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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

Abstract

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.

Keywords

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

Mathematics Subject Classification

90C25 90C51 90C53 90C30 

Notes

Acknowledgments

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)

References

  1. 1.
    Andersen, E.D., Andersen, K.D.: The MOSEK interior point optimization for linear programming: an implementation of the homogeneous algorithm. In: Frenk, H., Roos, K., Terlaky, T., Zhang, S. (eds.) High Performance Optimization, pp. 197–232. Kluwer, Boston (1999)Google Scholar
  2. 2.
    Andersen, E.D., Roos, C., Terlaky, T.: On implementing a primal–dual interior-point method for conic quadratic optimization. Math. Program. 95(2), 249–277 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Andersen, E.D., Ye, Y.: On a homogeneous algorithm for the monotone complementarity problem. Math. Program. 84(2), 375–399 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Ben-Tal, A., Nemirovski, A.S.: Lectures on Modern Convex Optimization: Analysis, Algorithms and Engineering Applications. SIAM, Philadelphia (2001)CrossRefGoogle Scholar
  5. 5.
    Boyd, S., Kim, S.J., Vandenberghe, L., Hassibi, A.: A tutorial on geometric programming. Optim. Eng. 8, 67–127 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Butcher, J.C.: Numerical Methods for Ordinary Differential Equations, 2nd edn. Wiley, New York (2008)CrossRefzbMATHGoogle Scholar
  7. 7.
    Chares, P.R.: Cones and interior-point algorithms for structured convex optimization involving powers and exponentials. PhD thesis, Uni. Catholique de Louvain (2009)Google Scholar
  8. 8.
    Grant, M., Boyd, S.: CVX: Matlab software for disciplined convex programming, version 1.21 (2010). http://cvxr.com/cvx
  9. 9.
    Güler, O.: Barrier functions in interior point methods. Math. Oper. Res. 21, 860–885 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Karmarkar, N.: A new polynomial-time algorithm for linear programming. Combinatorica 4, 373–395 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Luo, Z.Q., Sturm, J.F., Zhang, S.: Conic convex programming and self-dual embedding. Optim. Method Softw. 14, 169–218 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Mehrotra, S.: On the implementation of a primal–dual interior point method. SIAM J. Optim. 2, 575–601 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    MOSEK optimization software: developed by MOSEK ApS. www.mosek.com
  14. 14.
    Nesterov, Y.E.: Constructing self-concordant barriers for convex cones. CORE Discussion Paper (2006/30)Google Scholar
  15. 15.
    Nesterov, Y.E.: Towards nonsymmetric conic optimization. Optim. Method Softw. 27, 893–917 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Nesterov, Y.E., Nemirovski, A.S.: Interior-Point Polynomial Algorithms in Convex Programming. SIAM, Philadelphia (1994)CrossRefzbMATHGoogle Scholar
  17. 17.
    Nesterov, Y.E., Todd, M.J.: Self-scaled barriers and interior-point methods for convex programming. Math. Oper. Res. 22, 1–42 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Nesterov, Y.E., Todd, M.J.: Primal–dual interior-point methods for self-scaled cones. SIAM J. Optim. 8, 324–364 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Nesterov, Y.E., Todd, M.J., Ye, Y.: Infeasible-start primal–dual methods and infeasibility detectors for nonlinear programming problems. Math. Program. 84, 227–267 (1999)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Nocedal, J., Wright, S.J.: Numerical Optimization, 2nd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  21. 21.
    Renegar, J.: A Mathematical View of Interior-Point Methods in Convex Optimization. SIAM, Philadelphia (1987)Google Scholar
  22. 22.
    Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Method Softw. 12, 625–653 (1999)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Sturm, J.F.: Implementation of interior point methods for mixed semidefinite and second order cone optimization problems. Optim. Method Softw. 17, 1105–1154 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Tuncel, L.: Primal–dual symmetry and scale invariance of interior-point algorithms for convex optimization. Math. Oper. Res. 23, 708–718 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Tuncel, L.: Generalization of primal–dual interior-point methods to convex optimization problems in conic form. Found. Comput. Math. 1, 229–254 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Xu, X., Hung, P.F., Ye, Y.: A simplified homogeneous and self-dual linear programming algorithm and its implementation. Ann. Oper. Res. 62, 151–171 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Xue, G., Ye, Y.: An efficient algorithm for minimizing a sum of \(p\)-norms. SIAM J. Optim. 10, 551–579 (1999)CrossRefMathSciNetGoogle Scholar

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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