### Abstract

- Top of page
- Abstract
- 1. Introduction and motivation
- 2. Probabilistic stopping rule
- 3. GRASP and experimental environment
- 4. Normal approximation for GRASP iterations
- 5. Validation of the probabilistic stopping rule
- 6. Concluding remarks
- References

The main drawback of most metaheuristics is the absence of effective stopping criteria. Most implementations of such algorithms stop after performing a given maximum number of iterations or a given maximum number of consecutive iterations without improvement in the best-known solution value, or after the stabilization of the set of elite solutions found along the search. We propose effective probabilistic stopping rules for randomized metaheuristics such as GRASP (Greedy Randomized Adaptive Search Procedures). We show how the probability density function of the solution values obtained along the iterations of such algorithms can be used to implement stopping rules based on the tradeoff between solution quality and the time needed to find a solution that might improve the best solution found. We show experimentally that, in the particular case of GRASP heuristics, the solution values obtained along its iterations fit a normal distribution that may be used to give an online estimation of the number of solutions obtained in forthcoming iterations that might be at least as good as the incumbent. This estimation is used to validate the stopping rule based on the tradeoff between solution quality and the time needed to find a solution that might improve the incumbent. The robustness of this strategy is illustrated and validated by a thorough computational study reporting results obtained with GRASP implementations to four different combinatorial optimization problems.

### 1. Introduction and motivation

- Top of page
- Abstract
- 1. Introduction and motivation
- 2. Probabilistic stopping rule
- 3. GRASP and experimental environment
- 4. Normal approximation for GRASP iterations
- 5. Validation of the probabilistic stopping rule
- 6. Concluding remarks
- References

Metaheuristics are general high-level procedures that coordinate simple heuristics and rules to find good approximate solutions to computationally difficult combinatorial optimization problems. Among them, we find simulated annealing, tabu search, GRASP, VNS (Variable Neighborhood Search), genetic algorithms, scatter search, ant colonies, and others. They are based on distinct paradigms and offer different mechanisms to escape from locally optimal solutions, contrarily to greedy algorithms or local search methods. Metaheuristics are among the most effective solution strategies for solving combinatorial optimization problems in practice and they have been applied to a large variety of areas and situations. The customization (or instantiation) of some metaheuristic to a given problem yields a heuristic to the latter.

A number of principles and building blocks blended into different and often innovative strategies are common to different metaheuristics. Randomization plays a very important role in algorithm design. Metaheuristics such as simulated annealing, GRASP, VNS, and genetic algorithms rely on randomization to sample the search space. Randomization can also be used to break ties, so that different trajectories can be followed from the same initial solution in multistart methods or to sample fractions of large neighborhoods.

One particularly important use of randomization appears in the context of greedy randomized algorithms that are based on the same principle of pure greedy algorithms, but make use of randomization to build different solutions at different runs. Greedy randomized algorithms are used in the construction phase of GRASP heuristics or to create initial solutions to population-based metaheuristics such as genetic algorithms or scatter search.

Randomization is also a major component of metaheuristics such as simulated annealing and VNS, in which a solution in the neighborhood of the current one is randomly generated at each iteration.

The main drawback of most metaheuristics is the absence of effective stopping criteria. Most implementations of such algorithms stop after performing a given maximum number of iterations or a given maximum number of consecutive iterations without improvement in the best-known solution value, or after the stabilization of the set of elite solutions found along the search. In some cases, the algorithm may perform an exaggerated and non-necessary number of iterations, when the best solution is quickly found (as often happens in GRASP implementations). In other situations, the algorithm may stop just before the iteration that could find an optimal solution. Dual bounds may be used to implement quality-based stopping rules, but they are often hard to compute or very far from the optimal values, which make them unusable in both situations.

Although Bayesian stopping rules have been proposed in the past, they were not followed by too many applications or computational experiments and results. Bartkuté et al. (2006) and Bartkuté and Sakalauskas (2009) made use of order statistics, keeping the value of the *k*th best solution found. A probabilistic criterion is used to infer with some confidence that this value will not change further. The method proposed for estimating the optimal value with an associated confidence interval is implemented for optimality testing and stopping in continuous optimization and in a simulated annealing algorithm for the bin-packing problem. The authors observed that the confidence interval for the minimum value can be estimated with admissible accuracy when the number of iterations is increased.

Boender and Rinnooy Kan (1987) observed that the most efficient methods for global optimization are based on starting a local optimization routine from an appropriate subset of uniformly distributed starting points. As the number of local optima is frequently unknown in advance, it is a crucial problem when to stop the sequence of sampling and searching. By viewing a set of observed minima as a sample from a generalized multinomial distribution whose cells correspond to the local optima of the objective function, they obtain the posterior distribution of the number of local optima and of the relative size of their regions of attraction. This information is used to construct sequential Bayesian stopping rules which find the optimal tradeoff between solution quality and computational effort.

Dorea (1990) described a stochastic algorithm for estimating the global minimum of a function and derived two types of stopping rules. The first is based on the estimation of the region of attraction of the global minimum, whereas the second is based on the existence of an asymptotic distribution of properly normalized estimators. Hart (1998) described sequential stopping rules for several stochastic algorithms that estimate the global minimum of a function. Stopping rules are described for pure random search and stratified random search (which partitions the search domain into a finite set of subdomains, with samples being selected from every subdomain according to a fixed distribution). These stopping rules use an estimation of the probability measure of the ε-close points to terminate the algorithms when a specified confidence has been achieved. Numerical results indicate that these stopping rules require fewer samples and are more reliable than the previous stopping rules for these algorithms. They also show that the proposed stopping rules can perform as well as Bayesian stopping rules for multistart local search. The authors claimed an improvement on the results reported in Dorea (1990).

Orsenigo and Vercellis (2006) developed a Bayesian framework for stopping rules aimed at controlling the number of iterations in a GRASP heuristic. Two different prior distributions are proposed and stopping conditions are explicitly derived in analytical form. The authors claimed that the stopping rules lead to an optimal tradeoff between accuracy and computational effort, saving from unnecessary iterations and still achieving good approximations.

Stopping rules have also been discussed in Duin and Voss (1999) and Voss et al. (2005) in another context. The statistical estimation of optimal values for combinatorial optimization problems as a way to evaluate the performance of heuristics was also addressed in Rardin and Uzsoy (2001) and Serifoglu and Ulusoy (2004).

In this paper, we propose effective probabilistic stopping rules for randomized metaheuristics. In the next section, we show how an estimation of the probability density function of the solution values (obtained by a stochastic local search heuristic) can be used to implement stopping rules based on the tradeoff between solution quality and the time needed to find a solution that might improve the best solution found until the current iteration. In section 'GRASP and experimental environment', we give a template of GRASP heuristics for minimization problems and describe the test instances of the four combinatorial optimization problems that have been considered in the computational experiments: the 2-path network design problem, the *p*-median problem, the quadratic assignment problem, and the set *k*-covering problem. Next, we show experimentally in section 'Normal approximation for GRASP iterations' that, in the particular case of GRASP algorithms, the solution values obtained along its iterations fit a normal distribution. This result is validated by thorough numerical experiments on the four combinatorial optimization problems cited above. This approximation is used in section 'Validation of the probabilistic stopping rule' to give an online estimation of the number of solutions obtained in forthcoming iterations that might be at least as good as the best-known solution at the time of the current iteration. This estimation is used to validate the stopping rule based on the tradeoff between solution quality and the time needed to find a solution that might improve the incumbent. The robustness of this strategy is illustrated and validated by a computational study reporting results obtained with the GRASP implementations for the four selected problems. Concluding remarks are made in the last section, together with a discussion of extensions to other heuristics, including more general memory-based methods such as GRASP with path-relinking.

### 2. Probabilistic stopping rule

- Top of page
- Abstract
- 1. Introduction and motivation
- 2. Probabilistic stopping rule
- 3. GRASP and experimental environment
- 4. Normal approximation for GRASP iterations
- 5. Validation of the probabilistic stopping rule
- 6. Concluding remarks
- References

We show in this section that and can be used to give an online estimation of the number of solutions obtained in forthcoming iterations that might be at least as good as the best-known solution at the time of the current iteration. This estimation will be used to implement the stopping rules based on the time needed to find a solution that might improve the incumbent.

For any given threshold β, the GRASP iterations can be interrupted when becomes smaller than or equal to β, i.e., as soon as the probability of finding in the next iteration a solution at least as good as the current best becomes smaller than or equal to the threshold β. Therefore, the probability value may be used to estimate the number of iterations that must be performed by the algorithm to find a new solution that is at least as good as the currently best one. Since the user is able to account for the average time taken by each GRASP iteration, the threshold defining the stopping criterion can either be fixed or determined online, so as to limit the computation time when the probability of finding improving solutions becomes very small. This strategy will be validated in the next section for GRASP implementations.

### 4. Normal approximation for GRASP iterations

- Top of page
- Abstract
- 1. Introduction and motivation
- 2. Probabilistic stopping rule
- 3. GRASP and experimental environment
- 4. Normal approximation for GRASP iterations
- 5. Validation of the probabilistic stopping rule
- 6. Concluding remarks
- References

In this section, we assume that the solution values obtained by a GRASP procedure fit a normal distribution. This hypothesis is validated experimentally for all problems and test instances described in the previous section.

Let be a sample formed by all solution values obtained along *N* GRASP iterations. We assume that the null (*H*_{0}) and alternative (*H*_{1}) hypotheses are

*H*_{0}:the sample follows a normal distribution; and

*H*_{1}:the sample does not follow a normal distribution.

The chi-square test is most commonly used to determine if a given set of observations fits a specified distribution. It is very general and can be used to fit both discrete or continuous distributions (Jain, 1991).

If the null hypothesis holds, then *D* follows a chi-square distribution with degrees of freedom. Since the mean and the standard deviation is estimated from the sample, then two degrees of freedom are lost to compensate for that. The null hypothesis cannot be rejected at a level of significance α if *D* is less than the value tabulated for .

We show below that the solution values obtained along *N* GRASP iterations fit a normal distribution, for all problems and test instances presented in sections 'The 2-path network design problem'–'The set *k*-covering problem'. In all experiments, we used and , corresponding to a histogram with the intervals , , , , , , [ − 0.5, 0.0), [0.0, 0.5), [0.5, 1.0), [1.0, 1.5), [1.5, 2.0), [2.0, 2.5), [2.5, 3.0), and [3.0, ∞). For each instance, we illustrate the normal fittings after 100, 500, 1000, 5000, and 10,000 GRASP iterations.

Table 5 reports the application of the chi-square test to the four instances of the 2-path network design problem after iterations. We observe that already after as few as 50 iterations, the solution values obtained by the GRASP heuristic fit very close to a normal distribution.

Table 5. Chi-square test for confidence level: 2-path network design problemInstance | Iterations | *D* | |
---|

2pndp50 | 50 | 0.398049 | 17.275000 |

2pndp70 | 50 | 0.119183 | 17.275000 |

2pndp90 | 50 | 0.174208 | 17.275000 |

2pndp200 | 50 | 0.414327 | 17.275000 |

Table 6. Statistics for normal fittings: 2-path network design problemInstance | Iterations | Mean | Standard deviation | Skewness | Kurtosis |
---|

| 50 | 372.920000 | 7.583772 | 0.060352 | 3.065799 |

| 100 | 373.550000 | 7.235157 | −0.082404 | 2.897830 |

2pndp50 | 500 | 373.802000 | 7.318661 | −0.002923 | 2.942312 |

| 1000 | 373.854000 | 7.192127 | 0.044952 | 3.007478 |

| 5000 | 374.031400 | 7.442044 | 0.019068 | 3.065486 |

| 10,000 | 374.063500 | 7.487167 | −0.010021 | 3.068129 |

| 50 | 540.080000 | 9.180065 | 0.411839 | 2.775086 |

| 100 | 538.990000 | 8.584282 | 0.314778 | 2.821599 |

2pndp70 | 500 | 538.334000 | 8.789451 | 0.184305 | 3.146800 |

| 1000 | 537.967000 | 8.637703 | 0.099512 | 3.007691 |

| 5000 | 538.576600 | 8.638989 | 0.076935 | 3.016206 |

| 10,000 | 538.675600 | 8.713436 | 0.062057 | 2.969389 |

| 50 | 698.100000 | 9.353609 | −0.020075 | 2.932646 |

| 100 | 700.790000 | 9.891709 | −0.197567 | 2.612179 |

2pndp90 | 500 | 701.766000 | 9.248310 | −0.035663 | 2.883188 |

| 1000 | 702.023000 | 9.293141 | −0.120806 | 2.753207 |

| 5000 | 702.281000 | 9.149319 | 0.059303 | 2.896096 |

| 10,000 | 702.332600 | 9.196813 | 0.022076 | 2.938744 |

| 50 | 1599.240000 | 13.019309 | 0.690802 | 3.311439 |

| 100 | 1600.060000 | 14.179436 | 0.393329 | 2.685849 |

2pndp200 | 500 | 1597.626000 | 13.052744 | 0.157841 | 3.008731 |

| 1000 | 1597.727000 | 12.828035 | 0.083604 | 3.009355 |

| 5000 | 1598.313200 | 13.017984 | 0.057133 | 3.002759 |

| 10,000 | 1598.366100 | 13.066900 | 0.008450 | 3.019011 |

The skewness measures the symmetry of the original data, whereas the kurtosis measures the shape of the fitted distribution. Ideally, they should be equal to 0 and 3, respectively, in the case of a perfect normal fitting. We first notice that the mean is either stable or converges very quickly to a steady-state value when the number of iterations increases. Furthermore, the mean after 50 iterations is already very close to that of the normal fitting after 10,000 iterations. The skewness values are consistently very close to 0, while the measured kurtosis of the sample is always close to 3.

Figure 2 displays the normal distribution fitted for each instance and for each number of iterations. Together with the statistics reported above, these plots illustrate the robustness of the normal fittings to the solution values obtained along the iterations of the GRASP heuristic for the 2-path network design problem.

Table 7 reports the application of the chi-square test to the four instances of the *p*-median problem after iterations. As before, we observe that already after as few as 50 iterations the solution values obtained by the GRASP heuristic for this problem also fit very close to a normal distribution.

Table 7. Chi-square test for confidence level: *p*-median problemInstance | Iterations | *D* | |
---|

pmed10 | 50 | 0.196116 | 17.275000 |

pmed15 | 50 | 0.167526 | 17.275000 |

pmed25 | 50 | 0.249443 | 17.275000 |

pmed30 | 50 | 0.160131 | 17.275000 |

Table 8 gives the main statistics for each instance of the *p*-median problem and for increasing values of the number 100, 500, 1000, 5000, and 10,000 of GRASP iterations: mean, standard deviation, skewness, and kurtosis. As for the previous problem, we notice that the mean value converges or oscillates very slightly when the number of iterations increases. Furthermore, the mean after 50 iterations is already very close to that of the normal fitting after 10,000 iterations. Once again, the skewness values are consistently very close to 0, while the measured kurtosis of the sample is always close to 3.

Table 8. Statistics for normal fittings: *p*-median problemInstance | Iterations | Mean | Standard deviation | Skewness | Kurtosis |
---|

| 50 | 1622.020000 | 57.844097 | −0.179163 | 3.255009 |

| 100 | 1620.890000 | 59.932611 | −0.364414 | 3.304588 |

pmed10 | 500 | 1620.332000 | 63.484721 | 0.111186 | 3.142248 |

| 1000 | 1619.075000 | 64.402076 | 0.074091 | 2.964164 |

| 5000 | 1617.875200 | 63.499795 | 0.043152 | 2.951273 |

| 10,000 | 1618.415400 | 63.415181 | 0.087909 | 2.955408 |

| 50 | 2170.500000 | 58.880642 | −0.041262 | 1.949923 |

| 100 | 2168.450000 | 65.313609 | 0.270892 | 2.693553 |

pmed15 | 500 | 2173.060000 | 65.881958 | 0.202400 | 2.828056 |

| 1000 | 2173.484000 | 65.590272 | 0.129234 | 2.784433 |

| 5000 | 2174.860000 | 64.639604 | 0.086450 | 2.940204 |

| 10,000 | 2175.651600 | 65.101495 | 0.096328 | 2.954639 |

| 50 | 2277.780000 | 54.782220 | 0.330959 | 3.028905 |

| 100 | 2279.610000 | 58.034799 | 0.360133 | 3.466265 |

pmed25 | 500 | 2271.546000 | 56.029848 | 0.219415 | 3.311486 |

| 1000 | 2274.182000 | 56.915366 | 0.081878 | 3.068963 |

| 5000 | 2276.305200 | 56.985195 | −0.041096 | 3.108109 |

| 10,000 | 2277.151600 | 57.583524 | −0.041570 | 3.073374 |

| 50 | 2434.660000 | 57.809899 | −0.130383 | 2.961249 |

| 100 | 2446.560000 | 57.292464 | −0.259531 | 2.667470 |

pmed30 | 500 | 2444.638000 | 56.109134 | −0.189935 | 2.691882 |

| 1000 | 2441.465000 | 57.265005 | −0.053183 | 2.858399 |

| 5000 | 2441.340400 | 54.941836 | −0.013377 | 3.054188 |

| 10,000 | 2441.277700 | 54.978827 | 0.006407 | 3.066879 |

Figure 3 displays the normal distribution fitted for each instance and for each number of iterations. Once again, the statistics and plots in these tables and figure illustrate the robustness of the normal fittings to the solution values obtained along the iterations of the GRASP heuristic for the *p*-median problem.

Results obtained with the GRASP heuristic for the quadratic assignment problem are reported in Tables 9 and 10 and in Fig. 4. The same statistics and plots provided for the previous problems lead to similar findings: they illustrate the robustness of the normal fittings to the solution values obtained along the iterations of the GRASP heuristic for the quadratic assignment problem.

Table 9. Chi-square test for confidence level: quadratic assignment problemInstance | Iterations | *D* | |
---|

tai30a | 50 | 0.127260 | 17.275000 |

tai35a | 50 | 0.213226 | 17.275000 |

tai40a | 50 | 0.080164 | 17.275000 |

tai50a | 50 | 0.075752 | 17.275000 |

Table 10. Statistics for normal fittings: quadratic assignment problemInstance | Iterations | Mean | Standard deviation | Skewness | Kurtosis |
---|

| 50 | 1,907,129.960000 | 15,106.752548 | −0.068782 | 2.562099 |

| 100 | 1,906,149.760000 | 16,779.060456 | 0.112965 | 3.028193 |

tai30a | 500 | 1,907,924.412000 | 17,663.997163 | −0.005122 | 3.071314 |

| 1000 | 1,908,292.204000 | 17,241.785219 | −0.058100 | 2.982074 |

| 5000 | 1,907,542.144400 | 17,484.852454 | 0.077001 | 2.978316 |

| 10,000 | 1,907,411.800800 | 17,354.183037 | 0.044985 | 2.982363 |

| 50 | 2,544,227.480000 | 24,293.234765 | 0.260849 | 2.906127 |

| 100 | 2,541,730.980000 | 21,782.204670 | 0.374843 | 3.131055 |

tai35a | 500 | 2,541,151.156000 | 20,167.926106 | 0.098408 | 2.990821 |

| 1000 | 2,541,735.064000 | 20,809.432271 | 0.079094 | 3.073285 |

| 5000 | 2,541,625.512800 | 20,952.352020 | 0.057649 | 3.069945 |

| 10,000 | 2,541,104.138000 | 21,191.460956 | 0.055055 | 3.089498 |

| 50 | 3,289,069.880000 | 23,456.147422 | −0.043084 | 2.503491 |

| 100 | 3,287,766.120000 | 25,032.774604 | −0.162022 | 2.395749 |

tai40a | 500 | 3291,146.756000 | 24,690.208400 | 0.058502 | 2.745715 |

| 1000 | 3,290,388.372000 | 23,935.199864 | 0.020905 | 2.800432 |

| 5000 | 3,290,638.646000 | 24,404.060084 | −0.006498 | 2.942513 |

| 10,000 | 3,290,795.196400 | 24,493.025438 | 0.032977 | 2.980864 |

| 50 | 5,172,456.560000 | 32,336.075962 | −0.080097 | 2.622378 |

| 100 | 5,175,961.000000 | 31,507.868139 | −0.010372 | 2.451724 |

tai45a | 500 | 5,175,471.476000 | 32,557.814838 | 0.100482 | 2.705749 |

| 1000 | 5,175,322.794000 | 32,090.270871 | 0.027317 | 2.919791 |

| 5000 | 5,175,315.985600 | 31,907.168093 | 0.045656 | 2.968253 |

| 10,000 | 5,174,955.537200 | 31,883.827203 | 0.048446 | 2.981676 |

Finally, we report in Tables 11 and 12 and in Fig. 5 the results obtained with the GRASP heuristic for the set *k*-covering problem. The same statistics and plots already given to the other problems show that also for the set *k*-covering problem the normal fittings to the solution values obtained along the iterations of the GRASP heuristic are very robust.

Table 11. Chi-square test for confidence level: set *k*-covering problemInstance | Iterations | *D* | |
---|

scp42 | 50 | 0.119939 | 17.275000 |

scp47 | 50 | 0.147765 | 17.275000 |

scp55 | 50 | 0.164476 | 17.275000 |

scpa2 | 50 | 0.092947 | 17.275000 |

Table 12. Statistics for normal fittings: set *k*-covering problemInstance | Iterations | Mean | Standard deviation | Skewness | Kurtosis |
---|

| 50 | 1692.200000 | 122.108968 | 0.346549 | 2.485267 |

| 100 | 1707.790000 | 138.210513 | 0.747575 | 3.727116 |

scp42 | 500 | 1682.012000 | 129.047681 | 0.453641 | 3.395710 |

| 1000 | 1677.603000 | 127.209156 | 0.424774 | 3.437712 |

| 5000 | 1678.960800 | 129.853048 | 0.481598 | 3.395114 |

| 10,000 | 1678.848600 | 130.216475 | 0.478711 | 3.328128 |

| 50 | 1596.560000 | 133.676050 | 0.551858 | 2.722159 |

| 100 | 1604.100000 | 132.584200 | 0.589321 | 3.260413 |

scp47 | 500 | 1610.160000 | 135.050740 | 0.577331 | 3.677277 |

| 1000 | 1603.936000 | 134.173380 | 0.528879 | 3.560794 |

| 5000 | 1598.799600 | 133.743778 | 0.424347 | 3.147632 |

| 10,000 | 1600.043100 | 134.398671 | 0.428076 | 3.160244 |

| 50 | 1105.160000 | 149.749439 | 0.139493 | 2.671918 |

| 100 | 1115.010000 | 154.429304 | 0.585166 | 4.036000 |

scp55 | 500 | 1146.800000 | 157.817350 | 0.299096 | 3.059246 |

| 1000 | 1146.450000 | 155.945348 | 0.332401 | 3.045766 |

| 5000 | 1151.254200 | 164.425966 | 0.384420 | 3.099880 |

| 10,000 | 1154.463700 | 164.456147 | 0.397244 | 3.144651 |

| 50 | 1383.340000 | 169.200072 | 0.092324 | 2.373417 |

| 100 | 1395.780000 | 174.312282 | 0.085142 | 2.453907 |

scpa2 | 500 | 1392.900000 | 184.297428 | 0.070337 | 2.920209 |

| 1000 | 1395.541000 | 183.278641 | 0.210396 | 2.873949 |

| 5000 | 1398.725200 | 185.724296 | 0.326508 | 3.088006 |

| 10,000 | 1400.956100 | 186.330231 | 0.335011 | 3.087160 |

We conclude this section by observing that the null hypothesis cannot be rejected with of confidence. Therefore, we may approximate the solution values obtained along *N* iterations of a GRASP heuristic by a normal distribution that can be progressively fitted and improved as more iterations are performed and more information is available. This approximation will be used in the next section to validate the probabilistic stopping rule proposed in section 'Probabilistic stopping rule' for some GRASP heuristics.

### 5. Validation of the probabilistic stopping rule

- Top of page
- Abstract
- 1. Introduction and motivation
- 2. Probabilistic stopping rule
- 3. GRASP and experimental environment
- 4. Normal approximation for GRASP iterations
- 5. Validation of the probabilistic stopping rule
- 6. Concluding remarks
- References

However, we observe that although the normal already gives a good approximation to the random variable *X*, this estimation may be further improved. First, we suppose that and are lower and upper bounds, respectively, of the value obtained by the heuristic at each of its iterations. For all minimization test problems considered in this work, trivial bounds can be obtained, for example, by setting and equal to the sum of all positive costs, as if the corresponding variables were set to 1 and the others to 0. Better, and easily computable bounds reported in Table13, can be obtained as follows for each problem instance:

- 2-Path network design problem: set and to the sum over all
*K* demand pairs of the longest 2-path between each origin–destination pair. *p*-Median problem: set to the sum over all customers of the distance from each customer to its closest facility. Similarly, set to the sum over all customers of the distance from each customer to its most distant facility.- Quadratic assignment problem: bounds for the quadratic assignment problem have been collected from Burkard et al. (1997).
- Set
*k*-covering problem: set to the optimal value of the linear relaxation, whose value for each test instance is available in Pessôa et al. (2013). To compute , create a list associated with each row of the constraint matrix, formed by the *k* largest costs of the variables that cover this row. Next, build the set of variables formed by the union of the *m* individual lists and set to the sum of the costs of these variables.

Table 13. Lower and upper boundsProblem | Instance | | |
---|

| 2pndp50 | 0 | 6244 |

2-Path | 2pndp70 | 0 | 10,353 |

| 2pndp90 | 246 | 14,621 |

| 2pndp200 | 0 | 37,314 |

| pmed14 | 2968 | 5898 |

*p*-Median | pmed15 | 1729 | 9791 |

| pmed25 | 1828 | 16,477 |

| pmed30 | 1989 | 19,826 |

| tai30a | 1,706,855 | 8,596,620 |

QAP | tai35a | 2,216,627 | 11,803,330 |

| tai40a | 2,843,274 | 15,469,120 |

| tai50a | 4,390,920 | 24,275,700 |

| scp42 | 1205 | 37,132 |

Set *k*-covering | scp47 | 1115 | 36,570 |

| scp55 | 550 | 38,972 |

| scpa2 | 560 | 58,550 |

We consider the stopping rule proposed in section 'Probabilistic stopping rule': for any given threshold β, stop the iterations of the heuristic whenever . The iterations will be interrupted as soon as the probability of finding in the next iteration a solution at least as good as the current best becomes smaller than or equal to the threshold β.

This strategy will be validated for some GRASP implementations already considered in section'Normal approximation for GRASP iterations', using the fact that the probability can be used to give an online estimation of the number of solutions obtained in forthcoming iterations that might be at least as good as the best-known solution at the current iteration.

Table 14. 2-path network design and *p*-median problems: stopping criterion vs. estimated and counted number of solutions at least as good as the incumbent after additional iterations | | Threshold | Iteration | Probability | Estimation | Count |
---|

Problem | Instance | β | | | | |
---|

| | 10^{−1} | 3 | 0.079046 | 79,046 | 1843 |

| | 10^{−2} | 25 | 0.009970 | 9970 | 1843 |

| 2pndp50 | 10^{−3} | 318 | 0.000757 | 757 | 738 |

| | 10^{−4} | 4778 | 0.000001 | 1 | 0 |

| | 10^{−5} | 4778 | 0.000001 | 1 | 0 |

| | 10^{−1} | 3 | 0.078669 | 78,669 | 148,028 |

| | 10^{−2} | 102 | 0.008923 | 8923 | 9537 |

2pndp | 2pndp70 | 10^{−3} | 1870 | 0.000643 | 643 | 465 |

| | 10^{−4} | 32,771 | 0.000036 | 36 | 26 |

| | 10^{−5} | 49,633 | 0.000005 | 5 | 4 |

| | 10^{−1} | 4 | 0.085933 | 85,933 | 2066 |

| | 10^{−2} | 41 | 0.009257 | 9257 | 2066 |

| 2pndp90 | 10^{−3} | 722 | 0.000326 | 326 | 190 |

| | 10^{−4} | 5209 | 0.000015 | 15 | 7 |

| | 10^{−5} | 270,618 | 0.000001 | 1 | 0 |

| | 10^{−1} | 23 | 0.028989 | 28,989 | 32,151 |

| | 10^{−2} | 232 | 0.001821 | 1821 | 1539 |

| 2pndp200 | 10^{−3} | 556 | 0.000566 | 566 | 503 |

| | 10^{−4} | 5377 | 0.000100 | 100 | 95 |

| | 10^{−5} | 77,448 | 0.000001 | 1 | 1 |

| | 10^{−1} | 4 | 0.060647 | 60,647 | 79,535 |

| | 10^{−2} | 21 | 0.008542 | 8542 | 7507 |

| pmed14 | 10^{−3} | 608 | 0.000786 | 787 | 215 |

| | 10^{−4} | 217,169 | 6.93 | 69 | 5 |

| | 10^{−5} | 437,422 | 5.55 | 6 | 0 |

| | 10^{−1} | 5 | 0.069694 | 69,694 | 117,054 |

| | 10^{−2} | 56 | 0.009214 | 9214 | 16,968 |

*p*-Median | pmed15 | 10^{−3} | 3533 | 0.000626 | 626 | 311 |

| | 10^{−4} | 10,264 | 6.36 | 63 | 26 |

| | 10^{−5} | 235,853 | 9.99 | 10 | 3 |

| | 10^{−1} | 3 | 0.089011 | 89,011 | 12,428 |

| | 10^{−2} | 34 | 0.009309 | 9309 | 4176 |

| pmed25 | 10^{−3} | 1060 | 0.000998 | 998 | 1232 |

| | 10^{−4} | 2760 | 2.82 | 28 | 38 |

| | 10^{−5} | 81,382 | 4.84 | 5 | 4 |

| | 10^{−1} | 4 | 0.089941 | 89,941 | 120,598 |

| | 10^{−2} | 40 | 0.004635 | 4635 | 1426 |

| pmed30 | 10^{−3} | 320 | 0.000992 | 992 | 1133 |

| | 10^{−4} | 29,142 | 2.86 | 3 | 1 |

| | 10^{−5} | 29,142 | 2.86 | 3 | 1 |

Table 15. Quadratic assignment and set *k*-covering problems: stopping criterion vs. estimated and counted number of solutions at least as good as the incumbent after *N* = 1,000,000 additional iterations | | Threshold | Iteration | Probability | Estimation | Count |
---|

Problem | Instance | β | | | | |
---|

| | 10^{−1} | 4 | 0.045151 | 45,151 | 94,107 |

| tai30a | 10^{−2} | 82 | 0.003773 | 3773 | 3759 |

| | 10^{−3} | 755 | 0.000996 | 996 | 1031 |

| | 10^{−1} | 10 | 0.098051 | 98,051 | 39,090 |

| tai35a | 10^{−2} | 57 | 0.009754 | 9754 | 5389 |

Quadratic assignment | | 10^{−3} | 1440 | 0.000999 | 1000 | 1152 |

| | 10^{−1} | 4 | 0.064517 | 64,517 | 15,748 |

| tai40a | 10^{−2} | 21 | 0.009094 | 9094 | 15,748 |

| | 10^{−3} | 1212 | 0.000121 | 121 | 111 |

| | 10^{−1} | 3 | 0.078783 | 78,783 | 281,757 |

| tai50a | 10^{−2} | 35 | 0.009633 | 9633 | 10,214 |

| | 10^{−3} | 1737 | 0.000477 | 477 | 373 |

| | 10^{−1} | 6 | 0.090414 | 90,414 | 111,059 |

| scp42 | 10^{−2} | 78 | 0.009679 | 9679 | 5944 |

| | 10^{−3} | 283,340 | 0.000457 | 457 | 0 |

| | 10^{−1} | 7 | 0.088826 | 88,826 | 125,995 |

| scp47 | 10^{−2} | 142 | 0.008171 | 8171 | 1897 |

Set *k*-covering | | 10^{−3} | 58,788 | 0.000359 | 359 | 0 |

| | 10^{−1} | 4 | 0.058515 | 58,515 | 36,119 |

| scp55 | 10^{−2} | 221 | 0.004957 | 4957 | 604 |

| | 10^{−3} | 137,239 | 0.000999 | 1000 | 7 |

| | 10^{−1} | 3 | 0.078973 | 78,973 | 160,036 |

| scpa2 | 10^{−2} | 155 | 0.009952 | 9952 | 8496 |

| | 10^{−3} | 359 | 0.000199 | 199 | 0 |

Therefore, the probability may be used to estimate the number of iterations that must be performed by the algorithm to find a new solution at least as good as the currently best one. The threshold β used to implement the stopping criterion may either be fixed a priori as a parameter or iteratively computed. In the last case, since the user is able to account for the average time taken by each GRASP iteration, this threshold can be determined online so as to limit the computation time when the probability of finding improving solutions becomes very small and the time needed to find improving solutions could become very large.

Another promising avenue of research consists in investigating stopping rules based on estimating the number of iterations needed to improve the value of the best solution found by different amounts. Figure 8 displays the results obtained for instance 2pndp90 of the 2-path network design problem with the threshold β set at 10^{−3}. For each percent improvement that is sought in the objective function, the figure plots the expected additional number of iterations needed to find a solution that improves the best-known solution value by this amount. For instance, we may see that the expected number of iterations needed to improve by 0.5% the best solution value found at termination is 12,131. If one seeks a percent improvement of 1%, then the expected number of additional iterations to be performed increases to 54,153.

### 6. Concluding remarks

- Top of page
- Abstract
- 1. Introduction and motivation
- 2. Probabilistic stopping rule
- 3. GRASP and experimental environment
- 4. Normal approximation for GRASP iterations
- 5. Validation of the probabilistic stopping rule
- 6. Concluding remarks
- References

The main drawback of most metaheuristics is the absence of effective stopping criteria. Most implementations of such algorithms stop after performing a given maximum number of iterations or a given maximum number of consecutive iterations without improvement in the best-known solution value, or after the stabilization of a population of elite solutions. In some cases, the algorithm may perform an exaggerated and non-necessary number of iterations. In other situations, the algorithm may stop just before the iteration that could find a better, or even optimal, solution.

In this paper, we proposed effective probabilistic stopping rules for randomized metaheuristics. We first showed experimentally that the solution values obtained by a randomized heuristic such as GRASP fit a normal distribution. Next, we used this normal approximation to estimate the probability of finding in the next iteration a solution at least as good as the currently best-known solution. This probability also gives an estimate of the expected number of iterations that must be performed by the algorithm to find a new solution at least as good as the currently best one. The stopping rule is based on the tradeoff between solution quality and the time (or the number of additional iterations) needed to improve the best-known solution.

Since the average time consumed by each GRASP iteration is known, another promising avenue of research consists in investigating stopping rules based on estimating the amount of time needed to improve the best solution value for some amount and evaluating the tradeoff between these two quantities. Figure 8 illustrated this issue, displaying the expected additional number of iterations needed to improve the best solution value by different percentage values.

The robustness of the proposed strategy was illustrated and validated by a thorough computational study reporting results obtained with GRASP implementations to four combinatorial optimization problems. We are investigating extensions and applications of the proposed approach to other metaheuristics that rely on randomization to sample the search space, such as simulated annealing, VNS, genetic algorithms and, in particular, to implementations of GRASP with path-relinking.