Improved ant colony optimization algorithms for continuous ... - CoDE

4.4 Performance Evaluation of ACOMV-o and ACOMV-c 39
X2
−2 0 2 4 6
Natural
−2 0 2 4 6
X1
X2
−2 0 2 4 6
Random
−2 0 2 4 6
Figure 4.3: Randomly rotated ellipsoid function (β = 5) with discrete variable
x1 ∈ T, |T| = t = 30. The left plot presents the case in which the
natural ordering of the intervals is used, while the right one presents the
case in which a random ordering is used.
characteristics such as non-separable, ill-conditioned and multi-modal. Nonseparate
functions often exhibit intricate dependencies between decision
variables. Ill-conditioned functions, like fRosenbrockMV , often lead to premature
convergence. Multi-modal functions, like fAckleyMV , fRastriginMV and
fGriewankMV , serves to find effectively a search globally in a highly multimodal
topography [43]. For example, in the continuous study of [8], we
can see PSO performs well on the separable problems. However, on nonseparable
problems, PSO exhibits a strong performance decline, and PSO
also performs very poorly even on moderately ill-conditioned functions, let
alone in mixed-variable optimization cases. Therefore, the proposed artificial
mixed-variable benchmark functions are expected to lead a challenge
for different mixed-variable optimization algorithms. In anther aspect, the
flexible discrete intervals and dimensions of the proposed benchmark functions
are not only helpful for investigating the performance scalability of
mixed-variable optimization algorithms, but also provide a convenient environment
for automatic parameter tuning in mixed-variable optimization
solvers generalization, thereby facing unseen real-world complex engineering
optimization problems.
4.4 Performance Evaluation of ACOMV-o and
ACOMV-c
ACOMV-o and ACOMV-c represent a continuous relaxation approach and a
native mixed-variable optimization approach on handling discrete variables,
X1