Cutting-plane algorithm for estimation of sparse Cox proportional hazards models

Survival analysis is a family of statistical methods for analyzing event occurrence times. We adopt a mixed-integer optimization approach to estimation of sparse Cox proportional hazards (PH) models for survival analysis. Specifically, we propose a high-performance cutting-plane algorithm based on a reformulation of our sparse estimation problem into a bilevel optimization problem. 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 regularized estimation methods in terms of accuracy for both prediction and subset selection especially for low-dimensional datasets. Moreover, our quadratic approximation of the Fenchel conjugate function accelerates the cutting-plane algorithm and maintains high generalization performance of sparse Cox PH models.