Journal of Optimization Theory and Applications

, Volume 136, Issue 1, pp 87–103 | Cite as

Convergence of an Interior Point Algorithm for Continuous Minimax

Article

Abstract

We propose an algorithm for the constrained continuous minimax problem. The algorithm uses a quasi-Newton search direction, based on subgradient information, conditional on maximizers. The initial problem is transformed to an equivalent equality constrained problem, where the logarithmic barrier function is used to ensure feasibility. In the case of multiple maximizers, the algorithm adopts semi-infinite programming iterations toward epiconvergence. Satisfaction of the equality constraints is ensured by an adaptive quadratic penalty function. The algorithm is augmented by a discrete minimax procedure to compute the semi-infinite programming steps and ensure overall progress when required by the adaptive penalty procedure. Progress toward the solution is maintained using merit functions.

Keywords

Worst case analysis Continuous minimax algorithms Interior point methods Semi–infinite programming 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Rustem, B., Zakovic, S., Parpas, P.: An interior point algorithm for continuous minimax: implementation and computation. Comput. Optim. Appl. (2008, to appear) Google Scholar
  2. 2.
    Womersley, R.S., Fletcher, R.: An algorithm for composite nonsmooth optimization problems. J. Optim. Theory Appl. 48, 493–523 (1986) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Polak, E., Mayne, D.Q., Higgins, E.J.: A superlinearly convergent minimax algorithm for minimax problems. UCB/ERL M86/103, Department of Electrical Engineering, University California, Berkeley, CA (1988) Google Scholar
  4. 4.
    Rustem, B., Nguyen, Q.: An algorithm for inequality constrained discrete minimax. SIAM J. Optim. 8, 256–283 (1998) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Obasanjo, E., Rustem, B.: An interior point algorithm for nonlinear minimax problems (to appear) Google Scholar
  6. 6.
    Demyanov, V.F., Pvnyi, A.B.: Numerical methods for finding saddle points. USSR Comput. Math. Math. Phys. 12, 1099–1127 (1972) Google Scholar
  7. 7.
    Rustem, B., Howe, M.A.: Algorithms for Worst-case Design with Applications to Risk Management. Princeton University Press, Princeton (2001) Google Scholar
  8. 8.
    Panin, V.M.: Linearization method for continuous min–max problems. Kibernetika 2, 75–78 (1981) MathSciNetGoogle Scholar
  9. 9.
    Polak, E.: Optimization Algorithms and Consistent Approximations. Springer, Berlin (1997) MATHGoogle Scholar
  10. 10.
    Polak, E., Royset, J.O.: Algorithms for finite and semi-infinite min–max–min problems using adaptive smoothing techniques. J. Optim. Theory Appl. 119, 421–457 (2003) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Polak, E., Royset, J.O., Womersley, R.S.: Algorithms with adaptive smoothing for finite minimax problems. J. Optim. Theory Appl. 119, 459–484 (2003) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Sasai, H.: An interior penalty method for minimax problems with constraints. SIAM J. Control Optim. 12, 643–649 (1974) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Stein, O., Still, G.: Solving semi-infinite optimization problems with interior point techniques. SIAM J. Control Optim. 42, 769–788 (2003) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970) MATHGoogle Scholar
  15. 15.
    Powell, M.J.D.: A fast algorithm for nonlinearly constrained optimization calculations. In: Lecture Notes in Mathematics, vol. 630, pp. 144–157. Springer, Berlin (1978) Google Scholar
  16. 16.
    Hogan, W.W.: Point-to-set maps in mathematical programming. SIAM Rev. 15, 591–603 (1973) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Parpas, P., Rustem, B.: An algorithm for the global optimization of a class of continuous minimax problems. J. Optim. Theory Appl. (2008, to appear) Google Scholar
  18. 18.
    El-Bakry, A.S., Tapia, R.A., Tsuchiya, T., Zhang, Y.: On the formulation and theory of the Newton interior point method for nonlinear programming. J. Optim. Theory Appl. 89, 507–541 (1996) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Yamashita, H.: A globally convergent primal-dual interior point method for constrained optimization. Technical Report, Mathematical Systems Institute Inc, 2-5-3 Shinjuku, Shinjuku-ku, Tokyo, Japan (May 1995) Google Scholar
  20. 20.
    Akrotirianakis, I., Rustem, B.: A globally convergent interior point algorithm for general non-linear programming problems. J. Optim. Theory Appl. 125, 497–521 (2005) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of ComputingImperial CollegeLondonUK

Personalised recommendations