Geometric Selective Harmony Search
Introduction
In the last years several meta-heuristic algorithms based on the collective behavior of natural or artificial decentralized self-organized systems have been proposed [37], [11], [20], [50], [23], [51], [48], [27], [38], [39], [9], [45], [47], [4], [25], [29], [18]. As pointed out in [7], these algorithms can be viewed as powerful problem-solving systems that exploit sophisticated collective intelligence strategies. Composed of simple interacting agents, this intelligence lies in the networks of interactions among individuals and between individuals and the environment. The problems they solve [8], [33], [49] have important counterparts in engineering and computer science, as well as in several other different applicative domains.
Among these meta-heuristic algorithms, Harmony Search (HS) [12] is one of the youngest. As reported in [22], this approach is inspired by the musical performance process that takes place when a musician looks for a better state of harmony. In fact, in some senses, Jazz improvisation looks for musically pleasant harmony in a similar way to an optimization process, which searches for optimal solutions to a problem. In this perspective, the pitch of each musical instrument determines the aesthetic quality, just as the objective function value in an optimization problem is determined by the set of values assigned to each decision variable. The HS algorithm has been very successful on a wide set of different optimization problems [24], [16], [30], [28], [40], [42], presenting several advantages when compared to traditional optimization techniques, as discussed in [28]. Moreover, several studies have been proposed aimed at improving the performance of HS and gaining a better understanding of its dynamics, such as stochastic derivative [13], exploratory power [10], and parameter-setting-free [14] or adaptive [17] techniques.
In the last years a new variant of the basic HS algorithm has been proposed in order to improve its performance. In [31] authors proposed a new algorithm called Improved Harmony Search (IHS). IHS employs a novel method for generating new solutions that enhances accuracy and convergence rate of the standard HS algorithm. The authors discuss the impact of constant parameters on the HS algorithm and present a strategy for tuning these parameters. This algorithm has been used to successfully solve complex optimization problems as in [46], [22].
Two of the most recent studies concerning HS are [52], [2]. The authors of [52] proposed a HS search algorithm for the flexible job shop scheduling problem (FJSP). To make the HS algorithm adaptive to the FJSP, first the continuous harmony vector is converted into a kind of discrete two-vector code for the FJSP. Secondly, the harmony vector is mapped into a feasible active schedule through effectively decoding the transformed two-vector code, which could largely reduce the search space. Thirdly, an initialization scheme combining heuristic and random strategies is introduced to increase diversity in the initial harmony memory. Furthermore, a local search procedure is embedded in the HS algorithm to enhance the local exploitation ability, whereas HS is employed to perform exploration by evolving harmony vectors in the harmony memory. The authors of [2], proposed an hybrid HS algorithm to address the university course timetabling problem. In particular, the HS algorithm has been hybridized with hill climbing to improve local exploitation, and with an algorithm inspired by particle swarm optimization to improve exploration.
In this paper we propose a new HS algorithm called Geometric Selective Harmony Search (GSHS). GSHS is different from HS, IHS and existing selection-based HS variants in the following: (1) unlike HS and IHS, it uses a selection procedure that is typical of Genetic Algorithms (GAs) [19], [15] and Genetic Programming (GP) [26]; (2) the process of generating a new harmony makes use of a particular recombination operator that combines the information of two harmonies; (3) it uses a newly defined mutation operator. These modifications were inspired by the novel semantic genetic operators recently published in the GP literature [32], [43], which greatly improve the evolvability of better solutions thanks to their geometric properties. Specifically, geometric semantic crossover produces, by construction, an offspring that is not worse than the worst of its parents, and geometric semantic mutation causes a perturbation on the semantics of solutions, whose magnitude is controlled by a parameter. Regarding the usage of selection, it is typically coupled with the geometric operators, and an important element for good evolvability. Furthermore, recent studies have reported selection procedures as beneficial in HS algorithms [1], [3], including the one we use, tournament selection [21].
The rest of the paper is organized as follows: Section 2 presents the basic HS algorithm and Section 3 outlines the IHS algorithm and also other existing variants, both the ones that integrate and the ones that do not integrate a selection strategy. In Section 4 we define the new GSHS algorithm, pointing out the main differences to standard HS and to existing selection-based HS methods; Section 5 presents the test problems and the experimental settings, and discusses the obtained results. Finally, Section 6 concludes the paper, summarizing the contributions of this work and suggesting possible future work.
Section snippets
Harmony Search
Let us consider the following specification of an optimization problem: the objective is to minimize , subject to , with , . is an objective function, is the solution vector composed by decision variables is the number of decision variables, and is the set of possible values for each decision variable.
The standard HS algorithm introduced in [12] consists of five basic steps: (1) Initialize the HS parameters; (2) Initialize the harmony memory
Previous and related work
In this section we report the most important and the most used variants of the HS algorithm described in Section 2. In particular, Section 3.1 describes the IHS algorithm proposed in [31], also used in this paper for comparison with our own approach. The same section also presents some other variants of the HS algorithm. None of the algorithms reported in Section 3.1 apply any selection procedure to choose the candidate harmonies to be used in the search process. In Section 3.2 we describe
Geometric Selective Harmony Search
In this section we present our approach, the Geometric Selective Harmony Search (GSHS) algorithm. GSHS is similar to the HS algorithm, with the following main differences: (1) the memory consideration process involves the presence of a selection procedure, (2) the algorithm integrates a particular recombination operator that combines the information of two harmonies and (3) the algorithm includes a mutation operation that uses the PAR parameter.
In detail, during the memory consideration
Experimental study
The objective of our experiments is to compare the performance of different HS algorithms on a well-known suite of benchmark problems. We have chosen the 20 benchmarks of the CEC 2010 suite [41] because of the high variety of difficulty levels of the problems it contains, and because it has been used so often in the last few years that it is now regarded as a well established benchmark suite, widely accepted by the scientific community.
Five different algorithms have been compared with each
Conclusions
We have presented a new variant of the Harmony Search (HS) algorithm, called Geometric Selective Harmony Search (GSHS). The main differences between GSHS and the original HS are (1) the presence of a selection procedure inspired by the tournament selection of Genetic Algorithms and Genetic Programming, (2) the definition of a new memory consideration process that is based on the use of a recombination operator and (3) the definition of a new mutation operator. GSHS has been compared to the
Acknowledgments
This work was partially supported by projects EnviGP (PTDC/EIA-CCO/103363/2008) and MassGP (PTDC/EEI-CTP/2975/2012), FCT, Portugal, and by the French National Research Agency Project EMC (ANR-09-BLAN-0164).
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2023, Expert Systems with ApplicationsA novel harmony search algorithm and its application to data clustering
2020, Applied Soft Computing JournalCitation Excerpt :Several studies in literature changed the memory consideration process to improve the performance of HS. For instance, Castelli et al. [33] replaced the random harmony selection of memory consideration procedure with the tournament selection to select two harmonies with good fitness from HM in each iteration. Then, the information of corresponding pitches in these harmonies is combined through a linear recombination operator to produce the new pitch.
Review of harmony search with respect to algorithm structure
2019, Swarm and Evolutionary ComputationCitation Excerpt :Hasan et al. [74] proposed a hybrid harmony search algorithm by replacing all random consideration in HS with five different types of mutation operations. The selection operator of the Evolutionary Algorithm (EA) was integrated into HS algorithm, resulting a new algorithm known as Geometric Selective Harmony Search (GSHS) [75], which introduces a selection procedure and recombination operator in the memory consideration process, and a mutation operation in the pitch adjustment. The Improved Adaptive Harmony Search algorithm (IAHS) [76] added two main blocks in the algorithm structure: forward and backward schemes when initialising harmony memory and forward scheduling scheme when improvising a new harmony.
Simplified hybrid fireworks algorithm
2019, Knowledge-Based SystemsSelective Refining Harmony Search: A new optimization algorithm
2017, Expert Systems with ApplicationsCitation Excerpt :This section evaluates the performance of SRHS algorithm and compares it with several HS algorithms. To make comprehensive comparisons, we have implemented original HS algorithm (Geem et al., 2001) and some of its state-of-the-art variations including IHS (Mahdavi et al., 2007), SAHS (Wang & Huang, 2010), GBHS (Omran & Mahdavi, 2008), NGHS (Zou et al., 2010), and GSHS (Castelli et al., 2014). All of the algorithms were examined over 20 benchmark problems of the IEEE CEC 2010 (Tang, Xiaodong Li, & Zhenyu Yang, 2010), a well-established benchmark suite in the literature.
Tournament-based harmony search algorithm for non-convex economic load dispatch problem
2016, Applied Soft Computing JournalCitation Excerpt :Thus, the investigations of novel selection methods in the memory consideration for the HS algorithm are proposed in [3,13] and analyzed in [6], where five selection schemes were investigated: proportional, tournament, global best, linear rank, and exponential rank. Interestingly, the tournament-based HS algorithm achieved the best performance for the global optimization problems and adopted by other researchers as an efficient variant of the HS algorithm [25,75,49]. In this paper, the tournament-based HS (THS) algorithm is investigated for the ELD problem.