Multi-graph matching is a prominent structured prediction task, in which the predicted label is constrained to the space of cycle-consistent matchings. While direct loss minimization is an effective method for learning predictors over structured label spaces, it cannot be applied efficiently to the problem at hand, since executing a specialized solver across sets of matching predictions is computationally prohibitive. Moreover,there’s no supervision on the ground-truth matchings over cycle-consistent prediction sets.Our key insight is to strictly enforce the matching constraints in pairwise matching predictions and softly enforce the cycle-consistency constraintsby casting them as weighted loss terms, such that the severity of inconsistency with global predictions is tuned by a penalty parameter.Inspired by the classic penalty method, we prove that our method theoretically recovers the optimal multi-graph matching constrained solution.Our method's advantages are brought to light in experimental results on the popular keypoint matching task on the Pascal VOC and the Willow ObjectClass datasets.