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Outline

Outline. secrets equivalence between row operations & matrix multiplication simplex tableau in matrix form revised simplex method relationship with column generation. The Most Beautiful …. Maybe the Most Beautiful of All…. linear algebra. geometric properties. algebraic properties.

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Outline

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

  2. The Most Beautiful … 2

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

  4. To be at Home with the Material • 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., in some sense 4

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

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

  7. Equivalence Between Row Operations & Matrix Multiplication • what should Ebe to make “v basic in (3)”? v w xyb 7

  8. Simplex Tableau Minimization at some intermediate tableau with xB as basic variables initial tableau B-1 I B-1b B-1N initial tableau with columns of xB in the intermediate tableau separated out short form 8

  9. Simplex Procedure • 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 xenter as the xj with the smallest • 3 Identify the leaving variable xleave as xiwith the minimal ratio. Stop if the problem is unbounded; else continue. • 4 Identify aleave,enter from xenter and xleave. • 5 Pivot on element aleave,enter to update the whole tableau and go to step 1. 9

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

  11. Minimal Information for the Simplex Procedure • minimal information: the set of current basic variables xBto generate the WHOLE tableau • conceptually, from xB • known cB • known current basis Bcur and hence known (Bcur)-1 • any clever (i.e., lazy) method to get (Bnew)-1 from (Bcur)-1 without inverting Bnew every time? • the whole tableau from B-1 11

  12. Revised Simplex Algorithm • keeping track of xB and (Bcur)-1 • entering variable from reduced costs • leaving variable from minimum ratio test • finding (Bnew)-1 from (Bcur)-1 12

  13. Revised Simplex Algorithm • suppose we have the current basic variables xB,cur and the inverse of the basis (Bcur)-1 • known entities of the tableau: 13

  14. Revised Simplex Algorithm • to find the entering variable xe: calculate for non-basic variables • stop if all reduced costs are non-negative; else pick the first xj with negative reduced cost as the entering variable 14

  15. Revised Simplex Algorithm • to find the leaving variable xl • known column (Bcur)-1Aeof the entering variable xe • with known RHS, execution of minimal ratio test to determine the leaving variable xl (if available) • pivoting on al,e to turn column e into (0, .., 0, 1, 0.., 0)T, where “1” occurs at the lth row 15

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

  17. Revised Simplex Algorithm • 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

  18. Revised Simplex Algorithm • to find (Bnew)-1 from (Bcur)-1 • claim: (Bnew)-1 = E(Bcur)-1 row operations pre-multiplied by E 18

  19. Example of Revised Simplex Algorithm • max 2x1+x2 min 2x1x2, • s.t. –x1+x2 2, • x2 4, • x1+x2 8, • x1 6, • x1, x2 0. 19

  20. Solving the Exampleby Simplex Method 20

  21. Solving the Exampleby Simplex Method 21

  22. Solving the Exampleby Simplex Method 22

  23. Example of Revised Simplex Algorithm 23

  24. Relationship Between Revised Simplex and Column Generation • 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

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