Mixed-integer convex optimization problems (MICPs) are problems that become convex when all integrality constraints are relaxed. I will present recent advances in solving these problems to global optimality by constructing polyhedral relaxations in a higher-dimensional space. This work develops significant new connections between MICP and modeling with symmetric and nonsymmetric convex cones, a discovery that influenced the development of MOSEK version 9 with their support for exponential and power cones. I will present our own implementation of an iterative outer approximation algorithm and a branch-and-cut variant in the open-source MICP solver Pajarito.

This is joint work with Russell Bent, Chris Coey, Juan Pablo Vielma and Emre Yamangil.

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