introduction-lp-duality1

# In fact this deadlock indicates that the last

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Unformatted text preview: 11) we obtain: x3 x1 x5 z =1 =2 =1 = 13 + x2 − 2x2 + 5x2 − 3x2 + 3x4 − 2x6 − 2x4 + x6 + 2x4 − x4 − x6 . (9.13) It is now time to do the third iteration. First, we have to ﬁnd a variable of the right side of (9.13) whose increase would result in an increase of the objective z. But there is no such variable, as any increase of x2 , x4 or x6 would lower z. We are stuck. In fact, this deadlock indicates that the last solution is optimal. Why? The answer lies in the last line of (9.13): z = 13 − 3x2 − x4 − x6 . (9.14) The last solution (9.12) gives a value z = 13; proving that this solution is optimal boils down to prove that any feasible solution satisﬁes z ≤ 13. As any feasible solution x1 , x2 , . . . , x6 satisﬁes the inequalities x2 ≥ 0, x4 ≥ 0, x6 ≥ 0, then z ≤ 13 directly derives from (9.14). 9.2.2 The dictionaries More generally, given a problem Maximize ∑n=1 c j x j j Subject to: ∑n=1 ai j x j ≤ bi j xj ≥ 0 for all 1 ≤ i ≤ m for all 1 ≤ j ≤ n (9.15) we ﬁrst introduce the slack variables xn+1 , xn+2 , . . . , xn+m and we note the objective function z. That is, we deﬁne xn+i = bi − ∑n=1 ai j x j j z= ∑n=1 c j x j j for all 1 ≤ i ≤ m (9.16) In the framework of the Simplex Method, each feasible solution (x1 , x2 , . . . , xn ) of (9.15) is represented by n + m positive or null numbers x1 , x2 , . . . , xn+m , with xn+1 , xn+2 , . . . , xn+m deﬁned by (9.16). At each iteration, the Simplex Method goes from one feasible solution (x1 , x2 , . . . , xn+m ) to an other feasible solution (x1 , x2 , . . . , xn+m ), which is better in the sense that ¯¯ ¯ n ¯ ∑ c jx j > j=1 n ∑ c jx j. j=1 As we have seen in the example, it is convenient to associate a system of linear equations to each feasible solution. As a matter of fact, it allows to ﬁnd better solutions in an easy way. The technique is to translate the choices of the values of the variables of the right side of the system into the variables of the left side and in the objective function as well. These systems have been named dictionaries by J.E. Strum (1972). Thus, every dictionary associated to (9.15) is a system of equations whose variables xn+1 , xn+2 , . . . , xn+m and z are expressed in function of x1 , x2 , . . . , xn . These n + m + 1 variables are closely linked and every dictionary express these dependencies. 9.2. THE SIMPLEX METHOD 135 Property 9.1. Any feasible solution of the equations of a dictionary is also a feasible solution of (9.16) and vice versa. For example, for any choice of x1 , x2 , . . . , x6 and of z, the three following assertions are equivalent: • (x1 , x2 , . . . , x6 , z) is a feasible solution of (9.6); • (x1 , x2 , . . . , x6 , z) is a feasible solution of (9.11); • (x1 , x2 , . . . , x6 , z) is a feasible solution of (9.13). From this point of view, the three dictionaries (9.6), (9.11) and (9.13) contain the same information on the dependencies between the seven variables. However, each dictionary present this information in a speciﬁc way. (9.6) sugges...
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## This note was uploaded on 11/20/2013 for the course CS 101 taught by Professor Smith during the Fall '13 term at Mitchell Technical Institute.

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