Mathematical Programming

, Volume 150, Issue 2, pp 391–422 | Cite as

A homogeneous interior-point algorithm for nonsymmetric convex conic optimization

  • Anders SkajaaEmail author
  • Yinyu Ye
Full Length Paper Series A


A homogeneous interior-point algorithm for solving nonsymmetric convex conic optimization problems is presented. Starting each iteration from the vicinity of the central path, the method steps in the approximate tangent direction and then applies a correction phase to locate the next well-centered primal–dual point. Features of the algorithm include that it makes use only of the primal barrier function, that it is able to detect infeasibilities in the problem and that no phase-I method is needed. We prove convergence to \(\epsilon \)-accuracy in \({\mathcal {O}}(\sqrt{\nu } \log {(1/\epsilon )})\) iterations. To improve performance, the algorithm employs a new Runge–Kutta type second order search direction suitable for the general nonsymmetric conic problem. Moreover, quasi-Newton updating is used to reduce the number of factorizations needed, implemented so that data sparsity can still be exploited. Extensive and promising computational results are presented for the \(p\)-cone problem, the facility location problem, entropy maximization problems and geometric programs; all formulated as nonsymmetric convex conic optimization problems.


Convex optimization Nonsymmetric conic optimization   Homogeneous self-dual model Interior-point algorithm 

Mathematics Subject Classification

90C25 90C51 90C53 90C30 



The authors thank Erling D. Andersen and Joachim Dahl of Mosek ApS for lots of insights and for supplying us with test problems for the geometric programs and the entropy problems. The authors also thank the reviewers for many helpful comments.

Supplementary material

10107_2014_773_MOESM1_ESM.pdf (324 kb)
Supplementary material 1 (pdf 324 KB)


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2014

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Computer ScienceTechnical University of DenmarkKgs. LyngbyDenmark
  2. 2.Department of Management Science and EngineeringStanford UniversityStanfordUSA
  3. 3.The International Center of Management Science and EngineeringNanjing UniversityNanjingChina

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