Abstract
The natural aggregation algorithm (NAA) is a new efficient population-based optimizer. The NAA has a competent performance when compared to other well-established optimizers. However, a problem of concern is NAA lack of exploitation in its local search. In this article, we propose an improved version of NAA. The modifications made are: hypercubes with displacement and shrink mechanism applied in each shelter, we designed a new movement operator to search inside the hypercubes, an improved readjustment of the algorithm’s parameters and “leave shelter” formula of NAA, to better mimic the aggregation behavior. To prove the effectiveness of the modified hypercube natural aggregation algorithm (HYNAA), we compared with classics optimizers, such as PSO, DE and ABC, state of the art, such as CMA-ES, MSA and NAA himself with a benchmark of 28 functions. The said functions consist of five unimodal, 19 multimodal and four hybrids, and we compared them on 30, 50 and 100 dimensions. We also made extra comparisons against NAA in 500 and 1000 dimensions to contrast the ability of the hypercubes to reduce the dimensional complexity. Finally, we tested two trajectory optimization problems. Experimental results and statistical tests demonstrate that the performance of HYNAA is significantly better than that of other optimizers.
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Appendices
Appendix A
The set of benchmark test functions implemented in the experiments is described in Table 30, where \( f\left( {{\mathbf{x}}^{*} } \right) \) is the optimum value of the function, \( {\mathbf{x}}^{*} \) the optimum position and S\( \in {\mathbb{R}}^{n} \) the search space. The benchmark test functions are classified into unimodal, multimodal and composite (Tables 31, 32).
Appendix B
2.1 Multiple gravity-assist problems
In this section, we provide a comprehensive description of both multiple gravity-assist (MGA) problems we considered in the experiments. These problems represent interplanetary trajectories of a spacecraft equipped with chemical propulsion, able to thrust only during its planetocentric phases, and this model is useful because it allows to calculate trajectories in a small-dimensional optimization problem. The goal is to provide the best trajectory to obtain a cost-effective mission which consists of launching a spacecraft from a given astronomical body along a trajectory leading to another.
2.2 GTOC1
This is an MGA problem that gets it inspiration from the first edition of the Global Trajectory Optimization Competition, which consists of designing an optimal trajectory for a pioneering asteroid deflection mission. The rather long fly-by sequence is Earth, Venus, Earth, Venus, Earth, Jupiter and Saturn in retrograde orbit, and the final target is the asteroid TW229.
Minimize: Maximum change in the semimajor axis of the asteroid orbit following an anaelastic impact of the spacecraft with the asteroid. \( \frac{ - 1}{d}, \,{\text{where}}\,d = m_{\text{f}} *U*V \), \( m_{\text{f}} \) being the final mass, U the velocity of the asteroid and V the relative velocity of the asteroid and the spacecraft.
Subject to:
Variable | Lower bound | Upper bound | Unit type |
|---|---|---|---|
\( X_{1} \) | 3000 | 10,000 | Modified Julian date 2000 |
\( X_{2} \) | 14 | 400 | Days |
\( X_{3} \) | 14 | 2000 | Days |
\( X_{4} \) | 14 | 2000 | Days |
\( X_{5} \) | 14 | 2000 | Days |
\( X_{6} \) | 100 | 9000 | Days |
\( X_{7} \) | 366 | 9000 | Days |
\( X_{8} \) | 300 | 9000 | Days |
Constrains: The minimum allowed of the fly-by pericenters for each travel is considered, and for each km of violation, a penalty factor is applied (Figs. 7, 8).
General view of GTOC1 design problem
Close-up of GTOC1 design problem
Travel | Planet | Min. pericenter radius (km) |
|---|---|---|
\( Rpmin1 \) | Venus | 6351.8 |
\( Rpmin2 \) | Earth | 6778.1 |
\( Rpmin3 \) | Venus | 6351.8 |
\( Rpmin4 \) | Earth | 6778.1 |
\( Rpmin5 \) | Jupiter | 600,000 |
\( Rpmin6 \) | Saturn | 70,000 |
2.3 CASSINI1
This is an MGA problem that is related to the Cassini spacecraft trajectory design problem executed by the NASA and ended in September 15, 2017. The objective of this mission is to reach Saturn and to be captured by its gravity into an orbit, having the periapsis or smallest radial distance is considered PR = 108,950 km, and the eccentricity ECC = 0.98 which denotes an ellipse. The planetary fly-by sequence treated is Earth–Venus–Venus–Earth–Jupiter–Saturn (as the one used by Cassini spacecraft).
As the objective function, the total \( \Delta V \) accumulated during the mission is used, which includes the launch \( \Delta V \) and the various \( \Delta V \) one needs to give at the planets and upon arrival to perform the final orbit injection.
Minimize: \( {\text{total}} \Delta V \) (V launcher is not counted and penalties are added).
Penalties are activated when the constrains on the swing by altitudes are violated.
Subject to:
Decision variables | Lower bound | Upper bound | Unit type |
|---|---|---|---|
\( X_{1} \) | − 1000 | 0 | Modified Julian date 2000 |
\( X_{2} \) | 30 | 400 | Days |
\( X_{3} \) | 100 | 470 | Days |
\( X_{4} \) | 30 | 400 | Days |
\( X_{5} \) | 400 | 2000 | Days |
\( X_{6} \) | 1000 | 6000 | Days |
Constrains: The minimum allowed of the fly-by pericenters for each travel is considered, and for each km of violation, a penalty factor is applied (Figs. 9).
CASSINI1 design problem
Travel | Planet | Min. pericenter radius (km) |
|---|---|---|
\( Rpmin1 \) | Venus | 6351.8 |
\( Rpmin2 \) | Venus | 6351.8 |
\( Rpmin3 \) | Earth | 6778.1 |
\( Rpmin4 \) | Jupiter | 671,492 |
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Maciel, O., Valdivia, A., Oliva, D. et al. A novel hybrid metaheuristic optimization method: hypercube natural aggregation algorithm. Soft Comput 24, 8823–8856 (2020). https://doi.org/10.1007/s00500-019-04416-2
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DOI: https://doi.org/10.1007/s00500-019-04416-2