  The optimization problems with a sparsity constraint is a class of important
global optimization problems. A typical type of thresholding algorithms for
solving such a problem adopts the traditional full steepest descent direction
or Newton-like direction as a search direction to generate an iterate on which
a certain thresholding is performed. Traditional hard thresholding discards a
large part of a vector when the vector is dense. Thus a large part of important
information contained in a dense vector has been lost in such a thresholding
process. Recent study [Zhao, SIAM J Optim, 30(1), pp. 31-55, 2020] shows that
the hard thresholding should be applied to a compressible vector instead of a
dense vector to avoid a big loss of information. On the other hand, the optimal
$k$-thresholding as a novel thresholding technique may overcome the intrinsic
drawback of hard thresholding, and performs thresholding and objective function
minimization simultaneously. This motivates us to propose the so-called partial
gradient optimal thresholding method in this paper, which is an integration of
the partial gradient and the optimal $k$-thresholding technique. The solution
error bound and convergence for the proposed algorithms have been established
in this paper under suitable conditions. Application of our results to the
sparse optimization problems arising from signal recovery is also discussed.
Experiment results from synthetic data indicate that the proposed algorithm
called PGROTP is efficient and comparable to several existing algorithms.
