Spiral optimization algorithm

The spiral shares the global (blue) and intensive (red) behavior

In mathematics, the spiral optimization (SPO) algorithm is a metaheuristic inspired by spiral phenomena in nature.

The first SPO algorithm was proposed for two-dimensional unconstrained optimization based on two-dimensional spiral models. This was extended to n-dimensional problems by generalizing the two-dimensional spiral model to an n-dimensional spiral model. There are effective settings for the SPO algorithm: the periodic descent direction setting and the convergence setting.

m≥2 and the maximum iteration number x⋆(0)=xib(0), r(k) by a rule. 3) Update the search points: ib=argmini=1,…,m{f(xi(k+1))}. 5) Set x⋆(k). Otherwise, return to Step 2).

The search performance depends on setting the composite rotation matrix xi(0) (i=1,…,m). The following settings are new and effective.

This setting is an effective setting for high dimensional problems under the maximum iteration xi(0) (i=1,…,m) together ensure that the spiral models generate descent directions periodically. The condition of R(θ) as follows:(n−1)×(n−1) identity matrix and xi(0)∈ℝn dj,i(0)=xj(0)−xi(0). Note that this condition is almost all satisfied by a random placing and thus no check is actually fine.

• Set δ>0 such as kmax=∞. The settings of r(k) is as follows.
  • Set k⋆ is an iteration when the center is newly updated at Step 4) and k⋆ to the Algorithm:

  •(Step 1) k⋆=k+1.

  Future works

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  • The algorithms with the above settings are deterministic. Thus, incorporating some random operations make this algorithm powerful for global optimization. Cruz-Duarte et al. demonstrated it by including stochastic disturbances in spiral searching trajectories. However, this door remains open to further studies.
  • To find an appropriate balance between diversification and intensification spirals depending on the target problem class (including
    1. ↑ Tamura, K.; Yasuda, K. (2011). "Primary Study of Spiral Dynamics Inspired Optimization". IEEJ Transactions on Electrical and Electronic Engineering. 6 (S1): 98–100. doi:10.1002/tee.20628. S2CID 109093423.
    2. ↑ Tamura, K.; Yasuda, K. (2020). "The Spiral Optimization Algorithm: Convergence Conditions and Settings". IEEE Transactions on Systems, Man, and Cybernetics: Systems. 50 (1): 360–375. doi:10.1109/TSMC.2017.2695577. S2CID 126109444.
    3. ↑ Nasir, A. N. K.; Ismail, R.M.T.R.; Tokhi, M. O. (2016). "Adaptive spiral dynamics metaheuristic algorithm for global optimisation with application to modelling of a flexible system" (PDF). Applied Mathematical Modelling. 40 (9–10): 5442–5461. doi:10.1016/j.apm.2016.01.002.
    4. ↑ Sidarto, K. A.; Kania, A. (2015). "Finding all solutions of systems of nonlinear equations using spiral dynamics inspired optimization with clustering". Journal of Advanced Computational Intelligence and Intelligent Informatics. 19 (5): 697–707. doi:10.20965/jaciii.2015.p0697.
    5. ↑