This paper proposes a new formulation for the corrective security-constrained optimal power flow (SCOPF) problem with DC power flow constraints. The goal is to produce a generation schedule which has a minimal number of post-contingency corrections as well as a minimal amount of total MW rescheduled. In other words, the new SCOPF model effectively clears contingencies with corrective actions that have a minimal impact on system operations. The proposed SCOPF model utilizes sparse optimization techniques to achieve computational tractability for large-scale power systems. We also propose two efficient decomposition algorithms. Extensive computational experiments show the advantage of the proposed model and algorithms on several standard IEEE test systems and large-scale real-world power systems.