Contact-rich manipulation is challenging due to dynamically-changing physical constraints by the contact mode changes undergone during manipulation. This paper proposes a versatile local planning and control framework for contact-rich manipulation that determines the continuous control action under variable contact modes online. We model the physical characteristics of contact-rich manipulation by quasistatic dynamics and complementarity constraints. We then propose a linear complementarity quadratic program (LCQP) to efficiently determine the control action that implicitly includes the decisions on the contact modes under these constraints. In the LCQP, we relax the complementarity constraints to alleviate ill-conditioned problems that are typically caused by measure noises or model miss-matches. We conduct dynamical simulations on a 3D physical simulator and demonstrate that the proposed method can achieve various contact-rich manipulation tasks by determining the control action including the contact modes in real-time.