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# mma7 - Math 1b Practical March 9 2009...

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Math 1b Practical March 9, 2009 LinearProgramming[c,A,b] solves what I would call the transpose-dual-canonical form LP problem -- it finds a vector x that minimizes the value of cx subject to Ax >= b and x >= 0 . In In[1], I define LP[c,A,b] which solves the canonical form LP problem: maximize cx subject to Ax <= b , x >= 0. If A is as in Out[2], c = {1,1,1,1} and b = {11,8,9}, a vector x_0 that maximizes cx in the canonical form problem is given in Out[3] as {2.7, 0, 1.1, 1.5} . The value of cx is 5.3, as shown in Out[4]. A vector y_0 that minimizes yb in the dual canonical form problem is {0.1, 0.3, 0.2} . The value of yb is 5.3. This checks the Duality Theorem. Let A be the upper-left 3x4 submatrix of Out[7] and consider the two-person zero-sum game defined by A . An optimal strategy for player I is to pick row 1 with probability 1/7 and row 2 with probability 6/7. An optimal trategy for player II is to pick column 2 with probability 4/7 and column 4 with probability 3/7. For these strategies, the expected win for Player I is 25/7.

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