  This paper studies a unified framework for graph matching problems called the
constrained gradient method. Popular algorithms within this framework include
graduated assignment (GA), integer projected fixed-point method (IPFP), and
doubly stochastic projected fixed-point method (DSPFP). These algorithms differ
from the step size parameter and constrained operator. Our contributed adaptive
step size parameter can guarantee the underlying algorithms' convergence and
enhance their efficiency and accuracy. A preliminary analysis suggests that the
optimal step size parameter has a high probability of being 1 in fully
connected graph matching. Secondly, we propose a dynamic strategy for
softassign, a popular constrained operator, to address its sensitivity
concerning nodes' cardinality and risk of overflow. Combining the adaptive step
size parameter and the dynamical softassign, we propose a novel graph matching
algorithm: the softassign constrained gradient method. Various experiments
demonstrate that it is significantly faster than other state-of-the-art
algorithms based on the constrained gradient method with improved accuracy.
