Comparison of nature-inspired population-based algorithms on continuous optimisation problems

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Abstract

Eleven swarm-intelligence-based (SI) and bio-inspired (BI) algorithms are compared with four advanced adaptive differential evolution (DE) variants, the classic DE and the blind random search on two benchmark sets. One of the benchmark sets is the CEC 2011 collection of 22 real-world optimisation problems, the latter is the suite of 30 artificial optimisation problems defined for the competition of the algorithms within CEC 2014. The results of the experiments demonstrate the superiority of the adaptive DE variants both on real-world problems and the artificial CEC 2014 test suite at all the levels of dimension (10, 30, and 50). Some of the SI and BI algorithms perform even worse than the blind random search. The efficiency of the classic DE is comparable with the better performing SI and BI methods. The results entitle to form a recommendation for practitioners: Do not propose a pseudo-new algorithm but select from the optimisation algorithms supported by thorough research and good ranking at international competitions of optimisation algorithms.

Introduction

The performance of selected adaptive variants of differential evolution (DE) and some other nature-inspired algorithms are compared on standard single-objective optimisation problems. We have experimentally studied the performance of nature-inspired algorithms [1] on a collection of real-world optimisation problems [2]. It was found that none of the eleven nature-inspired algorithms included in the comparison are competitive with three recently proposed adaptive DE variants. Some of the nature-inspired algorithms performed even worse than the random blind search.

The aim of this paper is to show if so-called pseudo-new optimisation algorithms (inspired by some natural systems) are able to provide acceptable solutions in testing as well as real-world problems. Within this paper, the groups of swarm-intelligence-based and bio-inspired algorithms are termed nature-inspired methods. Many researchers select and also apply existing algorithms from this group of optimisation methods. The main questions that have inspired this paper were ‘Are the results obtained from frequently used optimisation algorithms acceptable?’ and ‘Is it better to use more complex algorithms developed in optimisation competitions and symposia?’. We have selected several representations of swarm-intelligence-based and bio-inspired algorithms to answer the question in an experimental way. It is clear that we analysed a limited selection pf algorithms from a very big ‘pool’ of possible methods, and we also only used two independent sets of problems. Therefore, the results provided estimate the real situation the paper deals with.

In this paper, we compare the nature-inspired algorithms with four novel adaptive DE variants on two benchmark sets. Namely a collection of real-world optimisation problems [2] and an artificial set defined for the CEC 2014 competition [3]. Originally, we intended to exclude those nature-inspired algorithms performing worse than the random blind search from this comparison, but preliminary experiments showed that the Firefly algorithm [4] (which was one of the worse performing [1]) performs well on some problems of the CEC 2014 benchmark set. Another loser in the comparison, the Bat algorithm [5], has a huge number of control parameters and it was formerly tested with the default parameter settings used in the public source code.1 Moreover, the original variant Monarch Butterfly optimiser (MBO [6]) of the EBOwithCMAR algorithm [7] (the winner of the CEC 2017 competition of optimisation algorithms) is also added to the experimental comparison. EBOwithCMAR employs the swarm-intelligence-based EBO algorithm [8] and EBO is a slightly enhanced version of the original MBO. Note that only the MBO algorithm (from this class of algorithms) is included in this study. An adaptive jSO algorithm [9], which was the best performing DE variant of the CEC 2017 competition, is also included in our experiments.

In this study, only problems defined in a real-value, continuous search area are tested. For the problems solved in a discrete search space, the use of other optimisation methods or some modifications to the presented methods are required. We consider the objective function f(x), x = (x1, x2, …, xD) ∈I RD defined on the search domain Ω limited by lower and upper boundaries, i.e. Ω=j=1D[aj,bj],aj<bj,j=1,2,,D. The global minimum point x, which satisfies condition f(x) ≤ f(x), x ∈Ω is the required solution with the minimal function value.

The rest of the paper is organised as follows. Section 2 briefly describes the algorithms selected for the experimental comparison. The experimental settings and methods applied to the statistical assessment are described in section 3. Experimental results on both benchmark sets are presented in section 4. The results achieved on two benchmark sets are compared and discussed in section 5. Section 6 concludes the paper with some final remarks.

Section snippets

Algorithms selected for experimental comparison

A survey of bio-inspired algorithms has been recently presented in the paper [10] and the book [4] were the main sources for the selection of swarm-intelligence-based and bio-inspired algorithms for this experimental comparison.

The other algorithms were selected using our experience from two experimental studies [11,12] and the paper [13], where the authors studied the performance of the various swarm and evolutionary algorithms dependent on the demands of time using real-world optimisation

Experimental setting

A test suite of 22 real-world problems selected for the CEC 2011 competition in the Special Session on Real-Parameter Numerical optimisation [2] is used as one benchmark in the experimental comparison. The functions in the benchmark differ in their computational complexity and in the dimension of the search space which varies from D = 1 to D = 240, the dimensionality of most problems exceeds D = 20. The labels of the test problems are taken from Ref. [2]. The test functions are described [2] in

Results

The results of the experimental comparison are presented separately for each benchmark set but the format of the presentation and the applied methods of statistical analysis are the same. More detailed results for both sets of problems and the estimated time complexity for all algorithms are in a supplementary file.

Comparison of results on two benchmark sets

The experimental results show that adaptive variants of differential evolution perform significantly better than the nature-inspired algorithms included into the comparison. Despite our expectation, the CoBiDE algorithm using covariance-matrix crossover (based on principal components) is outperformed by other two DE variants in the benchmark of real-world optimisation problems. The frequency of real-world problems with highly correlated coordinates of parameters is not as significant large as

Conclusion

The comparison of eleven popular swarm based and bio-inspired algorithms with a random blind search, and four variants of adaptive DE, clearly demonstrate the superiority of the adaptive DE variants with regard to efficiency.

At first, it is necessary to note that we only used only several problems and algorithms in our comparison and therefore the provided results are only a simulation and an estimate of the real situation. However, from the order of algorithms in the experiment, we can assume

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