### Table 3: Primal Dual algorithm

"... In PAGE 9: ... The running time increases with the accuracy needed. The next Theorem states the running time and the correctness of the algorithm shown in Table3 . The proof is omitted here due to lack of space, but is similar to the one in [31].... In PAGE 9: ... Theorem 4. The algorithm in Table3 computes a (1 ) 3 optimal solution to the ow scaling problem in time polynomial in Q; L; n and 1 , where Q is the number of com- modities, L is the number of constraining sets, and n is the number of nodes. 6.... ..."

### Table 9: Comparison of the primal barrier method and the primal-dual method for lNa13

2000

"... In PAGE 21: ... Compared with the primal barrier method in [And96b], there is a signi cant reduction in the number of itera- tions and in CPU time. This is shown in Table9 . The primal-dual algorithm also obtains signi cantly more zero norms in the optimal solution.... In PAGE 22: ... As shown in Table 10, the number of zero norms varies from 62 to 96 percent of the total number of terms. Comparison with [And96b] con rms the observations from Table9 : for the primal-dual method the iteration count is signi cantly reduced and increases very slowly with the problem size. The CPU time is reduced by a factor 4 or more, and we are able to solve larger instances of the problem.... ..."

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### Table 2. Overall e#0Eciency of the primal-dual cutting-plane method.

1996

"... In PAGE 16: ...Table2... In PAGE 16: ... All cuts mentioned in the second column of the table are appended to the restricted master problem but some of them are eliminated in later iterations because they are inactive #28see #5B12#5D for details#29. The remaining columns of Table2 report the number of iterations of the cutting-plane method #28Outer#29 and the number of interior-point iterations #28Inner#29. In the latter wehave distinguished the iterations needed to reach the approximate analytic center #28to be saved for the future warm start#29, AC, the iterations to reach the desired accuracy of solution to the restricted master problem, Opt, and their sum, respectively.... In PAGE 16: ... In the latter wehave distinguished the iterations needed to reach the approximate analytic center #28to be saved for the future warm start#29, AC, the iterations to reach the desired accuracy of solution to the restricted master problem, Opt, and their sum, respectively. From the results collected in Table2 one can see that we really deal with nontrivial warm- start examples. The sizes of the restricted master problems always reach tens of thousand columns with, on the average, thousands of new cuts to be accommodated at every reoptimiza- tion.... In PAGE 16: ... The sizes of the restricted master problems always reach tens of thousand columns with, on the average, thousands of new cuts to be accommodated at every reoptimiza- tion. The results collected in Table2 con#0Crm the overall good performance of the primal-dual analytic center cutting-plane method, but they do not givemuch insightinto the behavior of the warm-start procedure proposed in this paper. Such insight is given by the results reported in Table 3.... ..."

### TABLE I COMPARISON BETWEEN THE RESULT OF THE PROPOSED PRIMAL-DUAL ALGORITHM AND THE OPTIMAL SOLUTION

2006

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### Table 1. Primal-dual interior-point method: Summary of the algorithm.

"... In PAGE 20: ... Assume that yy gt; 0 and zy are distinct. Let fy i ; z i gi=1; be the sequence of iterates generated by the interior-point method as described in Table1 and de ne d = P i=1 y i z i . Assume that = n + 1 and that the sequence S = conv(z 1 ; : : : ; z ) is initialized by S0 and is converging.... ..."

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### Table 1: Primal-dual smooth multiplier method applied to MCPLIB problems (part 1).

1996

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### Table 2: Primal-dual smooth multiplier method applied to MCPLIB problems (part 2).

1996

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### Table 1: Primal-dual smooth multiplier method applied to MCPLIB problems (part 1).

1996

Cited by 2

### Table 2: Primal-dual smooth multiplier method applied to MCPLIB problems (part 2).

1996

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### Table 1: Rough comparison among the standard primal-dual interior-point method, conver- sion method and completion method.

2003

"... In PAGE 15: ... We assume that each data matrix Ap has only O(1) nonzero elements (p = 0; 1; : : : ; m). In Table1 , \other parts quot; includes the computations of dY 2 Sn(E; 0), dX 2 Sn(F; ?), the primal and dual step lengths, etc. Table 1: Rough comparison among the standard primal-dual interior-point method, conver- sion method and completion method.... ..."

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