  The optimal transport (OT) problem and its related problems have attracted
significant attention and have been extensively studied in various
applications. In this paper, we focus on a class of group-quadratic regularized
OT problems which aim to find solutions with specialized structures that are
advantageous in practical scenarios. To solve this class of problems, we
propose a corrected inexact proximal augmented Lagrangian method (ciPALM), with
the subproblems being solved by the semi-smooth Newton ({\sc Ssn}) method. We
establish that the proposed method exhibits appealing convergence properties
under mild conditions. Moreover, our ciPALM distinguishes itself from the
recently developed semismooth Newton-based inexact proximal augmented
Lagrangian ({\sc Snipal}) method for linear programming. Specifically, {\sc
Snipal} uses an absolute error criterion for the approximate minimization of
the subproblem for which a summable sequence of tolerance parameters needs to
be pre-specified for practical implementations. In contrast, our ciPALM adopts
a relative error criterion with a \textit{single} tolerance parameter, which
would be more friendly to tune from computational and implementation
perspectives. These favorable properties position our ciPALM as a promising
candidate for tackling large-scale problems. Various numerical studies validate
the effectiveness of employing a relative error criterion for the inexact
proximal augmented Lagrangian method, and also demonstrate that our ciPALM is
competitive for solving large-scale group-quadratic regularized OT problems.
