Differential Evolution Algorithm

Author: Tianjiang Says
Website: https://cislunarspace.cn

Definition

Differential Evolution (DE) is a population-based stochastic optimization algorithm that belongs to the family of evolutionary algorithms. It is particularly effective for continuous optimization problems and has been widely applied in orbital mechanics for trajectory design, orbit determination, and control optimization.

Core Operations

DE evolves a population of candidate solutions through three operations:

Mutation: For each candidate $\mathbf{x}_i$ , create a mutant vector:
$\mathbf{v}_i = \mathbf{x}_{r1} + F \cdot (\mathbf{x}_{r2} - \mathbf{x}_{r3})$
where $r1, r2, r3$ are random indices and $F$ is the mutation factor.
Crossover: Combine the mutant with the original to create a trial vector:
$u_{i,j} = \begin{cases} v_{i,j} & \text{if } rand < CR \\ x_{i,j} & \text{otherwise} \end{cases}$
where $CR$ is the crossover rate.
Selection: Keep the better of the original and trial vectors:
$\mathbf{x}_i^{(t+1)} = \begin{cases} \mathbf{u}_i & \text{if } f(\mathbf{u}_i) \leq f(\mathbf{x}_i) \\ \mathbf{x}_i & \text{otherwise} \end{cases}$

Applications in Orbital Mechanics

Application	Description
Initial value search	Finding initial conditions for DRO computation in ephemeris models
Orbit keeping optimization	Optimizing control parameters for station-keeping maneuvers
Transfer orbit design	Searching for low-energy transfer trajectories
Multi-objective optimization	Balancing fuel consumption, orbit accuracy, and observation coverage

Advantages

Global search capability: Population-based approach avoids local optima
Derivative-free: No gradient computation required
Few control parameters: Only $F$ and $CR$ need tuning
Parallelizable: Population members can be evaluated independently

References

Chen Yuju. DRO Orbit Design and Control Research for Cislunar Space Situation Awareness[D]. 2024.