  secrets  equivalence between row operations & matrix multiplication  simplex tableau in matrix form  revised simplex method  relationship with column generation

2 The Most Beautiful …

 linear algebra 3 Maybe the Most Beautiful of All… algebraic properties geometric properties matrix properties

 familiar with the equivalence  be lazy  keeping and working only with the essence  e.g., how much information to carry in solving  (sometimes) use logic, not eyes  e.g., 4 To be at Home with the Material in some sense

 let E = and A =  EA = 5 Equivalence Between Row Operations & Matrix Multiplication making w basic in (1) w x y b (1) (2) row operations: (a) (1) = (1)/3 (b) (2) = (2)-2(1) (1) (2)

 let E = and A =  EA = 6 Equivalence Between Row Operations & Matrix Multiplication w x y b making y basic in (2) (1) (2) row operations: (1) (2) (a) (2) = (2)/4 (b) (1) = (1)+8(2)

 what should E be to make “v basic in (3)”? 7 Equivalence Between Row Operations & Matrix Multiplication v w x y b

8 Simplex Tableau xxSxS -zRHS B. V.cTcT 010 xSxS AI0b initial tableau at some intermediate tableau with x B as basic variables (x B x N )xSxS -zRHS B. V.010 xSxS (B N)I0b initial tableau with columns of x B in the intermediate tableau separated out short form xRHS B. V. xBxB B -1 AB -1 b Minimization xBxB xNxN xSxS -zRHS B. V.1 xBxB 0 I B -1 N B -1 B -1 b

 an iteration before minimal:  1  Find the smallest if all are non-negative, the minimal has been found and stop; else continue.  2  Identify the entering variable x enter as the x j with the smallest  3  Identify the leaving variable x leave as x i with the minimal ratio. Stop if the problem is unbounded; else continue.  4  Identify a leave,enter from x enter and x leave.  5  Pivot on element a leave,enter to update the whole tableau and go to step 1 . 9 Simplex Procedure xRHS B. V. xBxB B -1 AB -1 b

 no guarantee that the smallest gives the least number of iterations  can arbitrarily pick an x j with negative reduced cost as the entering variable  no need to update the whole tableau 10 Inefficient Simplex Procedure opt.

 minimal information: the set of current basic variables x B to generate the WHOLE tableau  conceptually, from x B  known c B  known current basis B cur and hence known (B cur ) -1  any clever (i.e., lazy) method to get (B new ) -1 from (B cur ) -1 without inverting B new every time?  the whole tableau from B Minimal Information for the Simplex Procedure xRHS B. V. xBxB B -1 AB -1 b

 keeping track of x B and (B cur ) -1  entering variable from reduced costs  leaving variable from minimum ratio test  finding (B new ) -1 from (B cur ) Revised Simplex Algorithm xRHS B. V. xBxB B -1 AB -1 b

 suppose we have the current basic variables x B,cur and the inverse of the basis (B cur ) -1  known entities of the tableau: 13 Revised Simplex Algorithm x B,cur x N,cur -zRHS B. V.0?1 xBxB I?0(B cur ) -1 b

 to find the entering variable x e : calculate for non- basic variables  stop if all reduced costs are non-negative; else pick the first x j with negative reduced cost as the entering variable 14 Revised Simplex Algorithm x B,cur x N,cur zRHS B. V.0? xBxB I?0(B cur ) -1 b

 to find the leaving variable x l  known column (B cur ) -1 A  e of the entering variable x e  with known RHS, execution of minimal ratio test to determine the leaving variable x l (if available)  pivoting on a l,e to turn column e into (0,.., 0, 1, 0.., 0) T, where “1” occurs at the lth row 15 Revised Simplex Algorithm x B,cur xexe x N,cur -zRHS B. V.0?1 xBxB I(B cur ) -1 A  e ?0(B cur ) -1 b

 what should E be to make “v basic in (3)”? 16 Equivalence Between Row Operations & Matrix Multiplication v w x y b making v basic in (3) row operations: (a) (3) = (3)/2 (b) (2) = (2)+(3) (c) (1) = (1)-2(3) v w x y b elementary matrix E =

 to find the elementary matrix E that turns A  e into  row operations are equivalent to pre- multiplying by matrix E, where E = I except the lth column, 17 Revised Simplex Algorithm

 to find (B new ) -1 from (B cur ) -1  claim: (B new ) -1 = E(B cur ) Revised Simplex Algorithm xBxB xNxN xSxS -zRHS B. V.………1… xBxB I…(B cur ) xBxB xNxN xSxS -zRHS B. V.………1… xBxB I…(B new ) row operations pre-multiplied by E

 max 2x 1 +x 2  min  2x 1  x 2,  s.t. –x 1 +x 2  2,  x 2  4,  x 1 +x 2  8,  x 1  6,  x 1, x 2  Example of Revised Simplex Algorithm

20 Solving the Example by Simplex Method

21 Solving the Example by Simplex Method

22 Solving the Example by Simplex Method

23 Example of Revised Simplex Algorithm Example of Revised Simplex Algorithm

 revised simplex method  no need to generate the whole tableau  only generating columns when searching for first negative reduced cost  column generation method  generating column of non-basic variables only when necessary  usually with additional complexity to determine the best entering variable for a given situation 24 Relationship Between Revised Simplex and Column Generation