Journal of Global Optimization

, Volume 25, Issue 2, pp 141–155 | Cite as

Cutting Plane Algorithms for Nonlinear Semi-Definite Programming Problems with Applications

  • Hiroshi Konno
  • Naoya Kawadai
  • Hoang Tuy


We will propose an outer-approximation (cutting plane) method for minimizing a function fX subject to semi-definite constraints on the variables XRn. A number of efficient algorithms have been proposed when the objective function is linear. However, there are very few practical algorithms when the objective function is nonlinear. An algorithm to be proposed here is a kind of outer-approximation(cutting plane) method, which has been successfully applied to several low rank global optimization problems including generalized convex multiplicative programming problems and generalized linear fractional programming problems, etc. We will show that this algorithm works well when f is convex and n is relatively small. Also, we will provide the proof of its convergence under various technical assumptions.

Semi-definite programming Low rank nonconvex problem Cutting plane algorithm Outer-approximation Semi-infinite programming Ellipsoidal separation Quadratic regression Semi-definite logit model 


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  1. 1.
    Altman, E.I. and Nelson, A.D. (1968), Financial ratios, discriminant analysis and the prediction of corporate bankruptcy. Journal of Finance 23: 589-609.Google Scholar
  2. 2.
    Bertsimas, D. and Popescu, I. (1998), On the relation between option and stock prices: a convex optimization approach, Technical Paper, Sloan School of Managemanet, MIT.Google Scholar
  3. 3.
    Brace, A. and Womersley. R. S., Exact fit to the swaption volatility matrix using semidefinite programming. to appear in Mathematical Finance.Google Scholar
  4. 4.
    Bradley, P. S., Fayyad, U. M. and Mangasarian, O. L. (1999), Mathematical programming for data mining: formulations and challenges. INFORMS J. of Computing 11: 217-238.Google Scholar
  5. 5.
    Fujisawa, K. Kojima, M. and Nakata, K. (1999), SDPA User's Manual-Version 5.0, Research Paper on Mathematical and Computing Science, Tokyo Institute of Technology.Google Scholar
  6. 6.
    Gantmacher, F. R. (1959), The Theory of Matrices, Chelsea Publishing Co.Google Scholar
  7. 7.
    Helmberg, C., Rendl, F., Vanderbei, R. J. and Wolkowitz, H. (1996), An interior point method for semi-definite programming. SIAM J. on Optimizaiton 6: 342-361.Google Scholar
  8. 8.
    Horst, R. and Tuy, H. (1996), Global Optimization: Deterministic Approaches, 3rd ed. Springer, Berlin.Google Scholar
  9. 9.
    Ioffe, A.D., and Tihomirov, V.M. (1979), Theory of External Problems. North-Holland, Amsterdam.Google Scholar
  10. 10.
    Kelley, J.E. (1960), The cutting-plane method for solving convex programs. J. of the Society of Industrial Applied Mathematics, 8: 703-712.Google Scholar
  11. 11.
    Konno, H., Gotoh, J. and Uno, T. (2000), A cutting plane algorithm for semi-definite programming problems with applications. WP 00-05, Center for Research in Advanced Financial Technology. Tokyo Institute of Technology. (to appear in J. of Computational and Applied Mathematics)Google Scholar
  12. 12.
    Konno, H., Gotoh, J. and Uryasev, S. and Yuuki, A. (2001), Failure discrimination by semidefinite programming. WP01-02, Center for Research in Advanced Financial Technology, Tokyo Institute of Technology. (to appear in FEES 2001 (P. Pardalos, ed.))Google Scholar
  13. 13.
    Konno, H. and Kobayashi, H. (2000), Failure discrimination and rating of enterprises by semidefinite programming, Asia-Pacific Financial Markets, 7: 261-273.Google Scholar
  14. 14.
    Konno, H., Thach, P.T. and Tuy, H. (1997), Optimization over Low Rank Nonconvex Structures, Kluwer Academic Publishers, Dordrecht.Google Scholar
  15. 15.
    Konno, H. and Wu, D. (2001), Estimation of failure probability using semi-definite logit model. WP 01-04, Center for Research in Advanced Financial Technology.Google Scholar
  16. 16.
    Mangasarian, O., Street, W. and Wolberg, W. (1995), Breast cancer diagnoses and prognoses via linear programming. Operations Research, 43: 570-577.Google Scholar
  17. 17.
    Tuy, H. (1983), Outer approximation methods for solving concave minimization problems. Acta Mathematica Vietnamica, 8: 3-34.Google Scholar
  18. 18.
    Vandenberghe, L. and Boyd, S. (1998), Connections between semi-definite and semi-infinite programming. Semi-infinite Programming, 277-294, Kluwer Academic Publishers, Boston.Google Scholar
  19. 19.
    Vandenberghe, L., and Boyd, S. (1996), Semi-definite programming. SIAM Review 38: 49-95.Google Scholar
  20. 20.
    Veinott., A. F., Jr. (1967), The supporting hyperplane method for unimodal programming. Operations Research, 15: 147-152.Google Scholar
  21. 21.
    Wolkowitz, H., Saigal, R. and Vandenberghe, L. (2000), Handbook of Semi-Definite Programming: Theory, Algorithm and Application, Kluwer Academic Publishers, Dordrecht.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Hiroshi Konno
    • 1
  • Naoya Kawadai
    • 2
  • Hoang Tuy
    • 3
  1. 1.Department of Industrial and Systems EngineeringChuo-UniversityJapan
  2. 2.Department of Industrial Engineering and ManagementTokyo Institute of TechnologyJapan
  3. 3.Institute of MathematicsHanoiVietnam

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