IEEE Computational Intelligence Magazine - May 2022 - 63

has thus opened the door for EAs to wideranging
disciplines. However, it has been
shown that as the number of objective functions
increases (referred to as many-objective
optimization problems, or MaOPs for short),
the convergence rate of EAs may begin to
slow down due to severe weakening of selection
pressures [117]. It is to remedy this
shortcoming that motivates the revisit of MaOPs through the
lens of EMT.
Lemma 1 of [118] suggests that an MaOP could be simplified
into several MOPs, such that points on the Pareto front of
an MOP also belong to the Pareto front of the target MaOP.
Hence, the lemma naturally leads to a recipe for turning
MaOPs into MTO problem formulations through the generation
of a series of auxiliary multi-objective optimization tasks.
The known efficacy of EAs for MOPs could then be harnessed
in an EMT algorithm to solve the main MaOP, with intuitive
guarantees of useful inter-task information transfer. Notably, a
different but associated idea has already been studied in [119],
where a large-scale MaOP is transformed into MTO and
solved using the MFEA. The experimental results showed that,
with limited computational budget, the multitask approach
outperformed state-of-the-art baselines on benchmark MaOPs.
Similar to the recipe above, one can imagine that given a
multi-constrained problem (or combined multi-objective,
multi-constrained problem), simplified auxiliary tasks may be
generated by dropping out some of the constraints [120]. As
long as the a priori unknown active constraints are preserved, it is
likely that solutions evolved for the auxiliary tasks would transfer
beneficially to the main task at hand.
B. EMT in Multi-Fidelity Optimization
Multi-fidelity optimization is arguably a precise fit for MTO,
and, by extension, for EMT. A population of candidate solutions
is evolved to solve lower-fidelity tasks (with less accurate
but cheap function evaluations) jointly with the high-fidelity
(accurate but expensive) target problem instance-with the
goal of reducing the load on high-fidelity analysis. The lowerfidelity
tasks thus serve as catalysts to help quickly solve the target.
Given K tasks, where the K-th is the target, the MTO can
then be stated as:
{, ,, ,} =
xx xx
f
where the fi
and fK
low
)) ))
-
KK
12 11 2
1
argmax ff
ff
{( ),
low
f,( ),
low
Khigh
is their high-fidelity counterpart.
The setup of Eq. (4) has widespread practical applicability. It
has been alluded to previously in Section III, in connection
with data science pipelines (for small to big data transfers) and
safe UAV path planning. Engineering design also forms a major
application area, where low-fidelity models extensively used for
preliminary designs can be readily integrated into MTO frameworks.
An illustrative case study was carried out in [97], where
models with different levels of accuracy were combined in
K
low
xx
xx
()
high()},
(4)
's represent the low-fidelity objective functions,
The transformation of multi-X optimization into MTO
thus unpacks a new box of tools for practitioners,
going beyond the typical rendition of EMT for
simultaneously solving distinct problem instances.
MTO for the multi-objective optimization of beneficiation
processes; a variant of the MO-MFEA was utilized to this end.
Multitasking across local and global models in surrogate-assisted
optimization was considered in [121]. Furthermore, a generalized
EMT algorithm crafted for multi-fidelity problems in
particular was proposed in [122].
C. EMT in Multi-Level Optimization
Multi-level optimization is characterized by mathematical programs
whose constraints include a series of optimization problems
to be solved in a predetermined sequence. For simplicity,
our discussion here is limited to situations where only a single
such constraint exists, forming what is typically referred to as a
bilevel optimization problem [123]. A sample formulation of a
bilevel program is as follows:
minargmax (, ),ff (5)
!
x !
where fu
uuX
uu ll (, ),xx stxx xlu l
x X
))!
..
ll
is the upper-level objective function and fl
is the
lower-level objective function. The setup in Eq. (5) has manifold
real-world applicability, with examples in environmental
economics, optimal design, cybersecurity, and others [123].
In the regime of black-box search, solving Eq. (5) may
however give rise to computational bottlenecks in having to
repeatedly optimize lower-level problem instances corresponding
to different candidate solutions { ,,, }
xx x,, ,uu u123 f at the
upper level. It is in addressing this fundamental issue that EMT
is expected to excel. By viewing the lower-level through the
lens of EMT, a set of optimization tasks can be jointly solved as
part of a single MTO setting as:
xx forxx xxui uu
li, ,, ,,
x Xll
argmaxf (, ),
luil
)== f
!
12
{, ,}.
(6)
The recipe in Eq. (6) was first explored in [124], under the intuitive
assumption that similar upper-level candidate solutions would
lead to lower-level problem instances amenable to inter-task
transfers. An application optimizing the complete manufacturing
cycle of lightweight composites substantiated this intuition, giving
approximately 65% saving in computation time compared to a
standard evolutionary bilevel algorithm. In [125], the authors considered
solving expensive minimax optimization-derived by setting
ff
ul
= in Eq. (5)-via EMT. The resultant worst-case
formulation was used to model a robust airfoil design problem,
with experimental results showing that a surrogate-assisted MFEA
vastly outperformed all the baseline algorithms. (It is contended
that the success of [125] could be extended to multi-objective
minimax problems [126], [127] as well.)
MAY 2022 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 63
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