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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

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2 The Most Beautiful …

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

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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

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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)

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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)

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

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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

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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

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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.

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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 -1 11 Minimal Information for the Simplex Procedure xRHS B. V. xBxB B -1 AB -1 b

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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 ) -1 12 Revised Simplex Algorithm xRHS B. V. xBxB B -1 AB -1 b

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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

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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

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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

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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 =

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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

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to find (B new ) -1 from (B cur ) -1 claim: (B new ) -1 = E(B cur ) -1 18 Revised Simplex Algorithm xBxB xNxN xSxS -zRHS B. V.………1… xBxB I…(B cur ) -1 0... xBxB xNxN xSxS -zRHS B. V.………1… xBxB I…(B new ) -1 0... row operations pre-multiplied by E

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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 0. 19 Example of Revised Simplex Algorithm

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20 Solving the Example by Simplex Method

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21 Solving the Example by Simplex Method

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22 Solving the Example by Simplex Method

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23 Example of Revised Simplex Algorithm Example of Revised Simplex Algorithm

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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

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