### Table 1: Performance of cutting-plane algorithm and heuristics for design-routing problem ( = optimally solved).

1995

"... In PAGE 32: ... As a result, in all but one of the cases we ran the cutting-plane algorithm twice: once to get a lower bound (and an upper bound, see below) and then again with the bounds in place (in one of the cases the algorithm was run three times). Table1 summarizes our experience with the eight-node problems. For each problem, the column labeled \LP relax.... In PAGE 33: ...requiring a few hours in each case). A similar trick solved problem ring. Prob- lem central is quite easy and the cutting plane algorithm quickly found the integer optimum. The upper bounds in Table1 were obtained by us by branch-and-bound, except for the bound for problem quasiunif2 which was obtained by W. Cook, by running his branch-and-bound algorithm on the extended formulation (which required several days of computing).... ..."

Cited by 26

### Table 1: Results for Problems Solved to Optimality

1999

"... In PAGE 29: ...0 [1]. Table1 gives performance data for the 36 problems which it is capable of solving. The data were collected on a 333 MHz.... ..."

Cited by 4

### Table 1 Number of iterations to solve the optimal control problems. Optimal control Decoupled Coupled

1995

"... In PAGE 10: ....2. The number of linear systems needed is given in Table 1. While Table1 indicates that the decoupled approach is more e cient in terms of linear system solves, in applications with ill{conditioned Cy(x) the coupled ap- proach may be favorable. The reason is that in this case the decoupled approach may underestimate the size of Wk (sk)u vastly and, as a consequence, may require more... In PAGE 26: ...x = 20. The upper and lower bounds are bi = 0:01, ai = ?1000, i = 1; : : :; n ? m. We ran the exact and inexact TRIP SQP algorithms using decoupled and cou- pled approaches and reduced and full Hessians. The total number of iterations for each case is given in Table1 . The quantities f(x), kC(x)k, and kD(x)W(x)Trf(x)k are plotted in Figure 9.... In PAGE 28: ... The upper and lower bounds are bi = 5, ai = ?1000, i = 1; : : :; n ? m. The total number of iterations needed by the inexact TRIP SQP algorithms to solve this problem are presented in Table1 . In all situations but one, all the steps were accepted.... ..."

Cited by 14

### TABLE I Performance of the RBF-GA hybrid and the standard RBF algorithms in solving the XOR problem.

1998

Cited by 3

### Table 5.13: Problems solved in different evolutionary algorithms Method Unimodal Basic Multimodal Expanded Hybrid Composition

2006

### Table 6: Lagrangean algorithm and CPLEX results - Instance set III

"... In PAGE 17: ... Actually, the largest instances of this set overestimate the size of real cellular networks. Table6 reports the results of the Lagrangean algorithm and CPLEX. The meaning of each column is the same previously defined for table 2.... ..."

### Table 2. Solution Quality Comparison (Distinct Pieces)

1997

"... In PAGE 19: ...suggested that the Lagrangian method indeed provides guidance for the starting solutions of the IH. Table2 presents solution quality results from the algorithms. The best objective values found by all methods are shown.... In PAGE 19: ...ieces) to 23.13 seconds (Marker 8, 69 pieces). While the IP is faster for some smaller problems, the LH with IH is much faster (and, importantly, more predictable) for larger problems. Further, Table2 showed that this method appears to generate the optimal solution much of the time.... ..."

Cited by 1

### Table 1: Node counts and time for instances of multi-commodity network ow problems CPLEX CPLEX + CUTS

2007

"... In PAGE 126: ... CPLEX branch-and-bound was used to solve the two mixed integer programming formulations. Table1 1: Comparison of two formulations: lower and upper bounds were returned at the end of 300s of computation time (P1) (P2) Prob LB UB LB UB E10 10 0.00 0.... In PAGE 127: ... This shows that as an integer programming formulation, with no additional cuts or heuristics added, formulation (P 2) performs better than formulation (P 1). Table1 2: Comparison of two formulations: Node counts and solve times (P1) (P2) Prob Node Count Time Node Count Time E10 10 240 1.5 56 0.... In PAGE 128: ... The time limit was 300s, so if optimal solution is not found in the allotted time for a problem the corresponding entry for solve time is 300s and node count entry is the number of nodes explored in 300s. Table1 3: Comparison of two formulations with cutting planes and heuristics: lower and upper Bounds after 300s of computation time (P1) (P2) Prob LB UB LB UB E10 10 0.00 0.... In PAGE 128: ...00 0.00 Entries in bold represent that optimal solution was found in 300 second Looking at the results from Table1 3, we can see that, with the help of cuts and heuristics, formulation (P 1) was able to provide better results than (P 2). More problems were solved to optimality and for except one, the bounds provided for the problems not solved to optimality in allotted time by formulation (P 1) were stronger than formulation (P 2).... In PAGE 129: ...Table1 4: Comparison of two formulations with cutting Planes and heuristics: node counts and computation times (P1) (P2) Prob Node Count Time Node Count Time E10 10 0 4.01 0 0.... ..."

### Table 10: Solving with the start-value algorithm.

1997

"... In PAGE 22: ... During the test, the electricity price ci;e is 200 SEK/MWh. The results when solving the problem with the start-value algorithm are given in Table10 . Here C is the optimal production cost in kSEK, kx ? xstartk the distance between the optimal point x and the starting point xstart, eval the evaluation time in seconds, feval the number of function evaluations and flops the number of M ops used.... In PAGE 23: ... The results, which are the average of ve test runs using the di erent electricity prices, are given in Table 14. 8 Conclusions Comparing Table10 and Table 12 we can see that it is necessary to use a start-value algorithm to shorten the evaluation time and, in some cases, to ensure convergence. The start-value algorithm gives a heat power production _ Qi;k at its lower or upper bound, except for one unit.... In PAGE 24: ...e. to solve the short-term production planning problem in a reasonable time, the evaluation time eval must be shortened, see Table10 and Table 13. A better start-value algorithm possibly solves this problem.... ..."

Cited by 2

### Table 4 Number of solved pricing problems.

2004

"... In PAGE 25: ...ave been solved to proven optimality. The run-time is 160.68s on average, which is the fastest of our four B amp;P variants. In Table4 , the total number of pricing problems solved in each class and their sums are given for the B amp;P approaches. Furthermore, the bar charts shown in Fig.... In PAGE 29: ...Table4 we can observe that when using FFBC only (BPNoR) the number of solved pricing problems is lower than the one of BP where CPLEX(restricted 3-stage 2DKP) is used. Pricing using a more sophisticated heuristic, in this case exactly solving restricted 3-stage 2DKP, can therefore improve the overall results, see also Table 3.... In PAGE 29: ... These pricing problems can be denoted as easy ones. Looking at absolute numbers shows that CPLEX(restricted 3-stage 2DKP) successfully solved 21 500 pricing problems, which approximately corresponds to the increase of solved pricing problems when comparing BPNoR to BP in Table4 . The bar charts showing the relative success rates of the pric- ing algorithms indicate that the absolute number of easy pricing problems roughly remained the same.... ..."

Cited by 3