Advertisement

Graph Implementations for Nonsmooth Convex Programs

Conference paper
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 371)

Abstract

We describe graph implementations, a generic method for representing a convex function via its epigraph, described in a disciplined convex programming framework. This simple and natural idea allows a very wide variety of smooth and nonsmooth convex programs to be easily specified and efficiently solved, using interiorpoint methods for smooth or cone convex programs.

Keywords

Convex optimization nonsmooth optimization disciplined convex programming optimization modeling languages semidefinite programming second-order cone programming conic optimization nondifferentiable functions 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alizadeh, F., Goldfarb, D.: Second-order cone programming (January 2004)Google Scholar
  2. 2.
    Bertsekas, D.P.: Convex Analysis and Optimization. In: Nedić, A., Ozdaglar, A.E. (eds.) Athena Scientific (2003)Google Scholar
  3. 3.
    Bland, R., Goldfarb, D., Todd, M.: The ellipsoid method: A survey. Operations Research 29(6), 1039–1091 (1981)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Borwein, J., Lewis, A.: Convex Analysis and Nonlinear Optimization: Theory and Examples. Springer, Heidelberg (2000)zbMATHGoogle Scholar
  5. 5.
    Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization: Analysis, Algorithms and Engineering Applications. MPS/SIAM Series on Optimization. SIAM, Philadelphia (2001)Google Scholar
  6. 6.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge Univ. Press, Cambridge (2004), http://www.stanford.edu/~boyd/cvxbook.html zbMATHGoogle Scholar
  7. 7.
    Dantzig, G.: Linear Programming and Extensions. Princeton University Press, Princeton (1963)zbMATHGoogle Scholar
  8. 8.
    Grant, M., Boyd, S.: CVX: MATLAB® software for disciplined convex programming, version 1.1 (September 2007), http://www.stanford.edu/~boyd/cvx/
  9. 9.
    Grant, M., Boyd, S., Ye, Y.: Disciplined convex programming. In: Liberti, L., Maculan, N. (eds.) Global Optimization: from Theory to Implementation, Nonconvex Optimization and Its Applications, pp. 155–210. Springer Science+Business Media, Inc., New York (2006)Google Scholar
  10. 10.
    Grant, M.: Disciplined Convex Programming. PhD thesis, Department of Electrical Engineering, Stanford University (December 2004)Google Scholar
  11. 11.
    Löfberg, J.: YALMIP: A toolbox for modeling and optimization in MATLAB®. In: Proceedings of the CACSD Conference, Taipei, Taiwan (2004), http://control.ee.ethz.ch/~joloef/yalmip.php
  12. 12.
    Lobo, M., Vandenberghe, L., Boyd, S., Lebret, H.: Applications of second-order cone programming. Linear Algebra and its Applications 284, 193–228 (1998) (special issue on Signals and Image Processing)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    MOSEK ApS. Mosek (software package) (September 2007), http://www.mosek.com
  14. 14.
    Nesterov, Yu., Nemirovsky, A.: Interior-Point Polynomial Algorithms in Convex Programming: Theory and Algorithms of Studies in Applied Mathematics, vol. 13. SIAM Publications, Philadelphia, PA (1993)Google Scholar
  15. 15.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)zbMATHGoogle Scholar
  16. 16.
    Sturm, J.: Using SeDuMi 1.02, a MATLAB® toolbox for optimization over symmetric cones. Optimization Methods and Software 11, 625–653 (1999), Updated version available at http://sedumi.mcmaster.ca CrossRefMathSciNetGoogle Scholar
  17. 17.
    Toh, K., Todd, M., Tutuncu, R.: SDPT3 — a MATLAB® software package for semidefinite programming. Optimization Methods and Software 11, 545–581 (1999)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Vanderbei, R.: LOQO: An interior point code for quadratic programming. Optimization Methods and Software 11, 451–484 (1999)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Vandenberghe, L., Boyd, S.: Semidefinite programming. SIAM Review 38(1), 49–95 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    van Tiel, J.: Convex Analysis. An Introductory Text. John Wiley & Sons, Chichester (1984)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  1. 1.Stanford UniversityStanford

Personalised recommendations

Cite paper

Buy options