IEEE Computational Intelligence Magazine - May 2023 - 38

FIGURE 6. Geometric interpretation of the PBI method.
'QTCCD' canalsobecombinedinto " GVQTCCD. " ). The
fourth amino acid, 'C' of'CTYCSYLY',israndomlychosenand
replaced with 'D.' The offspring is 'CTYDSYLY.'
Note that the length ofthe offspring sequence should be in
the feasible range. Other splitting points can be chosen if the
crossover operator generates a peptide sequence with a length
outside the feasible range.
2) Environmental Selection
In decomposition-based algorithms, three scalarizing functions,
namely, weighted sum (WS), Tchebycheff (TCH), and penaltybased
boundary intersection (PBI), are often employed. Here, in
environmental selection, the PBI method is employed since it
works best experimentally. Please refer to Table SI in the Supplementary
Material. Table SI shows that the results obtained by the
WS and TCH methods are statistically significantly worse than
those obtained by the PBI method. The possible reason is that the
contour lines formed by the PBImethod canwellmatch the distribution
of the obtained non-dominated solutions. The PBI
method transforms multiple objectives into a scalarizing function
formulated as follows:
PBIðx j wÞ¼ d1 þ u d2
fff
d1 ¼
ðÞfffðxÞ z Tw
kwk
;
d2 ¼ fffffðxÞ z þ d1
w
kwk
;
(5Þ
(6Þ
(7)
where u is a parameter to address the importance ofd2, and z
is the ideal point. Since the Pareto front ofthe multi-objective
optimization problem is known (shown in Figure 3), z is set
to ð0; 0Þ. Figure 6 shows the geometric interpretation of the
PBI method. d2 is the perpendicular distance offffffðxÞ to w and
d1 is the length of the projection offff
fffðxÞ onto w. Therefore,
minimizing the scalarizing function will likely generate an
optimal solution corresponding to the weight vector.
In AMPEMO, a series of uniformly distributed weight
vectors are generated in the objective space. Figure 7 shows an
example where the number of weight vectors is seven. For
38 IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE | MAY 2023
3)Local Search
Properly integrating evolutionary algorithms with local search
can enhance the search ability, especially for solving combinatorial
optimization problems such as AMP design. This study
uses the local search operator in Algorithm 5. The local search
FIGURE 7. A series of uniformly distributed weight vectors are used in
environmental selection. The elite archive collects solutions withf1 <
andf2 ¼ 0.
each weight vector, the environmental selection operator
given in Algorithm 4 is employed to select peptides for the
population in the next generation and the elite archive. In
Algorithm 4, for the i-th weight vector, first, a duplicate x0
xi (Line 1) is produced since xi may be later replaced. Second,
if the PBI value of the candidate peptide _x is smaller than that
ofxi, xi is replaced by _x (Lines 2-4). Finally, ifx0
i (also _x) is elite
(i.e., on the green line segment in Figure 7) and is not in the
population P, it is added to the elite archive E (Lines 5-10).
Algorithm 4. Environmental Selection
Input:
Candidate peptide _x;
Index i;
i-th weight vector wi;
Population P;
Elite archive E;
Output:
i-th solution xi in P;
Elite archive E;
1: x0
i ¼ xi;
2: if PBIð _x j wiÞ < PBIðxi j wiÞ then
3: xi ¼ _x;
4: end
5: if x0
6: E ¼ E [fx0
i is elite and x0
ig;
i =2 P then
7: end
8: if _x is elite and _x =2 P then
9: E ¼ E [f_xg;
10: end
i of
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