Skip to main content
Log in

Modified proximal point algorithm for extended linear-quadratic programming

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

Extended linear-quadratic programming arises as a flexible modeling scheme in dynamic and stochastic optimization, which allows for penalty terms and facilitates the use of duality. Computationally it raises new challenges as well as new possibilities in large-scale applications. Recent efforts have been focused on the fully quadratic case ([15] and [23]), while relying on the fundamental proximal point algorithm (PPA) as a shell of “outer” iterations when the problem is not fully quadratic. In this paper, we focus on the nonfully quadratic cases by proposing some new variants of the fundamental PPA. We first construct a continuously differentiable saddle function S(u, v) through infimal convolution in such a way that the optimal primal-dual pairs of the original problem are just the saddle points of S(u, v) on the whole space. Then the original extended linear-quadratic-programming problem reduces to solving the nonlinear equation ∇S(u, v)=0. We then embed the fundamental PPA and some of its previous variants in the framework of a Newton-like iteration for this equation. After revealing the local quadratic structure of S near the solution, we derive new extensions of the fundamental PPA. In numerical tests, the modified iteration scheme based on the quasi-Newton update formula outperforms the fundamental PPA considerably.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J.V. Burke and M.C. Ferris, “Characterization of solution sets of convex programs,” Oper. Res. Letters, vol. 10, pp. 57–60, 1991.

    Google Scholar 

  2. J.E. Dennis Jr. and Jorge J. Moré, “Quasi-Newton methods, motivation and theory,” SIAM Rev., vol. 19, pp. 47–89, 1977.

    Google Scholar 

  3. J.E. Dennis Jr. and R.B. Schnabel, “Least change secant updates for Quasi-Newton methods,” SIAM Rev., vol. 21, pp. 443–459, 1979.

    Google Scholar 

  4. D. Gabay, “Application of the Method of Multipliers to Variational Inequalities,” in Augmented Lagrangian Methods: Applications to the Numerical Solutions of Boundary-Value Problems (M. Fortin and R. Golwinski, eds.), North-Holland, Amsterdam, 1983.

    Google Scholar 

  5. 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, 1988.

  6. F.J. Luque, “Asymptotic convergence analysis of the proximal point algorithm” SIAM J. Control Opt., vol. 22, pp. 277–293, 1984.

    Google Scholar 

  7. G.J. Minty, “Monotone (nonlinear) operators in Hilbert space,” Duke Math. J., vol. 29, pp. 341–346, 1962.

    Google Scholar 

  8. J.J. Moreau, “Proximité et dualité dans un espace Hilbertien,” Bull. Soc. Math. Fr., vol. 93, pp. 273–299, 1965.

    Google Scholar 

  9. S.M. Robinson, “Some continuity properties of polyhedral multifunctions,” Math. Programming Studies, vol. 14, pp. 206–214, 1981.

    Google Scholar 

  10. R.T. Rockafellar, Convex Analysis, Princeton Univ. Press, Princeton, NJ, 1970.

    Google Scholar 

  11. R.T. Rockafellar, “Monotone operators associated with saddle functions and minimax problems,” in Nonlinear Functional analysis, Part 1, (F.E. Browder, ed.), Symposia in Pure Math., vol. 18, Amer. Math Soc., Providence, R.I., 1970, pp. 241–250.

    Google Scholar 

  12. R.T. Rockafellar, “Monotone operators and the proximal point algorithm,” SIAM J. Control Opt., vol. 14, pp. 877–898, 1976.

    Google Scholar 

  13. R.T. Rockafellar, “A generalized approach to linear-quadratic programming,” in Proc. Int. Conf. on Numerical Optimization and Appl., Xi'an, China, pp. 58–66, 1986.

  14. R.T. Rockafellar, “Linear-quadratic programming and optimal control,” SIAM J. Control Opt., vol. 25, pp. 781–814, 1987.

    Google Scholar 

  15. R.T. Rockafellar, “Computational schemes for solving large-scale problems in extended linear-quadratic programming,” Math. Programming, vol. 48, pp. 447–474, 1990.

    Google Scholar 

  16. 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, 1990.

  17. R.T. Rockafellar and R.J.-B. Wets, “A Lagrangian finite generation technique for solving linear-quadratic problems in stochastic programming,” Math. Programming Studies, vol. 28, pp. 63–93, 1986.

    Google Scholar 

  18. R.T. Rockafellar and R.J.-B. Wets, “Linear-quadratic problems with stochastic penalities: 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, pp. 545–560, 1987.

  19. R.T. Rockafellar and R. J.-B. Wets, “Generalized linear-quadratic problems of deterministic and stochastic optimal control in discrete time,” SIAM J. Control Opt., vol. 28, pp. 810–822, 1990.

    Google Scholar 

  20. J.M. Wagner, Stochastic Programming with Recourse Applied to Groundwater Quality Management, PhD dissertation, M.I.T., Cambridge, MA, 1988.

    Google Scholar 

  21. S.E. Wright (with introduction by R.T. Rockafellar), “DYNFGM: Dynamic Finite Generation Method,” Department of Mathematics, University of Washington, 1989.

  22. C. Zhu, Methods for Large-Scale Extended Linear-Quadratic Programming, PhD dissertation, University of Washington, 1991.

  23. C. Zhu and R.T. Rockafellar, “Primal-dual projected gradient algorithms for extended linear-quadratic programming,” Submitted to SIAM J. Opt., 1990.

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhu, C. Modified proximal point algorithm for extended linear-quadratic programming. Comput Optim Applic 1, 185–205 (1992). https://doi.org/10.1007/BF00253806

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00253806

Keywords

Navigation