Evaluating a local genetic algorithm as context-independent local search operator for metaheuristics

Abstract

Local genetic algorithms have been designed with the aim of providing effective intensification. One of their most outstanding features is that they may help classical local search-based metaheuristics to improve their behavior. This paper focuses on experimentally investigating the role of a recent approach, the binary-coded local genetic algorithm (BLGA), as context-independent local search operator for three local search-based metaheuristics: random multi-start local search, iterated local search, and variable neighborhood search. These general-purpose models treat the objective function as a black box, allowing the search process to be context-independent. The results show that BLGA may provide an effective and efficient intensification, not only allowing these three metaheuristics to be enhanced, but also predicting successful applications in other local search-based algorithms. In addition, the empirical results reported here reveal relevant insights on the behavior of classical local search methods when they are performed as context-independent optimizers in these three well-known metaheuristics.

Appendices

Appendix 1: A test suite

The test suite that we have used for the experiments consists of 22 binary-coded test problems. They are described in the following sections.

1.1 Deceptive problem

In deceptive problems (Goldberg et al. 1989), there are certain schemata that guide the search toward some solution that is not globally competitive. The schemata that have the global optimum do not bear significance and so they may not proliferate during the genetic process. The used deceptive problem consists of the concatenation of k subproblems of length 3. The fitness for each 3-bit section of the string is given in Table 13. The overall fitness is the sum of the fitness of these deceptive subproblems.

Table 13 Deceptive order-3 problem

We have used a deceptive problem with 13 subproblems.

1.2 Trap problem

Trap problem (Thierens 2004) consists of misleading subfunctions of different lengths. Specifically, the fitness function $f(x)$ is constructed by adding subfunctions of length 1 ($F_1$), 2 ($F_2$), and 3 ($F_3$). Each subfunction has two optima: the optimal fitness value is obtained for an all-ones string, while the all-zeroes string represents a local optimum. The fitness of all other string in the subfunction is determined by the number of zeroes: the more zeroes, the higher the fitness value. This causes a large basin of attraction toward the local optimum. The fitness values for the subfunctions are specified in Table 14 where the columns indicate the number of ones in the subfunctions $F_1,$ $F_2,$ and $F_3.$ The fitness function $f(x)$ is composed of 4 $F_3$ subfunctions, 6 $F_2$ subfunctions, and 12 $F_1$ subfunctions. The overall length of the problem is thus 36. There are 2¹⁰ optima of which only one is the global optimum: the string with all ones having a fitness value of 220.

$$ \begin{aligned} f(x)=&\sum_{i=0}^3 F_3(x_{[3i:3i+2]}) + \sum_{i=0}^5 F_2(x_{[2i+12:2i+13]}) \\ \qquad+&\sum_{i=0}^{11} F_1(x_{24+i}) \\ \end{aligned} $$

Table 14 Fitness values of the subfunctions F _i

1.3 Max-Sat Problem

The satisfiability problem in propositional logic (SAT) (Smith et al. 2003) is the task to decide whether a given propositional formula has a model. More formally, given a set of m clauses $\{C_1,\ldots,C_m\}$ involving n boolean variables $X_1, \ldots, X_n$ the SAT problem is to decide whether an assignment of values to variables exists such that all clauses are simultaneously satisfied.

Max-Sat is the optimization variant of SAT and can be seen as a generalization of the SAT problem: given a propositional formula in conjunctive normal form (CNF), the Max-Sat problem then is to find a variable assignment that maximizes the number of satisfied clauses. It returns the percentage of satisfied clauses.

We have used two set of instances of the Max-Sat problem with 100 variables, 3 variables by clause, and 1,200 and 2,400 clauses, respectively. They have been obtained from De Jong et al. (1997). They are denoted as M-Sat(n, m, l), where l indicates the number of variables involved in each clause (3). Each run i, of every algorithm, uses a specific seed (${\rm seed}_i$) for generating the M-Sat(n, m, l) instance, i.e. ith execution of every algorithm uses the same ${\rm seed}_i,$ whereas jth execution uses ${\rm seed}_j.$

1.4 NK-landscapes

In the NK model (Kauffman 1989), N represents the number of genes in a haploid chromosome and K represents the number of linkages each gene has to other genes in the same chromosome. To compute the fitness of the entire chromosome, the fitness contribution from each locus is averaged as follows:

$$ f(s)={\frac{1}{N}} \sum_{i=1}^N f({\rm locus}_i) $$

(1)

where the fitness contribution of each locus, $f({\rm locus}_i),$ is determined by using the (binary) value of gene i together with values of the K interacting genes as an index into a table $T_i$ of size $2^{K+1}$ of randomly generated numbers uniformly distributed over the interval $[0, 1].$ For a given gene i, the set of K linked genes may be randomly selected or consist of the immediately adjacent genes.

We have used two set of instances of the NK-Landscape problem: one with $N = 48$ and $K = 4,$ and another with $N = 48$ and $K = 12.$ They are denoted as NKLand (N, K). They have been obtained from De Jong et al. (1997). Each run i, of every algorithm, uses a different seed (${\rm seed}_i$) for generating the NKLand (N, K) instance, i.e. the ith execution of every algorithm has used the same ${\rm seed}_i,$ whereas the jth execution has used ${\rm seed}_j.$

1.5 P-peak problems

P-peak problem generator (Spears 2000) creates instances with a certain number of peaks (the degree of multi-modality). For a problem with P peaks, P bit strings of length L are randomly generated. Each of these string is a peak (a local optima) in the landscape. Different heights can be assigned to different peaks based on various schemes (equal height, linear, logarithm-based, and so on). To evaluate an arbitrary solution S, first locate the nearest peak in Hamming space, call it ${\rm Peak}_n(S).$ Then, the fitness of s is the number of bits the string has in common with ${\rm Peak}_n(S),$ divided by L, and scaled by the height of the nearest peak. In case there is a tie when finding the nearest peak, the highest peak is chosen.

We have used different groups of P-Peak instances denoted as PPeaks(P,  L). Each run i, of every algorithm, uses a different seed (${\rm seed}_i$) for generating the PPeaks(P , L) instance. Linear scheme have been used for assigning heights to peaks in $[0.6, 1].$

1.6 Max-cut problem

The Max-cut problem (Karp 1972) is define as follows: Let an undirected and connected graph $G=(V,E),$ where $V=\{1,2,\ldots,n\}$ and $E \subset\{(i,j) : 1 \leq i < j \leq n\},$ be given. Let the edge weights $w_{ij} = w_{ji}$ be given such that $w_{ij}=0$ $\forall (i,j) \not \in E,$ and in particular, let $w_{ii}=0.$ The max-cut problem is to find a bipartition $(V_1,V_2)$ of V so that the sum of the weights of the edges between $V_1$ and $V_2$ is maximized.

We have used 6 instances of the max-cut problem (G10, G12, G17, G18, G19 G43), obtained from Helmberg and Rendl (2000).

1.7 Unconstrained binary quadratic programming problem

The objective of the Unconstrained Binary Quadratic Programming (BQP) (Beasley 1998) is to find, given a symmetric rational $n \times n$ matrix $Q=(Q_{ij}),$ a binary vector of length n that maximizes the following quantity:

$$ f(x)=x^t Q x= \sum_{i=1}^n \sum_{j=1}^n q_{ij} x_i x_j, \quad x_i \in \{0,1\} $$

(2)

We have used four instances with different values for n. They have been taken from the OR-Library (Beasley 1990). They are the first instances of the BQP problems in the files ‘bqp50’, ‘bqp100’, ‘bqp250’, and ‘bqp500’. They are BQP(50), BQP(100), BQP(250), and BQP(500), respectively.

Appendix 2: Results

Tables 15 and 16 show the fitness values obtained by the algorithms studied in the empirical analysis in Sects. 5, 6, and 7. Best results of every group of algorithms (RMLS, ILS, and VNS) are boldfaced. Table 17 displays the number of successful restarts obtained by the algorithms in Sect. 7.1. Best results of every group with the same LS method are marked.

Table 15 Results of the RMLS, ILS-0.1, and ILS-0.25 algorithms with each refinement procedure

Table 16 Results of the ILS-0.5, ILS-0.75, and VNS algorithms with each refinement procedure

Table 17 Number of successful restarts for each LS-based MH

Appendix 3: Statistical analysis

In this section, we explain the basic functionality of each non-parametric test applied in this study together with the aim pursued with its use:

• Friedman test: Although we will not use this test, because of its conservative undesirably effect, we describe it because it is the basis of the following one. Friedman test is a non-parametric equivalent of test of repeated-measures ANOVA. It computes the ranking of the observed results for each algorithm ($r_j$ for the algorithm j with k algorithms) for each function, assigning to the best of them the ranking 1, and to the worst the ranking k. Under the null hypothesis, formed from supposing that the results of the algorithms are equivalent and, therefore, their average rankings are also similar, the Friedman statistic

  $$ \chi^2_F = {\frac{12N}{k(k + 1)}} \left [\sum_j R^2_j - {\frac{k(k+1)^2}{4}} \right] $$

  (3)

  is distributed according to $\chi^2_F$ with $k - 1$ degrees of freedom, being $R_j = 1 / N \sum_i r^i_j,$ and N the number of functions. The critical values for the Friedman statistic coincide with the established in the $\chi^2$ distribution when $N > 10$ and $k > 5.$ In a contrary case, the exact values can be seen in Zar (1999).

• Iman and Davenport test (Iman and Davenport 1980): It is a metric derived from the Friedman statistic given that this last metric produces a conservative undesirably effect. The statistic is

  $$ F_F = {\frac{(N - 1)\chi^2_F}{N(k - 1) - \chi^2_F}} $$

  (4)

  and it is distributed according to a F distribution with $k - 1$ and $(k - 1)(N - 1)$ degrees of freedom.

• Holm method (Holm 1979): If the null hypothesis is rejected in Iman–Davenport test, we can proceed with a post-hoc test. The test of Holm is applied when we want to compare a control algorithm (the one with the best average Friedman ranking) opposite to the remainders. Holm test sequentially checks the hypotheses ordered according to their significance. We will denote the p-values ordered by $p_1, p_2, \ldots,$ in the way that $p_1 \leq p_2 \leq \cdots \leq p_{k-1}.$ Holm method compares each $p_i$ with $\alpha / (k - i)$ starting from the most significant p-value. If $p_1$ is below than $\alpha / (k - 1),$ the corresponding hypothesis is rejected and it leaves us to compare $p_2$ with $\alpha / (k - 2).$ If the second hypothesis is rejected, we continue with the process. As soon as a certain hypothesis cannot be rejected, all the remaining hypotheses are maintained as accepted. The statistic for comparing algorithm i with algorithm j is:

  $$ z = (R_i - R_j) / \sqrt{{\frac{k (k + 1)}{6N}}} $$

  (5)

  The value of z is used for finding the corresponding probability from the table of the normal distribution, which is compared with the corresponding value of $\alpha.$

• Wilcoxon signed rank test: This is the analogous of the paired t test in non-parametrical statistical procedures; therefore, it is a pairwise test that aims to detect significant differences between the results of two algorithms. Let $d_i$ be the difference between the performance scores of two algorithms on the ith out of N functions (we have normalized the results on every function to be in $[ 0, 1 ]$ according to the best and worst results obtained by all the algorithms). The differences are ranked according to their absolute values; average ranks are assigned in case of ties. Let $R^+$ be the sum of ranks for the functions on which the second algorithm outperformed the first, and $R^-$ the sum of ranks for the opposite. Ranks of $d_i = 0$ are split evenly among the sums; if there is an odd number of them, one is ignored:

  $$ R^+= \sum_{d_i > 0} {\rm rank}(d_i) + 1 / 2 \sum_{d_i = 0} {\rm rank}(d_i) $$

  (6)
  $$ R^-= \sum_{d_i < 0} {\rm rank}(d_i) + 1 / 2 \sum_{d_i = 0} {\rm rank}(d_i) $$

  (7)

  Let T be the smallest of the sums, $T = min(R^+,R^-).$ If T is less than or equal to the value of the distribution of Wilcoxon for N degrees of freedom [Table B.12 in Zar (1999)], the null hypothesis of equality of means is rejected.