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

8 Ant Colony Optimization
Algorithm 2 Outline of ACOR
Input: k, m, D, q, ξ, and termination criterion.
Output: The best solution found
Initialize and evaluate k solutions
// Sort solutions and store them in the archive
T = Sort(S1 · · · Sk)
while Termination criterion is not satisfied do
// Generate m new solutions
for l = 1 to m do
// Construct solution
for i = 1 to D do
according to weights
Select Gaussian g i j
Sample Gaussian g i j with parameters µi j , σi j
end for
Store and evaluate newly generated solution
end for
// Sort solutions and select the best k
T = Best(Sort(S1 · · · Sk+m), k)
end while
coordinates of all the solutions in the archive are rotated and recalculated
according to this new orthogonal base. At the end of the solution construction
process, the chosen values of the temporary variables are converted back
into the original coordinate system.
2.2 Further Investigation on ACOR
The original implementation of ACOR is in R [48] that is a language and
environment for statistical computing and graphics. For a higher execution
efficiency, we developed a C++ implementation. Figure 2.1 and 2.2 shows
the coincident performance of two different implementation and illustrates
the validity of C++ implementation. Moreover, Figure 2.1 is shown on
both non-rotated and rotated functions to demonstrate the positive effect
of variable correlation handling method in ACOR on rotated functions. The
formula of the non-rotated and rotated functions are given in Table 2.1.
When ACOR constructs a solution, a Gram-Schmidt process is used for
the new orthogonal basis of the ant’s coordinate system. Although, it helps
to handle variable correlation, the calculation of the Gram-Schmidt process
for each variable for each constructive step incurs a very high computational
demand. When the dimension of the objective function increases, the time
used by ACOR with this variable correlation handling increases rapidly. For
this reason, we also developed a C++ implementation of ACOR that does
not consider variable correlation handling part (it is referred as Sep-ACOR