Open Access

In recent work, we provide computational arguments for expanding the class of proper cones recognized by conic optimization solvers, to permit simpler, smaller, more natural conic formulations. We define an exotic cone as a proper cone for which we can implement a small set of tractable (i.e. fast, numerically stable, analytic) oracles for a logarithmically homogeneous self-concordant barrier for the cone or for its dual cone. Our extensible, open-source conic interior point solver, Hypatia, allows modeling and solving any conic problem over a Cartesian product of exotic cones. In this paper, we introduce Hypatia’s interior point algorithm, which generalizes that of Skajaa and Ye (Math. Program. 150(2):391–422, 2015) by handling exotic cones without tractable primal oracles. To improve iteration count and solve time in practice, we propose four enhancements to the interior point stepping procedure of Skajaa and Ye: (1) loosening the central path proximity conditions, (2) adjusting the directions using a third order directional derivative barrier oracle, (3) performing a backtracking search on a curve, and (4) combining the prediction and centering directions. We implement 23 useful exotic cones in Hypatia. We summarize the complexity of computing oracles for these cones and show that our new third order oracle is not a bottleneck. From 37 applied examples, we generate a diverse benchmark set of 379 problems. Our computational testing shows that each stepping enhancement improves Hypatia’s iteration count and solve time. Altogether, the enhancements reduce the geometric means of iteration count and solve time by over 80% and 70% respectively.

We face the issue of finding alternative paradigms for the resolution of generic Mixed Integer Programs (MIP), by considering the perspective option of general purpose solvers which switch to decomposition methods when pertinent. Currently, the main blocking factor in their design is the problem of automatic decomposition of MIPs, that is to produce good MIP decompositions algorithmically, looking only at the algebraic structure of the MIP instance. We propose to employ Dantzig–Wolfe reformulation and machine learning methods to obtain a fully data driven automatic decomposition framework. We also design strategies and introduce algorithmic techniques in order to make such a framework computationally effective. An extensive experimental analysis shows our framework to grant substantial improvements, in terms of both solutions quality and computing time, with respect to state-of-the-art automatic decomposition techniques. It also allows us to gain insights into the relative impact of different techniques. As a side product of our research, we provide a dataset of more than 31 thousand random decompositions of MIPLIB instances, with 121 features, including computations of their root node relaxation.