IEEE Computational Intelligence Magazine - November 2021 - 27

this paper, compared to the diversity of the solutions, the convergence
information is more important since we consider
improving the convergence performance of different tasks, so
the attainment function metric is utilized instead of HV.
2) Multi-Step Nonlinear Regression
As revealed by the proposed Theorem, MTO-DRA is intended
to resolve equation (9) by deriving the derivative of the
current best function g(t) using numerical differentiation
D
()
t
gt
D
implicitly. Conversely, aiming at handling the generalized
implicit objective function (10) in an explicit way, GRA constructs
the explicit mathematical form of the current best function
g(t) by nonlinear regression and subsequently derives its
corresponding derivative for computing the IoI vector. Generally,
the current best function g(t) indicates the best solution in
the first t generations, and hence g(t) can satisfy two attributes:
❏ Non-increasing Assume that the optimization problem is
to minimize a given objective, then current best function g(t)
should not obtain a worse solution when algorithm iteration
t proceeds. However, for the attainment function proposed in
equation (13), there can be the case that g(t) obtains a larger
value of attainment function with larger t, since the attainment
function only considers the convergence information
but not the diversity information. In spite of the special case
aforementioned, the general g(t) value should still be smaller
with increasing t from a long-term perspective.
❏ Convexity With increasing algorithm iteration, it is natural
that the changing magnitude, |( )|,gt2
er than zero, its second-order derivative 2
should get smaller. For
()
()
a non-increasing function whose derivative 2gt is no larg2gt
should be
larger than zero, which indicates that g(t) should be a convex
function.
Considering the attributes of the current best function g(t),
GRA explicitly constructs the function form as equation (14):
gt at ba=+ 12, 00t
()
log
,
(14)
where a and b are the system parameters and the function form
observes the two predefined properties. To derive the exact form
of g(t), one needs to estimate the system parameters a and b by
nonlinear regression, which is detailed in Algorithm 4. Notably,
the parameters [, ]ab are different for varying tasks. Before conducting
nonlinear regression, for each task k, the obtained
computational resource [, ,, ]
,,
attainment function [( ()), (( )),,kk f (( ))]IS tkM,
IS tI St
kk kM12
12
,,
,
tt tf and corresponding
should
be stored. For a stable estimate of MOO solution quality, instead
of computing derivative by two time steps like MTO-DRA,
Algorithm 4 Multi-step nonlinear regression.
Data: Sample data, (, (( ));
1 (( ), (), ..., ();,,...,)
! (( ( )), (( )), ..., (( )))T
! loglog
M
2 YI St IS tI StM
12
3 /*Least Square Method*/
4 [, ]( )
ab tt tY
-
TT1
!
Result: System parameters a and b
tt12 logtt 11 1 T
GRA employs M time steps of pairs, (, ())tI S . To be specific, these
pairs are recorded in a time window of size MK ,G))D
in tk,1 for task k is the generation amount obtained by task k in
overall generations [, D
where1
KG], tk,2
amount obtained in [, ]DD+KG KG12
represents the generation
, etc. Moreover, since the
objective space of varying tasks can be rather different, the solution
space of each task should be normalized when calculating
the attainment function in equation (13). The normalization of
task k can be described as equation (15):
f == (15)
,, ,
fijkl
ijk l
,, ,
maxfijkl
jl,
,, ,
where i, j, k, l represent the i-th objective, the j-th time step,
the k-th task, and the l-th MOO solution point, respectively.
With time window of size MK G))D and normalized
attainment function for each task in each time step, the
explicit current best function
gtk () for each task k can be
computed by the logarithmic least square method as in line 4
of Algorithm 4.
3) MOO-aware IoI Calculation
With the proposed normalized attainment function in
Section III.D.1), the current best function g(t) can evaluate
MOO solution quality in an online fashion. Equipped with the
multi-step nonlinear regression in Section III.D.2), GRA can
construct the explicit form of g(t) in the generalized implicit
objective function of equation (10). Additionally, with the
incorporation of the transfer information discussed in Section
III.C, we can redesign the IoI calculation process as illustrated
in Algorithm 5. During re-evaluation of IoI vector, the
current best function
jM
,, ,...,
12
gtk () for each task k is constructed first,
and then importance factor ka is computed for incorporating
transfer information recorded in rmp matrix, as equation (11).
Based on this online information, IoI vector can be evaluated
using the softmax function with parameter S.
Remark 2. It is notable that the computation process of IoI vector
in GRA is different from that of MTO-DRA. First, transfer information
a is taken into account. Second, the current best function g(t) is
constructed explicitly rather than implicitly analyzed with numerical
differentiation. Third, multiple steps of information are considered in
GRA rather than only two steps in MTO-DRA.
Algorithm 5 Re-evaluation of IoI vector.
Data: Random mating parameter matrix rmp, Time window size
parameter M, Task size K, Time interval
Result: New generated IoI vector
1 for k 1! to K do
2
3
t IS t11 22 tM
t ,; ;, IS tM
IS t (( )) f (( )))
[, ]ab Mkk ! -step nonlinear regression Algorithm 4
a k is computed as equation(11)
4 end
5 /*Compute IoI Vector*/
6 for k 1! to K do
7
IoIk
8 end
!
/
k
K
DG , Softmax parameter S
exp a )Sa
kk
)
=1 exp a )Sa
)
()
()
kk
NOVEMBER 2021 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 27
IEEE Computational Intelligence Magazine - November 2021

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