Facial Reduction Algorithms for Conic Optimization Problems

Hayato Waki, Masakazu Muramatsu

    Research output: Contribution to journalArticlepeer-review

    41 Citations (Scopus)


    In the conic optimization problems, it is well-known that a positive duality gap may occur, and that solving such a problem is numerically difficult or unstable. For such a case, we propose a facial reduction algorithm to find a primal-dual pair of conic optimization problems having the zero duality gap and the optimal value equal to one of the original primal or dual problems. The conic expansion approach is also known as a method to find such a primal-dual pair, and in this paper we clarify the relationship between our facial reduction algorithm and the conic expansion approach. Our analysis shows that, although they can be regarded as dual to each other, our facial reduction algorithm has ability to produce a finer sequence of faces of the cone including the feasible region. A simple proof of the convergence of our facial reduction algorithm for the conic optimization is presented. We also observe that our facial reduction algorithm has a practical impact by showing numerical experiments for graph partition problems; our facial reduction algorithm in fact enhances the numerical stability in those problems.

    Original languageEnglish
    Pages (from-to)188-215
    Number of pages28
    JournalJournal of Optimization Theory and Applications
    Issue number1
    Publication statusPublished - Jul 2013

    All Science Journal Classification (ASJC) codes

    • Control and Optimization
    • Applied Mathematics
    • Management Science and Operations Research


    Dive into the research topics of 'Facial Reduction Algorithms for Conic Optimization Problems'. Together they form a unique fingerprint.

    Cite this