Abstract
In constraint optimization problems set in continuous spaces, a feasible search space may consist of many disjoint regions and the global optimal solution might be within any of them. Thus, locating these feasible regions (as many as possible, ideally all of them) is of great importance. In this chapter, we introduce niching techniques that have been studied in connection with multimodal optimization for locating feasible regions, rather than for finding different local optima. One of the successful niching techniques was based on the particle swarm optimizer (PSO) with a specific topology, called nonoverlapping topology, where the swarm was divided into several nonoverlapping sub-swarms. Earlier studies have shown that PSO with such nonoverlapping topology, with a small number of particles in each sub-swarm, is quite effective in locating different local optima if the number of dimensions is small (up to 8). However, its performance drops rapidly when the number of dimensions grows. First, a new PSO, called mutation linear PSO, MLPSO, is proposed. This algorithm is effective in locating different local optima when the number of dimensions grows. MLPSO is applied to optimization problems with up to 50 dimensions, and its results in locating different local optima are compared with earlier algorithms. Second, we incorporate a constraint handling technique into MLPSO; this variant is called EMLPSO. We test different topologies of EMLPSO and evaluate them in terms of locating feasible regions when they are applied to constraint optimization problems with up to 30 dimensions. The results of this test show that the new method with nonoverlapping topology with small swarm size in each sub-swarm performs better in terms of locating different feasible regions in comparison to other topologies, such as the global best topology and the ring topology.
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Notes
- 1.
Note that this selection can be performed by a direct decision (the better solution is selected) or by some analysis to find out the potential of the solutions. However, in either approach, the concept of being better needs to be defined.
- 2.
The term “potentially disjoint feasible regions” refers to disjoint feasible regions and the regions that are connected with narrow passages. Also, note that without information about the topology of the search space, it is not possible to claim that the found solutions are in disjoint feasible regions.
- 3.
In general, personal best can be a set of best positions, but all PSO types listed in this paper use single personal best.
- 4.
These two coefficients control the effect of personal and global best vectors on the movement of particles and they play an important role in the convergence of the algorithm. They are usually determined by a practitioner or by the dynamic of particles’ movement.
- 5.
Alternatively, these two random matrices are often considered as two random vectors. In this case, the multiplication of these random vectors by PI and SI is element-wise.
- 6.
Niching is the ability of the algorithm to locate different optima rather than only one local optima. The niching concept is used usually in the multi-modal optimization.
- 7.
A particle \(i\) is connected to particle \( j \) if it is aware of the personal best location of the particle \( j \).
- 8.
Note that the GCPSO is another variant of PSO (introduced in Sect. 8.3) that does not have the swarm size issue. However, it is not a good choice for niching using the nonoverlapping topology. The reason is that, in GCPSO, the only particle which is able to move after stagnation is the global best particle. All other particles stay unchanged until this particle is improved. As the global best particle is only in one of the sub-swarms (the sub-swarms do not overlap with each other), this particle cannot share its information (personal best) with particles in the other sub-swarms. Thus, all other sub-swarms stay in the stagnation situation and only one of the sub-swarms may continue searching. This leads to ineffective niching behavior, as only one of the sub-swarms converges to a local optimum.
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Acknowledgments
This work was partially funded by the ARC Discovery Grants DP0985723, DP1096053, and DP130104395, as well as by the grant N N519 5788038 from the Polish Ministry of Science and Higher Education (MNiSW).
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Bonyadi, M.R., Michalewicz, Z. (2015). Locating Potentially Disjoint Feasible Regions of a Search Space with a Particle Swarm Optimizer. In: Datta, R., Deb, K. (eds) Evolutionary Constrained Optimization. Infosys Science Foundation Series(). Springer, New Delhi. https://doi.org/10.1007/978-81-322-2184-5_8
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