### Table 4.7: Large-Scale Max Cut Problem

2004

### Table 2: Results of sample experiments carried out using Matlabr implemen- tation of MCS: MAX-CUT problem.

in Constructing Test Functions for Global Optimization Using Continuous Formulations of Graph Problems

2005

"... In PAGE 10: ... These graphs, as well as johnson and hamming families arise in coding theory (see also [33]). The results of applying MCS to MAX-CUT formulation (16) are given in Table2 . Similar to Table 1, the flrst two columns in this table represent graph name and its number of vertices, while the other two columns contain the opti- mal objective value (\Opt quot;) and the terminal MCS objective value (\MCS quot;).... In PAGE 10: ...ere not solved to optimality. However, even some of the smaller problems (i.e., 5 and 8-vertex graphs in Table2 ) appeared to be challenging for MCS. In addition to the proposed examples, there are numerous larger-scale test in- stances for the maximum clique and maximum independent set problems which could be used for constructing large-scale test functions for global optimization.... ..."

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### Table 2. Max-Cut problems solved on 1 processor.

"... In PAGE 5: ...2 Results We report here some preliminary computational results. In Table2 , we show results on Max-Cut problems solved in serial. The first two columns tabulate the prob- lem name and the number of serious / total iterations (Ser/Tot).... ..."

### Table 3. Comparison of tabu search with different Nmax

"... In PAGE 11: ... This implies that a large tabusize is too restrictive to generate good solutions for the problem. Table3 shows the performance of tabu search with different value of Nmax. Based on the experiments by various Nmax, we see that the stopping criteria Nmax with 2n is appropriate for all the problems.... In PAGE 11: ... Table 2. Comparison of tabu search with different tabusize Table3 . Comparison of tabu search with different Nmax In Table 4, for the problems with 10, 20, and 30 nodes, the computational results of tabu search are shown with a recency based short term memory, a frequency based long term memory and optimal solution.... ..."

### Table 2. Comparison of tabu search with different tabusize

"... In PAGE 11: ... The stopping rule Nmax represents the number of consecutive iterations allowed for the search to continue without cost improvement. The experimental results are illustrated in Table2 and 3. The RCBA algorithm was employed as an initial heuristic for all experiments.... In PAGE 11: ... In particular, a fixed or variable tabusize including its scale should be determined through various experimental simulations. In Table2 , fixed tabusize n/4, n/2 and a uniformly distributed tabusize ranged from n/4 to n/2 are compared with the initial solution for each problem. From the table, we conclude that tabusize n/4 is appropriate for the problem structure considered in this paper.... In PAGE 11: ... Based on the experiments by various Nmax, we see that the stopping criteria Nmax with 2n is appropriate for all the problems. Table2 . Comparison of tabu search with different tabusize Table 3.... ..."

### Table 3: Performance of Tabu Search and Genetic Algorithms

"... In PAGE 9: ...ilk test for normality at an alpha level of 0.1. The results of the tabu search were compared to the genetic algorithm operating on the new facil- ity locations as described in [11]. The results of these experiments are given in Table3 .The last column of Table 3 indicates the alpha level of significance of not rejecting the hypothesis that the means of the solutions found by both the GA and tabu search are equal.... In PAGE 9: ... The results of these experiments are given in Table 3.The last column of Table3 indicates the alpha level of significance of not rejecting the hypothesis that the means of the solutions found by both the GA and tabu search are equal. The solutions found by both the GA and tabu search were tested for normality using the Shapiro-Wilk test.... In PAGE 9: ...1 are indicated in the table. As Table3 shows, the GA finds significantly better solutions for the larger problems, i.e.... In PAGE 16: ... As indicated in Table 5, the average number of iterations for tabu search to map these basins quickly grows very large. This is consistent with the results presented in Table3 . Tabu search is unable to escape from the large local minima, while the GA, with its strong random search component is able to easily jump between local minima.... ..."

### Table 3: Performance of Tabu Search and Genetic Algorithms

"... In PAGE 9: ...ilk test for normality at an alpha level of 0.1. The results of the tabu search were compared to the genetic algorithm operating on the new facil- ity locations as described in [11]. The results of these experiments are given in Table3 .The last column of Table 3 indicates the alpha level of significance of not rejecting the hypothesis that the means of the solutions found by both the GA and tabu search are equal.... In PAGE 9: ... The results of these experiments are given in Table 3.The last column of Table3 indicates the alpha level of significance of not rejecting the hypothesis that the means of the solutions found by both the GA and tabu search are equal. The solutions found by both the GA and tabu search were tested for normality using the Shapiro-Wilk test.... In PAGE 9: ...1 are indicated in the table. As Table3 shows, the GA finds significantly better solutions for the larger problems, i.e.... In PAGE 16: ... As indicated in Table 5, the average number of iterations for tabu search to map these basins quickly grows very large. This is consistent with the results presented in Table3 . Tabu search is unable to escape from the large local minima, while the GA, with its strong random search component is able to easily jump between local minima.... ..."

### Table 4. Results of tabu search in problems with 10, 20 and 30 nodes

"... In PAGE 11: ... Comparison of tabu search with different tabusize Table 3. Comparison of tabu search with different Nmax In Table4 , for the problems with 10, 20, and 30 nodes, the computational results of tabu search are shown with a recency based short term memory, a frequency based long term memory and optimal solution. In the table, optimal solutions are obtained by solving each integer problem by running the CPLEX solver9.... In PAGE 11: ... We note that the computational running time for the tabu search increases linearly as the number of nodes gets larger, but the computation time by CPLEX increases exponentially. Table4 . Results of tabu search in problems with 10, 20 and 30 nodes Table 5 shows the computational results of tabu search in problems with 40 and 50 nodes.... ..."

### Table 1: Number of Function Evaluations of selected problems using DE, DE-QN, SA and TS

"... In PAGE 11: ... By trial and error, incorporating QN into DE at 500 iterations was found to be the most computationally efficient. As seen from Table1 , the number of function evaluations required for solving the benchmark problems by DE-QN is considerably reduced compared to just using DE alone, indicating the local optimizer (QN) is able to locate the global minimum efficiently when DE has done the hard work at the beginning. Thus, DE is successfully combined with QN to reduce the computational time while maintaining the reliability.... ..."

### TABLE II RESULTS OF TABU SEARCH AND REROUTING OPTIMIZATION HEURISTICS FOR SOLVING THE REVENUE MAXIMIZATION PROBLEM IN THE ITALIAN NETWORK.

2003

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