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Solving Natural Conic Formulations with Hypatia.jl

Author

Listed:
  • Chris Coey

    (Operations Research Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139)

  • Lea Kapelevich

    (Operations Research Center, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139)

  • Juan Pablo Vielma

    (Google Research and MIT Sloan School of Management, Cambridge, Massachusetts 02142)

Abstract

Many convex optimization problems can be represented through conic extended formulations (EFs) using only the small number of standard cones recognized by advanced conic solvers such as MOSEK 9. However, EFs are often significantly larger and more complex than equivalent conic natural formulations (NFs) represented using the much broader class of exotic cones . We define an exotic cone as a proper cone for which we can implement easily computable logarithmically homogeneous self-concordant barrier oracles for either the cone or its dual cone. Our goal is to establish whether a generic conic interior point solver supporting NFs can outperform an advanced conic solver specialized for EFs across a variety of applied problems. We introduce Hypatia, a highly configurable open-source conic primal-dual interior point solver written in Julia and accessible through JuMP. Hypatia has a generic interface for exotic cones, some of which we define here. For seven applied problems, we introduce NFs using these cones and construct EFs that are necessarily larger and more complex. Our computational experiments demonstrate the advantages, especially in terms of solve time and memory usage, of solving the NFs with Hypatia compared with solving the EFs with either Hypatia or MOSEK 9.

Suggested Citation

  • Chris Coey & Lea Kapelevich & Juan Pablo Vielma, 2022. "Solving Natural Conic Formulations with Hypatia.jl," INFORMS Journal on Computing, INFORMS, vol. 34(5), pages 2686-2699, September.
    • Handle: RePEc:inm:orijoc:v:34:y:2022:i:5:p:2686-2699
    DOI: 10.1287/ijoc.2022.1202
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    References listed on IDEAS

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