Community-based graph clustering is one of the most popular topics in the analysis of complex social networks. This type of clustering involves grouping vertices that are considered to share more connections, whereas vertices in different groups share fewer connections. A successful clustering result forms densely connected induced subgraphs. This paper studies a specific form of graph clustering problems that can be formulated as semi-assignment problems, where the objective function exhibits block properties. We reformulate these problems as sparse-constrained optimization problems and relax them to continuous optimization models. We then apply the quadratic penalty method and the quadratic penalty regularized method to the relaxation problem, respectively. Extensive numerical experiments demonstrate that both methods effectively solve graph clustering tasks for both synthetic and real-world network datasets. For small-scale problems, the quadratic penalty regularized method demonstrates greater efficiency, whereas the quadratic penalty method proves more suitable for large-scale cases.