Global best harmony search algorithm with control parameters co-evolution based on PSO and its application to constrained optimal problems
Introduction
Harmony search (HS) is a new meta-heuristic algorithm firstly proposed by Geem et al. [1], which imitates the music improvisation process where the musicians continue to experiment and change their instruments’ pitches in order to find a better state of harmony. HS was first adapted into engineering optimization problems by Geem et al. [1]. In the HS algorithm, the solution vector is analogous to the harmony in music, and the local and global search schemes are analogous to musician’s improvisations. In musical composition, musicians try to find a musically pleasing harmony by the notes which are stored in their memories whereas a global optimal solution is found by the set of values assigned to each design variable in the optimization process. So far HS algorithm has been successfully applied to many practical optimization problems, such as parameter estimation of the nonlinear Muskingum model, vehicle routing, bandwidth-delay-constrained least-cost multicast routing, design optimization of water distribution networks, combined heat and power economic dispatch and others [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18].
There are also some shortcomings in the HS algorithm. Especially, it is not good at local search, and its control parameters, which make a significant impact on its search performance, are hard to be determined and tuned. There were several new variants of HS proposed to overcome its shortcomings. The IHS [19] based the basic HS algorithm, which use dynamical values for its control parameters, was proposed by Mahdavi. The GHS [20], which borrow the concept from swarm intelligence, was presented by Omran and Mahdavi. The SGHS [21], which introduce self-adaptive control parameter strategy, was proposed by Pan et al. The new variants’ experiment results show that their performances are better. But the above modified algorithms use the control parameters which are dynamically tuned with increasing Generations, it is difficult to obtain the optimum values of the control parameters due to the randomicity of HS and the different complexity of the optimized problems and then hard to get the best results for different problems. Therefore, a new variant called global best harmony search algorithm with control parameters co-evolution based on particle swarm optimization (PSO-CE-GHS) is proposed in this paper. In PSO-CE-GHS, control parameters are encoded to be a symbiotic individual of original individual, and PSO is applied to co-evolve the symbiotic population. Thus, with the evolution of the original population in PSO-CE-GHS, the symbiotic population is dynamically and self-adaptively adjusted and the real-time optimum control parameters are obtained for different problems. The results of experiment show that the proposed algorithm outperforms the mentioned HS, IHS, GHS and SGHS when they are applied to optimize 16 benchmark functions.
The remainder of the paper is organized as follows. In Section 2, a brief introduction of the previous mentioned HS, HIS, GHS and SGHS is summarized. The proposed algorithm is presented in Section 3. Results and analysis of the experiments are presented and discussed in Section 4. In Section 5, the proposed algorithm is applied to two constrained optimal problems. Finally, the conclusion of the paper is given in Section 6.
Section snippets
HS algorithm
In the HS, each solution is analogous to a ‘‘harmony” and represented by an -dimension real vector, it is measured in terms of the objective function. There are three basic phases called initialization, improvisation of a harmony vector and updating the harmony memory () in the HS, it works as follows:
Global best harmony search algorithm with control parameters co-evolution based on PSO
In this section, a global best harmony search algorithm with control parameters co-evolution based on PSO is proposed. The obvious difference between PSO-CE-GHS and SGHS is that another optimization algorithm, the known PSO, is employed to dynamically adjust two control parameters of and along with the optimization of harmony search. The improvisation scheme of PSO-CE-GHS is the same as the SGHS.
Experimental results
This section compares the performance of PSO-CE-GHS with that of the HS, IHS, GHS and SGHS algorithms based on a group of 14 benchmark functions which provide a balance of unimodal and multimodal functions as follows:
For each of these functions, the goal is to find the global minimum .
- A.
Sphere function, defined as where global optimum = 0 and = 0 for .
- B.
Schwefel’s problem 2.22, defined as where global optimum = 0 and = 0
Constrained optimal problems
In this paper, two constrained optimal problems are taken to show the validity and effectiveness of the PSO-CE-GHS algorithm. The two constrained optimal problems are solved through penalty function. The objective function is subjected to inequality constraints gi(x)0, ( = 1, 2, … , ) and the function = + ( [23], is the current generation and is the max generation in this paper) is defined so that the constrained optimal problems are
Conclusions
A global best harmony search algorithm with control parameters co-evolution based PSO is proposed in this paper. The proposed algorithm employed PSO to guide its two significant control parameters to converge to proper value for different problems and different searching generation. The result obtained from benchmark functions and constrained optimal problems showed that the proposed algorithm was more powerful than the existing variants of HS and suitable for the constrained optimal problems.
Acknowledgments
The authors gratefully acknowledge the supports from the following foundations: 973 project of China (2013CB733600). National Natural Science Foundation of China (21176073), Doctoral Fund of Ministry of Education of China (20090074110005), Program for New Century Excellent Talents in University (NCET-09-0346) and “Shu Guang” project (09SG29).
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