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Keywords:

  • applied probability;
  • artificial intelligence;
  • combinatorial optimization;
  • experimental results;
  • heuristics;
  • local search;
  • metaheuristics

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction and motivation
  4. 2. Probabilistic stopping rule
  5. 3. GRASP and experimental environment
  6. 4. Normal approximation for GRASP iterations
  7. 5. Validation of the probabilistic stopping rule
  8. 6. Concluding remarks
  9. 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

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

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

We denote by X the random variable associated with the objective function value the local minimum obtained at each GRASP iteration. The probability density function and the cumulative probability distribution of the random variable X are given by inline image and inline image, respectively. Let inline image be the solution value obtained at iteration k and inline image be a sample formed by the solution values obtained along the k-first iterations. We will use later in this paper, the estimated mean and standard deviation of the sample inline image, which will be denoted by inline image and inline image, respectively. Furthermore, let inline image and inline image be estimates of inline image and inline image, respectively, obtained after the k-first GRASP iterations.

We show in this section that inline image and inline image 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.

Let inline image be the value of the best solution found along the k-first iterations of the heuristic. Therefore, the probability of finding a solution value smaller than or equal to inline image in the next iteration can be estimated by

  • display math(1)

For sake of computational efficiency, the value of inline image may be recomputed periodically or whenever the value of the best-known solution improves, and not at every iteration of the heuristic.

For any given threshold β, the GRASP iterations can be interrupted when inline image 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 inline image 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.

3. GRASP and experimental environment

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

In this section, we give a template for a GRASP heuristic and we describe the optimization problems and test instances that have been used in all computational experiments reported in this paper.

3.1. A template for GRASP

We consider in what follows the combinatorial optimization problem of minimizing inline image over all solutions inline image, which is defined by a ground set inline image, a set of feasible solutions inline image, and an objective function inline image. The ground set E, the objective function inline image, and the constraints defining the set of feasible solutions F are defined and specific for each problem. We seek an optimal solution inline image such that inline image.

GRASP (Feo and Resende, 1995) is a multistart metaheuristic, in which each iteration consists of two phases: construction and local search. The construction phase builds a feasible solution. The local search phase investigates the neighborhood of the latter, until a local minimum is found. The best overall solution is kept as the result; see Festa and Resende (2009a, b), Resende and Ribeiro (2003, 2005, 2010, 2013) for surveys on GRASP and its extensions and applications.

The pseudocode in Fig. 1 gives a template illustrating the main blocks of a GRASP procedure for minimization, in which inline image iterations are performed and inline image is used as the initial seed for the pseudorandom number generator.

image

Figure 1. Template of a GRASP heuristic for minimization.

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An especially appealing characteristic of GRASP is the ease with which it can be implemented. Few parameters need to be set and tuned, and therefore development can focus on implementing efficient data structures to assure quick iterations. Basic implementations of GRASP rely exclusively on two parameters: the stopping criterion (which is usually set as a predefined number of iterations) and the parameter used to limit the size of the restricted candidate list within the greedy randomized algorithm used by the construction phase. In spite of its simplicity and ease of implementation, GRASP is a very effective metaheuristic and produces the best-known solutions for many problems, see Festa and Resende (2009a, b) for extensive surveys of applications of GRASP. The four combinatorial optimization problems and the test instances used in our computational experiments are reported below.

3.2. The 2-path network design problem

Given a connected undirected graph inline image with non-negative weights associated with its edges, together with a set formed by K pairs of origin–destination nodes, the 2-path network design problem consists in finding a minimum weighted subset of edges containing a path formed by at most two edges between every origin–destination pair. Applications can be found in the design of communication networks, in which paths with few edges are sought to enforce high reliability and small delays. Its decision version was proved to be NP-complete by Dahl and Johannessen (2004). The GRASP heuristic that has been used in the computational experiments with the 2-path network design problem was originally presented in Ribeiro and Rosseti (2002, 2007). The main characteristics of the four instances involved in the experiments are summarized in Table 1.

Table 1. Test instances of the 2-path network design problem
Instanceinline imageinline imageK
2pndp50501225500
2pndp70702415700
2pndp90904005900
2pndp20020019,9002000

3.3. The p-median problem

Given a set F of m potential facilities, a set U of n customers, a distance function inline image, and a constant inline image, the p-median problem consists in determining which p facilities to open so as to minimize the sum of the distances from each costumer to its closest open facility. It is a well-known NP-hard problem (Kariv and Hakimi, 1979), with numerous applications to location Tansel et al. (1983) and clustering (Rao, 1971; Vinod, 1969) problems. The GRASP heuristic that has been used in the computational experiments with the p-median problem was originally presented in Resende and Werneck (2004). The main characteristics of the four instances involved in the experiments are summarized in Table 2.

Table 2. Test instances of the p-median problem
Instancemnp
pmed1020080067
pmed153001800100
pmed255005000167
pmed306007200200

3.4. The quadratic assignment problem

Given n facilities and n locations represented, respectively, by the sets inline image and inline image, the quadratic assignment problem proposed by Koopmans and Beckmann (1957) consists in determining to which location each facility must be assigned. Let inline image be a matrix where each of its entries inline image represents the flow between facilities inline image and inline image. Let inline image be a matrix where each of its entries inline image represents the distance between locations inline image and inline image. Let inline image be an assignment with cost inline image. We seek a permutation vector inline image that minimizes the assignment cost inline image, where inline image stands for the set of all permutations of inline image. The quadratic assignment problem is well known to be strongly NP-hard (Sahni and Gonzalez, 1976). The GRASP heuristic that has been used in the computational experiments with the quadratic problem was originally presented in Oliveira et al. (2004). The main characteristics of the four instances involved in the experiments are summarized in Table 3.

Table 3. Test instances of the quadratic assignment problem
Instancen
tai30a30
tai35a35
tai40a40
tai50a50

3.5. The set k-covering problem

Given a set inline image of objects, let inline image be a collection of subsets of I, with a non-negative cost inline image associated with each subset inline image, for inline image. A subset inline image is a cover of I if inline image. The cost of a cover inline image is inline image. The “set covering problem” consists in finding a minimum cost cover inline image. The “set multicovering problem” is a generalization of the set covering problem, in which each object inline image must be covered by at least inline image elements of inline image. A special case of the set multicovering problem arises when inline image, for all inline image. Following Vazirani (2004), we refer to this problem as the “set k-covering problem.” The problem finds applications in the design of communication networks and in computational biology. The GRASP heuristic that has been used in the computational experiments with the quadratic problem was originally presented in Pessôa et al. (2013). The main characteristics of the four instances involved in the experiments are summarized in Table 4.

Table 4. Test instances of the set k-covering problem
Instancemnk
scp4220010002
scp4720010002
scp5520020002
scpa230030002

4. Normal approximation for GRASP iterations

  1. Top of page
  2. Abstract
  3. 1. Introduction and motivation
  4. 2. Probabilistic stopping rule
  5. 3. GRASP and experimental environment
  6. 4. Normal approximation for GRASP iterations
  7. 5. Validation of the probabilistic stopping rule
  8. 6. Concluding remarks
  9. 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 inline image be a sample formed by all solution values obtained along N GRASP iterations. We assume that the null (H0) and alternative (H1) hypotheses are

H0:

the sample inline image follows a normal distribution; and

H1:

the sample inline image 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).

First, a histogram of the sample data is estimated. Next, the observed frequencies are compared with those obtained from the specified density function. If the histogram is formed by k cells, let inline image and inline image be the observed and expected frequencies for the ith cell, with inline image. The test starts by computing

  • display math(2)

If the null hypothesis holds, then D follows a chi-square distribution with inline image 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 inline image.

Let m and S be, respectively, the average and the standard deviation of the sample inline image. A normalized sample inline image is obtained by subtracting the average m from each value inline image and dividing the result by the standard deviation S, for inline image. Then, the null hypothesis stating that the original sample fits a normal distribution with mean m and standard deviation S is equivalent to compare the normalized sample with the N(0, 1) distribution.

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 inline image and inline image, corresponding to a histogram with the intervals inline image, inline image, inline image, inline image, inline image, inline image, [ − 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 inline image 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 inline image 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 inline image confidence level: 2-path network design problem
InstanceIterationsDinline image
2pndp50500.39804917.275000
2pndp70500.11918317.275000
2pndp90500.17420817.275000
2pndp200500.41432717.275000

To further illustrate that this close fitting is maintained when the number of iterations increase, we present in Table 6 the main statistics for each instance and for increasing values of the number inline image 100, 500, 1000, 5000, and 10,000 of GRASP iterations: mean, standard deviation, skewness (η3), and kurtosis (η4). The skewness and the kurtosis are computed as follows (Evans et al. 2000):

  • math image(3)
  • math image(4)
Table 6. Statistics for normal fittings: 2-path network design problem
InstanceIterationsMeanStandard deviationSkewnessKurtosis
 50372.9200007.5837720.0603523.065799
 100373.5500007.235157−0.0824042.897830
2pndp50500373.8020007.318661−0.0029232.942312
 1000373.8540007.1921270.0449523.007478
 5000374.0314007.4420440.0190683.065486
 10,000374.0635007.487167−0.0100213.068129
 50540.0800009.1800650.4118392.775086
 100538.9900008.5842820.3147782.821599
2pndp70500538.3340008.7894510.1843053.146800
 1000537.9670008.6377030.0995123.007691
 5000538.5766008.6389890.0769353.016206
 10,000538.6756008.7134360.0620572.969389
 50698.1000009.353609−0.0200752.932646
 100700.7900009.891709−0.1975672.612179
2pndp90500701.7660009.248310−0.0356632.883188
 1000702.0230009.293141−0.1208062.753207
 5000702.2810009.1493190.0593032.896096
 10,000702.3326009.1968130.0220762.938744
 501599.24000013.0193090.6908023.311439
 1001600.06000014.1794360.3933292.685849
2pndp2005001597.62600013.0527440.1578413.008731
 10001597.72700012.8280350.0836043.009355
 50001598.31320013.0179840.0571333.002759
 10,0001598.36610013.0669000.0084503.019011

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.

image

Figure 2. Normal distributions: fitted probability density functions for the 2-path network design problem.

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Table 7 reports the application of the chi-square test to the four instances of the p-median problem after inline image 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 inline image confidence level: p-median problem
InstanceIterationsDinline image
pmed10500.19611617.275000
pmed15500.16752617.275000
pmed25500.24944317.275000
pmed30500.16013117.275000

Table 8 gives the main statistics for each instance of the p-median problem and for increasing values of the number inline image 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 problem
InstanceIterationsMeanStandard deviationSkewnessKurtosis
 501622.02000057.844097−0.1791633.255009
 1001620.89000059.932611−0.3644143.304588
pmed105001620.33200063.4847210.1111863.142248
inline image10001619.07500064.4020760.0740912.964164
 50001617.87520063.4997950.0431522.951273
 10,0001618.41540063.4151810.0879092.955408
 502170.50000058.880642−0.0412621.949923
 1002168.45000065.3136090.2708922.693553
pmed155002173.06000065.8819580.2024002.828056
inline image10002173.48400065.5902720.1292342.784433
 50002174.86000064.6396040.0864502.940204
 10,0002175.65160065.1014950.0963282.954639
 502277.78000054.7822200.3309593.028905
 1002279.61000058.0347990.3601333.466265
pmed255002271.54600056.0298480.2194153.311486
inline image10002274.18200056.9153660.0818783.068963
 50002276.30520056.985195−0.0410963.108109
 10,0002277.15160057.583524−0.0415703.073374
 502434.66000057.809899−0.1303832.961249
 1002446.56000057.292464−0.2595312.667470
pmed305002444.63800056.109134−0.1899352.691882
inline image10002441.46500057.265005−0.0531832.858399
 50002441.34040054.941836−0.0133773.054188
 10,0002441.27770054.9788270.0064073.066879

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.

image

Figure 3. Normal distributions: fitted probability density functions for the p-median problem.

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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 inline image confidence level: quadratic assignment problem
InstanceIterationsDinline image
tai30a500.12726017.275000
tai35a500.21322617.275000
tai40a500.08016417.275000
tai50a500.07575217.275000
Table 10. Statistics for normal fittings: quadratic assignment problem
InstanceIterationsMeanStandard deviationSkewnessKurtosis
 501,907,129.96000015,106.752548−0.0687822.562099
 1001,906,149.76000016,779.0604560.1129653.028193
tai30a5001,907,924.41200017,663.997163−0.0051223.071314
 10001,908,292.20400017,241.785219−0.0581002.982074
 50001,907,542.14440017,484.8524540.0770012.978316
 10,0001,907,411.80080017,354.1830370.0449852.982363
 502,544,227.48000024,293.2347650.2608492.906127
 1002,541,730.98000021,782.2046700.3748433.131055
tai35a5002,541,151.15600020,167.9261060.0984082.990821
 10002,541,735.06400020,809.4322710.0790943.073285
 50002,541,625.51280020,952.3520200.0576493.069945
 10,0002,541,104.13800021,191.4609560.0550553.089498
 503,289,069.88000023,456.147422−0.0430842.503491
 1003,287,766.12000025,032.774604−0.1620222.395749
tai40a5003291,146.75600024,690.2084000.0585022.745715
 10003,290,388.37200023,935.1998640.0209052.800432
 50003,290,638.64600024,404.060084−0.0064982.942513
 10,0003,290,795.19640024,493.0254380.0329772.980864
 505,172,456.56000032,336.075962−0.0800972.622378
 1005,175,961.00000031,507.868139−0.0103722.451724
tai45a5005,175,471.47600032,557.8148380.1004822.705749
 10005,175,322.79400032,090.2708710.0273172.919791
 50005,175,315.98560031,907.1680930.0456562.968253
 10,0005,174,955.53720031,883.8272030.0484462.981676
image

Figure 4. Normal distributions: quadratic assignment problem.

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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 inline image confidence level: set k-covering problem
InstanceIterationsDinline image
scp42500.11993917.275000
scp47500.14776517.275000
scp55500.16447617.275000
scpa2500.09294717.275000
Table 12. Statistics for normal fittings: set k-covering problem
InstanceIterationsMeanStandard deviationSkewnessKurtosis
 501692.200000122.1089680.3465492.485267
 1001707.790000138.2105130.7475753.727116
scp425001682.012000129.0476810.4536413.395710
 10001677.603000127.2091560.4247743.437712
 50001678.960800129.8530480.4815983.395114
 10,0001678.848600130.2164750.4787113.328128
 501596.560000133.6760500.5518582.722159
 1001604.100000132.5842000.5893213.260413
scp475001610.160000135.0507400.5773313.677277
 10001603.936000134.1733800.5288793.560794
 50001598.799600133.7437780.4243473.147632
 10,0001600.043100134.3986710.4280763.160244
 501105.160000149.7494390.1394932.671918
 1001115.010000154.4293040.5851664.036000
scp555001146.800000157.8173500.2990963.059246
 10001146.450000155.9453480.3324013.045766
 50001151.254200164.4259660.3844203.099880
 10,0001154.463700164.4561470.3972443.144651
 501383.340000169.2000720.0923242.373417
 1001395.780000174.3122820.0851422.453907
scpa25001392.900000184.2974280.0703372.920209
 10001395.541000183.2786410.2103962.873949
 50001398.725200185.7242960.3265083.088006
 10,0001400.956100186.3302310.3350113.087160
image

Figure 5. Normal distributions: set k-covering problem.

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We conclude this section by observing that the null hypothesis cannot be rejected with inline image 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

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

We recall from section 'Probabilistic stopping rule' that X is the random variable representing the values of the objective function associated with the local minima obtained by each GRASP iteration. The sample inline image is formed by the solution values obtained along the k-first iterations of the heuristic, whose estimated mean and standard deviation are inline image and inline image, respectively. Denoting by inline image the value of the best solution found along the k-first iterations of the heuristic, the probability of finding a solution value smaller than or equal to inline image in the next iteration can then be estimated by Equation (1):

  • display math

We have shown in the previous section that, in the case of implementations of the GRASP metaheuristic, the random variable X can be approximated by a normal distribution inline image with average inline image and standard deviation inline image, whose probability density function and cumulative probability distribution are, respectively, inline image and inline image.

However, we observe that although the normal inline image already gives a good approximation to the random variable X, this estimation may be further improved. First, we suppose that inline image and inline image 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 inline image and inline image 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 inline image and inline image to the sum over all K demand pairs of the longest 2-path between each origin–destination pair.
  • p-Median problem: set inline image to the sum over all customers of the distance from each customer to its closest facility. Similarly, set inline image 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 inline image to the optimal value of the linear relaxation, whose value for each test instance is available in Pessôa et al. (2013). To compute inline image, 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 inline image to the sum of the costs of these variables.
Table 13. Lower and upper bounds
ProblemInstanceinline imageinline image
 2pndp5006244
2-Path2pndp70010,353
 2pndp9024614,621
 2pndp200037,314
 pmed1429685898
p-Medianpmed1517299791
 pmed25182816,477
 pmed30198919,826
 tai30a1,706,8558,596,620
QAPtai35a2,216,62711,803,330
 tai40a2,843,27415,469,120
 tai50a4,390,92024,275,700
 scp42120537,132
Set k-coveringscp47111536,570
 scp5555038,972
 scpa256058,550

Since there may be no solution whose solution value is smaller than inline image or greater than inline image, the normal approximation of the random variable X can be improved to a doubly truncated normal distribution at inline image in the left and at inline image in the right. Let

  • display math(5)

be the probability density function of the normal approximation inline image. The probability density function inline image of the truncated normal approximation is given below, depending on the existence of each of the lower and upper bounds.

  • If there exists only a lower bound, then we have a left-truncated normal at inline image:
    • display math(6)
  • If there exists only an upper bound, then we have a right-truncated normal at inline image:
    • display math(7)
  • If there exist both lower and upper bounds, then we have a doubly truncated normal at inline image in the left and at inline image in the right:
    • display math(8)

The three cases above are illustrated in Fig. 6, where inline image and inline image, which depicts the probability density functions of the truncated normal distributions. We remark that the bounds inline image and inline image can eventually be further elaborated and improved. Furthermore, we notice that the tighter the bounds are, the better will be the doubly truncated normal approximation.

image

Figure 6. Probability density functions of the truncated normal distributions. (a) Left-truncated normal (0,1) at inline image (b) right-truncated normal (0,1) at inline image; (c) doubly truncated normal (0,1) at inline image in the left and inline image in the right.

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Therefore, if inline image denotes the value of the best solution found along the k-first iterations of the heuristic, the probability of finding a solution value smaller than or equal to inline image in the next iteration can be better estimated by using the cumulative probability function of the truncated normal distribution, which is given by

  • display math(9)

We consider the stopping rule proposed in section 'Probabilistic stopping rule': for any given threshold β, stop the iterations of the heuristic whenever inline image. 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 inline image 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.

To validate and assess the effectiveness of the proposed stopping rule based on the estimation of inline image, we have devised and performed the following experiment for each problem and test instance already considered in section 'Normal approximation for GRASP iterations'. For each value of the threshold β, we run the GRASP heuristic until inline image becomes less than or equal to β. Let us denote by inline image the iteration counter where this condition is met and by inline image the best-known solution value at this time. At this point, we may estimate by inline image the number of solutions whose value will be at least as good as inline image if N additional GRASP iterations are performed. We empirically set N = 1,000,000. Next, we perform N additional iterations and we count the number inline image of solutions whose value is smaller than or equal to inline image.

The computational results displayed in Tables 14 and 15 show that inline image is a good estimation for the number inline image of solutions found after N additional iterations whose value is smaller than or equal to the best solution value at the time the algorithm would stop for each threshold value β. Using the threshold inline image is not appropriate, since at this point we are usually still very far from the optimal value and inline image does not give a good estimate of the probability of finding a solution at least as good as the best known at this time. We also observe that for both the quadratic assignment and the set k-covering problems, whose results are depicted in Table 15, it has not been possible to reach a solution satisfying the threshold inline image for any of their instances.

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 inline image additional iterations
  ThresholdIterationProbabilityEstimationCount
ProblemInstanceβinline imageinline imageinline imageinline image
  10−130.07904679,0461843
  10−2250.00997099701843
 2pndp5010−33180.000757757738
  10−447780.00000110
  10−547780.00000110
  10−130.07866978,669148,028
  10−21020.00892389239537
2pndp2pndp7010−318700.000643643465
  10−432,7710.0000363626
  10−549,6330.00000554
  10−140.08593385,9332066
  10−2410.00925792572066
 2pndp9010−37220.000326326190
  10−452090.000015157
  10−5270,6180.00000110
  10−1230.02898928,98932,151
  10−22320.00182118211539
 2pndp20010−35560.000566566503
  10−453770.00010010095
  10−577,4480.00000111
  10−140.06064760,64779,535
  10−2210.00854285427507
 pmed1410−36080.000786787215
  10−4217,1696.93inline image695
  10−5437,4225.55inline image60
  10−150.06969469,694117,054
  10−2560.009214921416,968
p-Medianpmed1510−335330.000626626311
  10−410,2646.36inline image6326
  10−5235,8539.99inline image103
  10−130.08901189,01112,428
  10−2340.00930993094176
 pmed2510−310600.0009989981232
  10−427602.82inline image2838
  10−581,3824.84inline image54
  10−140.08994189,941120,598
  10−2400.00463546351426
 pmed3010−33200.0009929921133
  10−429,1422.86inline image31
  10−529,1422.86inline image31
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
  ThresholdIterationProbabilityEstimationCount
ProblemInstanceβinline imageinline imageinline imageinline image
  10−140.04515145,15194,107
 tai30a10−2820.00377337733759
  10−37550.0009969961031
  10−1100.09805198,05139,090
 tai35a10−2570.00975497545389
Quadratic assignment 10−314400.00099910001152
  10−140.06451764,51715,748
 tai40a10−2210.009094909415,748
  10−312120.000121121111
  10−130.07878378,783281,757
 tai50a10−2350.009633963310,214
  10−317370.000477477373
  10−160.09041490,414111,059
 scp4210−2780.00967996795944
  10−3283,3400.0004574570
  10−170.08882688,826125,995
 scp4710−21420.00817181711897
Set k-covering 10−358,7880.0003593590
  10−140.05851558,51536,119
 scp5510−22210.0049574957604
  10−3137,2390.00099910007
  10−130.07897378,973160,036
 scpa210−21550.00995299528496
  10−33590.0001991990

Therefore, the probability inline image 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.

The pseudocode in Fig. 7 extends the previous template of a GRASP procedure for minimization, implementing the termination rule based on stopping the GRASP iterations whenever the probability inline image of improving the best-known solution value gets smaller than or equal to β. Lines 11 and 12 update the sample inline image and the best-known solution value inline image at each iteration k. The mean inline image and the standard deviation inline image of the fitted normal distribution in iteration k are estimated in line 13. The probability of finding a solution whose value is better than the currently best-known solution value is computed in line 14 and used in the stopping criterion implemented in line 15.

image

Figure 7. Template of a GRASP heuristic for minimization with the probabilistic stopping criterion.

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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.

image

Figure 8. Estimated number of additional iterations needed to improve the best solution value.

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6. Concluding remarks

  1. Top of page
  2. Abstract
  3. 1. Introduction and motivation
  4. 2. Probabilistic stopping rule
  5. 3. GRASP and experimental environment
  6. 4. Normal approximation for GRASP iterations
  7. 5. Validation of the probabilistic stopping rule
  8. 6. Concluding remarks
  9. 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.

References

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