secn6 - 6 The simplex method We next formalize the method...

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6 The simplex method We next formalize the method we developed in the previous section. We again consider a general linear program in standard equality form: maximize c T x subject to Ax = b x 0 . As before, we introduce a new variable z to keep track of the objective value, so the system of equations defining the linear program is (6.1) z - c T x = 0 Ax = b. We break down each iteration of the solution algorithm, which is called the simplex method , into a sequence of steps. Step 0: Initialization. We (somehow) find an initial feasible basis B , form a list N consisting of its complement, and find the corresponding tableau: (6.2) z - j N ¯ c j x j = ¯ v x i + j N ¯ a ij x j = ¯ b i ( i B ) . This system is equivalent to the original constraints (6.1), and from it, we can read off the corresponding basic feasible solution: basic variables x i = ¯ b i ( i B ) nonbasic variables x j = 0 ( j N ) objective value z = ¯ v. Step 1: Check optimality. If each reduced cost satisfies ¯ c j 0 (for all indices j N ), then we stop: the current solution is optimal . To see this, notice that any feasible solution satisfies z = ¯ v + j N ¯ c j x j ¯ v, since ¯ c j 0 and x j 0 (for all j N ). Thus no feasible solution can have objective value larger than ¯ v . But the current solution attains this value, so it must be optimal. Step 2: Choose entering index. If we don’t stop in Step 1, we can find an entering index k N with corresponding reduced cost ¯ c k > 0. Consider now the effect of increasing the value of the corresponding variable x k (called the 36
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entering variable ) away from its current value zero to a new value t 0. At the next iteration of the simplex method, if the value of x k is nonzero, the index k must have entered the basis: hence the term “entering index”. If we keep the values of other nonbasic variables fixed at zero, the objective value z = ¯ v + j N ¯ c j x j will increase, helping our aim of trying to maximize it. We need to satisfy all
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