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Jun, et al.

crossover, and selection operations inspired by natural In summary, Nadam and DE exhibit complementary
selection. In the mutation stage, DE perturbs the current optimization characteristics. Nadam is suitable
individual using difference vectors randomly selected for continuous optimization problems with known
from the population to form an exploratory mutation gradients and offers excellent local convergence speed,
vector. During the crossover stage, the mutation vector while DE is capable of conducting large-scale global
is combined with the current individual according to a searches without requiring gradient data. Combining
specified probability, ensuring diversity in the search the two allows for the generation of high-quality global
process. In the selection stage, the algorithm retains initial solutions via DE, followed by accelerated local
individuals with better fitness values to proceed to the convergence in the solution space through Nadam.
next generation. This strategy enables DE to exhibit This hybrid strategy improves both the accuracy
strong global search capabilities without relying on and efficiency of the overall optimization process. It
gradient information, making it particularly suitable for enhances robustness, reduces the risk of becoming
complex, non-convex, or non-differentiable problems. trapped in local optima, and is of great significance
However, the local development ability of DE is for solving complex, high-dimensional optimization
relatively weak. Once the solution space begins to problems.
24
converge, it often lacks a fine search mechanism.
Moreover, the computational cost associated with 3.2. Design of hybrid algorithms and IEM
population-based evolution can be relatively high. To fully leverage the strengths of Nadam and DE,
Specifically, in the mutation stage, three distinct this study proposes a hybrid optimization algorithm
individuals, x , x , x are selected from the population that integrates the fast local convergence capability of
r1
r2
r3
to construct a difference vector. The mutation vector is Nadam with the strong global search ability of DE. In
generated as follows: this framework, the overall population was divided into
two subpopulations: one employed the Nadam algorithm
v = x + F⋅(x -x ) (V) for gradient-driven local development, while the other
r3
r2
i
r1
Where F∈ [0,2] is the scaling factor that controls applied the DE algorithm to perform global exploration.
the amplitude of the differential disturbance. In the To facilitate dynamic collaboration between the two,
crossover operation, the mutation vector is mixed with an IEM was introduced to regularly exchange optimal
the current individual x using a crossover probability solutions between subpopulations during the algorithm
i
CR∈[0,1] to produce the test vector u : operation process, promoting effective integration of
i
global and local search strategies.
v , ifrand CRorj j The core idea of this IEM lies in a strategy termed

u ij, rand (VI) “exchanging the superior for the inferior.” Specifically,
ij,
x , otherwise
ij,
at regular intervals (for example, every five generations),
the current optimal individual is exchanged between
Here, j rand is a randomly selected index that ensures the two subpopulations. This promotes information
that at least one dimension is inherited from the mutation flow across the solution space, prevents the algorithm
vector. Finally, u and x are compared using the fitness from converging to local optima, and simultaneously
i
i
function. The individuals with better performance are enhances both search diversity and accuracy. The basic
retained for the next generation: workflow of the IEM is shown in Figure 1.
f x
u , iff u Within each exchange cycle, the individual with

x i t1 i i i (VII) the current best performance (i.e., the solution with
x , otherwise the lowest objective function value) was first selected
i
from the Nadam subpopulation. This elite individual
The DE algorithm is widely used in various complex replaced the poorest-performing individual in the
optimization problems due to its simple structure and DE subpopulation. This step ensures that the DE
flexible implementation. It performs particularly well in subpopulation can inherit the local region optimization
scenarios where the search space is complex and gradient achievements obtained by Nadam, enabling it to
information is difficult to obtain. However, the pure DE continue global search and mining in this region.
method has limited local search ability, and its computational Conversely, the best-performing individual from the
cost can be relatively high due to the population evolution DE subpopulation was selected to replace the worst-
process, especially in high-dimensional spaces. performing individual in the Nadam subpopulation.

Volume 22 Issue 5 (2025) 68 doi: 10.36922/AJWEP025210165