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The optimal transport (OT) problem and its related problems have attractedsignificant attention and have been extensively studied in variousapplications. In this paper, we focus on a class of group-quadratic regularizedOT problems which aim to find solutions with specialized structures that areadvantageous in practical scenarios. To solve this class of problems, wepropose a corrected inexact proximal augmented Lagrangian method (ciPALM), withthe subproblems being solved by the semi-smooth Newton ({\sc Ssn}) method. Weestablish that the proposed method exhibits appealing convergence propertiesunder mild conditions. Moreover, our ciPALM distinguishes itself from therecently developed semismooth Newton-based inexact proximal augmentedLagrangian ({\sc Snipal}) method for linear programming. Specifically, {\scSnipal} uses an absolute error criterion for the approximate minimization ofthe subproblem for which a summable sequence of tolerance parameters needs tobe pre-specified for practical implementations. In contrast, our ciPALM adoptsa relative error criterion with a \textit{single} tolerance parameter, whichwould be more friendly to tune from computational and implementationperspectives. These favorable properties position our ciPALM as a promisingcandidate for tackling large-scale problems. Various numerical studies validatethe effectiveness of employing a relative error criterion for the inexactproximal augmented Lagrangian method, and also demonstrate that our ciPALM iscompetitive for solving large-scale group-quadratic regularized OT problems.

A Corrected Inexact Proximal Augmented Lagrangian Method with a Relative  Error Criterion for a Class of Group-quadratic Regularized Optimal Transport  Problems