Abstract: Objective function-based clustering is a class of important clustering analysis techniques, of which almost all the algorithms are built by optimization of non-convex objective. Thus, these algorithms can hardly get global optimal solution and are sensitive to the provided initialization. Recently, convex clustering has been proposed by optimizing a convex objective function, not only does it overcome the insufficiency illustrated above, but it also obtains a relatively stable solution. It has been proven that clustering performance can be improved effectively by combining useful auxiliary information (typically must-links and/or cannot-links) obtained from reality with the corresponding objective. To the best of our knowledge, all such semi-supervised objective function-based clustering algorithms are based on non-convex objective, semi-supervised convex clustering has not been proposed yet. Thus, we attempt to combine pairwise constraints with convex clustering. However, the existing methods usually make the original convex objectives lose their convexity, which add constraint penalty terms to the objective function. In order to deal with such problem, we introduce a novel semi-supervised convex clustering model by using the weakly-supervised information. In particular, the key idea is to change distance metric instead of adding constraint penalty terms to the objective function. As a result, the proposed method not only maintains the advantages of convex clustering, but also improves the performance of convex clustering.