Computer Vision

2014 Edition
| Editors: Katsushi Ikeuchi

Semidefinite Programming

  • Chunhua Shen
  • Anton van den Hengel
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-31439-6_688

Synonyms

Definition

Semidefinite programming is a subtopic of convex optimization. Convex optimization refers to minimization of a convex function subject to a set of convex constraints. Semidefinite programming involves minimization of a linear objective function over the intersection of linear constraints and the cone of positive semidefinite matrices. Clearly, semidefinite programming is a special case of convex optimization.

Background

Many computer vision problems can be formulated as convex optimization problems. The main advantage of convex optimization is that if a local minimum exists, then it is also a global minimum. In other words, the convexity guarantees to attain the global optimum if it exists.

In a semidefinite programming problem, one minimizes a linear function subject to the constraint that an affine combination of symmetric matrices is positive semidefinite. Semidefinite programming unifies a few standard problems...

This is a preview of subscription content, log in to check access.

References

  1. 1.
    Boyd S, Vandenberghe L (2004) Convex optimization. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  2. 2.
    Nesterov Y, Nemirovsky A (1988) A general approach to polynomial time algorithms design for convex programming. Technical report, USSR Acad Sci, Moscow, USSRGoogle Scholar
  3. 3.
    Borchers B (1999) CSDP, a C library for semidefinite programming. Optim Methods Softw 11(1):613–623MathSciNetCrossRefGoogle Scholar
  4. 4.
    Toh K, Todd M, Tutuncu R (1999) SDPT3—a matlab software package for semidefinite programming. Optim Methods Softw 11(1–4):545–581MathSciNetCrossRefGoogle Scholar
  5. 5.
    Sturm JF (1999) Using SeDuMi 1.02, a matlab toolbox for optimization over symmetric cones. Optim Methods Softw 11(1–4):625–653MathSciNetCrossRefGoogle Scholar
  6. 6.
    . Grant M, Boyd S (2011) CVX: matlab software for disciplined convex programming, version 1.21. http://cvxr.com/
  7. 7.
    Löfberg J (2004) YALMIP: a toolbox for modeling and optimization in MATLAB. Proceedings of the IEEE symposium on computer-aided control system design, Taipei, TaiwanGoogle Scholar
  8. 8.
    Wen Z, Goldfarb D, Yin W (2009) Alternating direction augmented lagrangian methods for semidefinite programming. Math Program Comput 2(3–4):203–230MathSciNetGoogle Scholar
  9. 9.
    Goemans MX, Williamson DP (1994) $.879$-approximation algorithms for max cut and max 2SAT. Proceedings of the ACM symposium on theory of computing. ACM, New York, pp 422–431Google Scholar
  10. 10.
    Keuchel J, Schnörr C, Schellewald C, Cremers D (2003) Binary partitioning, perceptual grouping, and restoration with semidefinite programming. IEEE Trans. Pattern Anal Mach Intell 25(11):1364–1379CrossRefGoogle Scholar
  11. 11.
    Weinberger KQ, Saul LK (2006) Unsupervised learning of image manifolds by semidefinite programming. Int J Comput Vision 70(1):77–90CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Chunhua Shen
    • 1
  • Anton van den Hengel
    • 1
  1. 1.School of Computer Science, The University of AdelaideAdelaide, SAAustralia