unboundedness - Introduction to Optimization (550.361)...

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Introduction to Optimization (550.361) Direction of Unboundedness Example 1 Use the simplex method to solve max z = 3 x 1 - x 2 subject to - 3 x 1 + 3 x 2 + x 3 = 6 - 8 x 1 + 4 x 2 + x 4 = 4 x j 0 j The current basic variables are { x 3 , x 4 } . The current solution is X = (0 , 0 , 6 , 4) and the current objective function value is z = 0. We see that if we select x 1 to enter the basis, we should be able to improve the objective value. The issue is, what value should we pick for x 1 ? If we select x 1 too large, one of the remaining variables could become negative. Let’s set x 1 = α and keep x 2 nonbasic (i.e., x 2 = 0). Then the equations above say that z new = 3 α and the new variable values are given by x 1 = α x 2 = 0 x 3 = 6 + 3 α x 4 = 4 + 8 α = 0 0 6 4 + α 1 0 3 8 . That is X new = X old + α d . Here d is the direction we wish to use to travel away from X old and α is the stepsize (distance) we move in that direction. The issue of how large to make
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This note was uploaded on 01/21/2011 for the course IEOR 162 taught by Professor Zhang during the Fall '07 term at University of California, Berkeley.

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unboundedness - Introduction to Optimization (550.361)...

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