IEEE Computational Intelligence Magazine - May 2022 - 51

half-dozen broad categories together with
representative experimental studies from
past publications. Although by no means
comprehensive, these six examples showcase
the computational advantages that
EMT could bring to areas such as the evolution
of embodied intelligence, the path
planning of unmanned vehicles, or lastmile
logistics optimization, to name just a few.
... by viewing the process of multitasking through the
lens of Eq. (3), it is possible to adaptively control the
extent of skills transfer between tasks by tuning the
coefficients of the mixture models.
❏ Beyond the six application domains, the paper presents recipes
by which certain problem formulations of applied interest-those
that cut across disciplines-may be transformed
in the new light of EMT. The formulations fall under the
umbrella of multi-X EC [4], unveiling avenues by which a
population's implicit parallelism, augmented by the capacity
to multitask, could be further leveraged for practical problem-solving.
These
discussions are intended to highlight the potential usecases
of EMT methods known today. In addition, it is hoped to
spark future research toward innovative multitask algorithms
crafted specifically for real-world deployment.
The rest of the paper is organized as follows. Section II
introduces multitask optimization, formulates EMT from a
probabilistic viewpoint, and presents a brief methodological
overview. Section III sets out the half-dozen broad categories
summarizing practical exemplars of EMT. Future applied
research prospects of multitasking are then discussed in Section
IV. Section V concludes the paper.
II. Background
In this section, the preliminaries of multitask optimization
are presented, a general probabilistic formulation for evolutionary
multitasking is introduced, and representative algorithmic
approaches from the literature are discussed-thus
laying the foundation for the study of real-world applications
of EMT.
A. The Multitask Optimization Problem
Multitask optimization (MTO) comprises multiple problem
instances to be jointly solved. Without loss of generality, MTO
consisting of K tasks can be defined as1:
xxi
x Xii
where x , Xi
)
i and fi
)== f
!
ii
argmax (),, ,, ,fi K12for
(1)
represent the optimal solution, search
space, and objective/fitness function of the ith task, respectively.
Typically, optimization includes additional constraint functions,
but these have been omitted in Eq. (1) for simplicity.
The motivation behind MTO is to facilitate the transfer of
learned skills from one task to another, enhancing overall optimization
performance. For such transfer to take place, a unified
1 Only single-objective maximization is depicted for brevity. The concept of MTO
readily extends to multiple multi-objective optimization tasks [33], or even a mixture
of single- and multi-objective optimization tasks [34]. For minimization, fitness functions
are simply multiplied by -1.
space X encoding candidate solutions for all K tasks is usually
defined (alternative approaches are discussed in Section II-C).
Let the encoding be achieved by an invertible mapping function
} i
for the ith task, such that }iiX .X "
:
-1
}i
solutions from the unified space back to a task-specific search
space is given as :.X X
" i Early works utilized the randomkey
representation [8] for solution mapping. More recently, solution
representation learning strategies have been derived [20],
[21], forming common highways by which building-blocks of
knowledge derived from heterogeneous tasks (i.e., with differing
search spaces) can be transmitted and recombined.
B. A Probabilistic Formulation of EMT
Defining an optimization task, with fitness function :
f
p () # () () .
$$
ii ii i
iix Xi
i X Ri
" ,
in terms of the expected fitness under a probability distribution
function gives [35]:
max fp dxx x
In probabilistic model-based EAs, ()p xii
#fd f
tion does not change the optimization outcome.
Consider MTO with K tasks. These are encoded in a unified
Xi
space X with a set of probabilistic models {( ), (),,pp
p ()}xK
12
xx f
corresponding to task-specific (sub-)populations. One
way to pose EMT is then by generalizing Eq. (2) over all K
tasks using probability mixture models as [14]:
K
{, (),, }/# (( )) [( )]$$ $
-1
wp ij
ij j x 6
..
max fw pd
wi
st R 1
j=
i =1 X
K
wi j
1
ij = ,,
ij $ 60,, ,
i } xx x,
6
ij j
R 1
K
=
ij
(3)
where wij 's are scalar coefficients indicating how individual
models are assimilated into the mixture.
Note that Eq. (3) would be exactly solved when all probabilistic
models converge to the respective optimal Dirac delta
functions () (( ))
p xx x
j
j
)) X , and w 0ij
d} j
=in
=
for all ij .!
Hence, the reformulation is in alignment with the definition of
MTO in Eq. (1). More importantly however, by viewing the process
of multitasking through the lens of Eq. (3), it is possible to adaptively
control the extent of skills transfer between tasks by tuning the coefficients
of the mixture models. Precisely, if candidate solutions
evolved for the jth task-i.e., drawn from the model ()
are found to be performant for the ith task as well, then the
pxj
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MAY 2022 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 51
(2)
represents the underlying
search distribution of a population of evolving solutions [36].
Note, if xi
tered at xi
() () ().
d -=i ii i
ii ixx xx x
) is the true optimum, then a Dirac delta function cen)
optimizes Eq. (2); i.e., () ()p xx x=-d
)) As such, probabilistic reforma))
since
i ii i
Then, the decoding of
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