IEEE Computational Intelligence Magazine - November 2021 - 35

the best performance compared to the other two algorithms.
Notably, although GRA achieves the best performance in
task 2, it still allocates some of the resources for task 2 since the
MOO solver is generally unaware of the global optima, and
this behavior can guarantee the stable allocation ability of GRA
due to the softmax function normalization for IoI vector. Furthermore,
in the last several generations, task 1 also tends to
converge, and hence task 1 and task 2 can generally attain nearly
average allocated resources.
H. A Case Study on Real-World Application:
Hyperspectral Unmixing
By collecting spectra of numerous bands, the hyperspectral
remote sensors can capture the hyperspectral images whose
pixels contain more detailed information than the color
images. However, due to the limitation of sensor resolution,
the obtained pixels in the hyperspectral images may be the
mixture of several materials [50]. To tackle this problem,
hyperspectral unmixing aims at specifying spectra (endmembers)
and estimating fractional abundances for the given pixels
[51] to decompose the mixed materials. When given the spectral
library, the actual effective spectra for a hyperspectral pixel
can be relatively sparse, and therefore the hyperspectral
unmixing problem can be resolved through sparse reconstruction
techniques.
Formally speaking, given a vector of hyperspectral image
pixels Y of size M 1# and a spectral library A of size M × N,
where M and N stand for the size of spectral bands and spectral
signatures, respectively, hyperspectral unmixing is intended
to obtain the sparse fractional abundance vector X of size
N 1#
, and the unmixing process takes the following equation
(18):
YAX=
(18)
To adopt the evolutionary multi-objective algorithms in this
application, an effective approach has been proposed in [50],
which is to optimize the sparsity of vector X and the reconstruction
error simultaneously, as shown in equation (19):

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