Survival analysis is a family of statistical methods for analyzing event occurrence times. In this paper, we address the mixed-integer optimization approach to sparse estimation of the Cox proportional-hazards model for survival analysis. Specifically, we propose a high-performance cutting-plane algorithm based on reformulation of bilevel optimization for sparse estimation. This algorithm solves the upper-level problem using cutting planes that are generated from the dual lower-level problem to approximate an upper-level nonlinear objective function. To solve the dual lower-level problem efficiently, we devise a quadratic approximation of the Fenchel conjugate of the loss function. We also develop a computationally efficient least-squares method for adjusting quadratic approximations to fit each dataset. Computational results demonstrate that our method outperforms the L1-regularized estimation method in terms of accuracy for both prediction and subset selection. Moreover, our quadratic approximation of the Fenchel conjugate function accelerates the cutting-plane algorithm and improves the generalization performance of sparse Cox proportional-hazards models.