Our ability to solve mixed-integer linear optimization problems (MILO) has increased dramatically over the past 25 years. In particular, current branch-and-bound solvers for MILO are over one million times faster than their counterparts in 1991 due to software improvements alone. Thus, problems considered intractable two decades ago can now be solved to optimality in minutes or seconds. However, many decision-making problems under uncertainty cannot be adequately modeled as MILO. Such problems can often be modeled as mixed-integer second-order cone optimization problems (MISOCO), but solvers for MISOCO are still struggling to solve the problems that arise in practice.

In this talk we present new improvements for MISOCO based on the study of the set defined by a single conic constraint with binary and continuous variables and separable quadratic term. We give a complete convex-hull description of the considered set and show how to use the resulting valid inequalities for non-separable quadratic terms. We also show how to strengthen the inequalities using additional constraints of the optimization problem. The proposed inequalities can be used in a broad class of problems, including value-at-risk minimization and assortment problems, and result in significant performance improvements.