Mathematical Programming

, Volume 67, Issue 1–3, pp 53–76 | Cite as

Solving large-scale minimax problems with the primal—dual steepest descent algorithm

  • Ciyou Zhu
Article
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Abstract

This paper shows that the primal-dual steepest descent algorithm developed by Zhu and Rockafellar for large-scale extended linear—quadratic programming can be used in solving constrained minimax problems related to a generalC2 saddle function. It is proved that the algorithm converges linearly from the very beginning of the iteration if the related saddle function is strongly convex—concave uniformly and the cross elements between the convex part and the concave part of the variables in its Hessian are bounded on the feasible region. Better bounds for the asymptotic rates of convergence are also obtained. The minimax problems where the saddle function has linear cross terms between the convex part and the concave part of the variables are discussed specifically as a generalization of the extended linear—quadratic programming. Some fundamental features of these problems are laid out and analyzed.

Keywords

Minimax problem Saddle function Large-scale numerical optimization Primal—dual projected gradient algorithm 
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References

  1. [1]
    C. Zhu and R.T. Rockafellar, “Primal-dual projected gradient algorithms for extended linear—quadratic programming,”SIAM Journal on Optimization 3 (4) (1993) 751–783.Google Scholar
  2. [2]
    C. Zhu, “On the primal—dual steepest descent algorithm for extended linear—quadratic programming,” technical report, Department of Mathematical Sciences, The Johns Hopkins University 1992. Accepted for publication inSIAM Journal on Optimization, 1993.Google Scholar
  3. [3]
    R.T. Rockafellar and R.J-B Wets, “A Lagrangian finite generation technique for solving linear-quadratic problems in stochastic programming,”Mathematical Programming Studies 28 (1986) 63–93.Google Scholar
  4. [4]
    R.T. Rockafellar and R. J-B Wets, “Linear-quadratic problems with stochastic penalties: the finite generation algorithm,” in:Numerical Techniques for Stochastic Optimization Problems, Y. Ermoliev and R. J-B Wets (eds.), Springer-Verlag Lecture Notes in Control and Information Sciences No. 81 (Springer-Verlag, Berlin, 1987) pp. 545–560.Google Scholar
  5. [5]
    R.T. Rockafellar, “A generalized approach to linear-quadratic programming,” in:Proceedings International Conference on Numerical Optimization and Applications (Xi'an, China, 1986) pp. 58–66.Google Scholar
  6. [6]
    R.T. Rockafellar, “Linear-quadratic programming and optimal control,”SIAM Journal on Control and Optimization 25 (1987) 781–814.Google Scholar
  7. [7]
    R.T. Rockafellar and R. J-B Wets, ‘Generalized linear—quadratic problems of deterministic and stochastic optimal control in discrete time,”SIAM Journal on Control and Optimization 28 (1990) 810–822.Google Scholar
  8. [8]
    R.T. Rockafellar, “Computational schemes for solving large-scale problems in extended linear—quadratic programming,”Mathematical Programming 48 (3) (1990) 447–474.Google Scholar
  9. [9]
    R.T. Rockafellar, “Large-scale extended linear-quadratic programming and multistage optimization,” in:Proc. Fifth Mexico-U.S. Workshop on Numerical Analysis, S. Gomez, J.-P. Hennart, R. Tapia, eds. (SIAM Publications, Philadelphia, 1990).Google Scholar
  10. [10]
    A. King, An implementation of the Lagrangian finite generation method, in:Numerical Techniques for Stochastic Programming Problems, Y. Ermoliev and R. J-B Wets (eds.) (Springer-Verlag, Berlin, 1988).Google Scholar
  11. [11]
    J.M. Wagner, Stochastic programming with recourse applied to groundwater quality management, doctoral dissertation, M.I.T, 1988.Google Scholar
  12. [12]
    R.T. Rockafellar,Convex Analysis (Princeton University Press, Princeton, NJ, 1970).Google Scholar
  13. [13]
    G. H.-G. Chen and R.T. Rockafellar, “Forward—backward splitting methods in Lagrangian optimization,” technical report, Department of Applied Math, University of Washington, May 1992.Google Scholar
  14. [14]
    C. Zhu, “Asymptotic convergence analysis of the splitting algorithm,” technical report, Department of Mathematical Sciences, The Johns Hopkins University 1991. Accepted for publication byMathematics of Operations Research, 1993.Google Scholar
  15. [15]
    C. Zhu, “Modified proximal point algorithm for extended linear—quadratic programming,”Computational Optimization and Applications 1 (1992) 185–205.Google Scholar
  16. [16]
    J.M. Ortega and W.C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970).Google Scholar

Copyright information

© The Mathematical Programming Society, Inc 1994

Authors and Affiliations

  • Ciyou Zhu
    • 1
  1. 1.Math/Computer Science DivisionArgonne National LaboratoryArgonneUSA

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