# Particle swarm optimization with thresheld convergence

## Summary (2 min read)

### II. BACKGROUND

- The first work by the authors that used threshold functions to control the rate of convergence was also an application to particle swarm optimization [2] .
- Using a threshold function to ensure that new pbest positions are kept a minimum distance from all existing pbest positions is not more efficient than standard crowding -it still requires p distance calculations.
- In [2] , this was implemented by ensuring that a particle would not update its pbest position to be within the threshold of its attracting lbest position -a requirement which only needs a single distance measurement to enforce.
- This initial implementation has some similarities to niching (e.g. [6] ) -as the pbest positions are kept a minimum distance apart, they encourage exploration around a more diverse group of attraction basins.
- Compared to niching and crowding, thresheld convergence prevents both convergence and local search.

### III. PARTICLE SWARM OPTIMIZATION WITH THRESHELD CONVERGENCE

- The development of particle swarm optimization (PSO) includes inspirations from "bird flocking, fish schooling, and swarming theory in particular" [8] .
- Rather than a simple line search between a current position and a best position, the velocity and momentum of each particle encourage a more explorative search path.
- Algorithm 2 then shows the new update condition for particle swarm optimization with thresheld convergence.
- In general, the results with thresheld convergence are better and more consistent on the functions with global structure (BBOB set 4) than without global structure (BBOB set 5).
- Of note are the Gallagher functions [11] (BBOB 21 and 22) -the previous work with threshold functions [2] had large improvements on these functions while the current results have essentially no change in performance.

### IV. AN ADAPTIVE THRESHOLD FUNCTION

- The key parameter affecting the performance of thresheld convergence is α.
- The basic premise is that a threshold value that is too high will prevent any improvements from being made.
- It appears that the adaptive threshold function does have some ability to find and maintain an appropriate threshold value regardless of the initial value of α.

### The results in Table IV

- In Fig. 3 , the threshold value drops off more slowly, but it does not approach the scheduled threshold function with α = 0.10 (which led to the best performance on BBOB 18) until very late in the search process.
- In general, there are no visible plateaus (except the initial stage where pbests are being improved from their very poor initial random solutions), so the idea of an ideal threshold value may have to be revisited.

### V. ANOTHER ADAPTATION

- The addition of thresheld convergence to particle swarm optimization increases the distances among the pbest positions.
- Due to the distances among the pbest attractors, the particles can reaccelerate, also known as (Note.
- The results in Table V again show percent difference (%diff = (b-a)/b) of mean performance between particle swarm optimization with thresheld convergence (a) and standard PSO (b).

### VI. DISCUSSION

- There are many ways to improve the performance of PSO from the standard baseline version [1] .
- Convergence occurs in PSO when a particle with zero speed has the same position as all of its pbest attractors.
- As previously discussed in Section III, it appears that "niching effect" may be more suitable for the Gallagher functions (BBOB 21 and 22) than the current implementation of thresheld convergence.
- In general, simple modifications (e.g. the switch from a GBest/star topology to an LBest/ring topology [1] ) are more likely to gain widespread adoption than more complex modifications (e.g. niching [6] ).
- The large potential benefits, computational efficiency, and general ease of adding thresheld convergence make improved threshold functions a promising area for further research.

### VII. SUMMARY

- The addition of thresheld convergence to particle swarm optimization can lead to large performance improvements on multi-modal functions with adequate global structure.
- A simple, effective, and robust adaptive threshold function has been developed to replace the originally developed scheduled threshold functions.
- The simplicity and effectiveness of the proposed modifications make thresheld convergence a promising area for further research.

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72 citations

22 citations

### Cites background from "Particle swarm optimization with th..."

...A new solutions replaces the personal attractor only if its distance to the local attractor is larger than the threshold [3]....

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...Thresheld convergence has proven useful to increase the search performance when applied to rapidly converging search techniques such as simulated annealing (SA) [2], particle swarm optimization (PSO) [3] and differential evolution (DE) [4]....

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...In PSO, the minimum step is enforced between the local and personal attractors (best solutions found)....

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...Thresheld convergence has proven useful to increase the search performance when applied to rapidly converging search techniques such as simulated annealing (SA) [2], particle swarm optimization (PSO) [3] and differential evolution (DE) [4]....

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18 citations

18 citations

### Cites methods from "Particle swarm optimization with th..."

...They are updated by a rule similar to that used in previous attempts to control convergence for PSO [20] and DE [21] in which an initial threshold is selected that then decays over the course of the search process....

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...The implementation of standard PSO [3] is the same as that described more fully in [20]....

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11 citations

### Cites background from "Particle swarm optimization with th..."

...In [7, 20, 47], termination criteria of EAs has been analyzed and the threshold strategy has been addressed....

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##### References

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### "Particle swarm optimization with th..." refers background or result in this paper

...This unexpected result is inconsistent with previous work involving threshold functions [2][5][7] which showed broad benefits across the full range of multi-modal functions (i....

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...Recent work with the use of threshold functions to control the rate of convergence has led to the new technique of “thresheld convergence” [7]....

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35,104 citations

### "Particle swarm optimization with th..." refers background in this paper

...switching from a GBest/star topology [8] to an LBest/ring topology [1])....

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...The development of particle swarm optimization (PSO) includes inspirations from “bird flocking, fish schooling, and swarming theory in particular” [8]....

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##### Frequently Asked Questions (2)

###### Q2. What are the future works in "Particle swarm optimization with thresheld convergence" ?

Future research will study the Gallagher functions more closely with an emphasis on achieving the simultaneous benefits of niching and thresheld convergence. Future research will also study the effects of each parameter more closely ( e. g. α, vf, and the threshold decay factor ). The large potential benefits, computational efficiency, and general ease of adding thresheld convergence make improved threshold functions a promising area for further research. This variation suggests that more improvements can be achieved through the development of improved ( adaptive ) threshold functions.