Skip to main content
Log in

Primal-dual algorithms and infinite-dimensional Jordan algebras of finite rank

  • Published:
Mathematical Programming Submit manuscript

Abstract.

We consider primal-dual algorithms for certain types of infinite-dimensional optimization problems. Our approach is based on the generalization of the technique of finite-dimensional Euclidean Jordan algebras to the case of infinite-dimensional JB-algebras of finite rank. This generalization enables us to develop polynomial-time primal-dual algorithms for ``infinite-dimensional second-order cone programs.'' We consider as an example a long-step primal-dual algorithm based on the Nesterov-Todd direction. It is shown that this algorithm can be generalized along with complexity estimates to the infinite-dimensional situation under consideration. An application is given to an important problem of control theory: multi-criteria analytic design of the linear regulator. The calculation of the Nesterov-Todd direction requires in this case solving one matrix differential Riccati equation plus solving a finite-dimensional system of linear algebraic equations on each iteration. The number of equations and unknown variables of this algebraic system is m+1, where m is a number of quadratic performance criteria.

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. Faraut, J., Koráyi, A.: Anal. on Symmetric Cones. Oxford University Press, New York, 1994

  2. Faybusovich, L.: Euclidean Jordan algebras and interior-point algorithms. Positivity 1, 331–357 (1997)

    Google Scholar 

  3. Faybusovich, L.: Linear systems in Jordan algebras and primal-dual interior-point algorithms. J. Comput. Appl. Math. 86, 149–175 (1997)

    Google Scholar 

  4. Faybusovich, L.: A Jordan-algebraic approach to potential-reduction algorithms. Math. Z. 239, 117–129 (2002)

    Google Scholar 

  5. Faybusovich, L., Arana, R.: A long-step primal-dual algorithm for the symmetric programming problem. Syst. Control Lett. 43, 3–7 (2001)

    Google Scholar 

  6. Faybusovich, L., Moore, J.B.: Infinite-dimensional quadratic optimization: interior-point methods and control applications. Appl. Math. Optim. 36, 43–66 (1997)

    Google Scholar 

  7. Faybusovich, L., Mouktonglang, T.: Finite-Rank Perturbation of the Linear-Quadratic control problem, preprint 2002. (To appear in in Proceedings of American Control Conference, Denver 2003.)

  8. Wilhelm Kaup: Jordan algebras and holomorphy. Functional analysis, holomorphy, and approximation theory (Proc. Sem., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1978), pp. 341–365, Lecture Notes in Mathematics, Vol. 843, Springer, Berlin, 1981

  9. Monteiro, R.D.C., Tsuchiya, T.: Polynomial convergence of primal-dual algorithms for the second-order cone program based on the MZ-family of directions. Math. Prog. 88(1), 61–83 (2000)

    Google Scholar 

  10. Monteiro, R.D.C., Zhang, Y.: A unified analysis for a class of long-step primal-dual path-following interior-point algorithms for semidefinite programming. Math. Prog. 81(3), 281–299 (1998)

    Google Scholar 

  11. Muramatsu, M.: On a commutative class of search directions for linear programming over symmetric cones. J. Optim. Theor. its Appl. 112(3), 595–625 (2002)

    Google Scholar 

  12. Nemirovskii, A., Scheinberg, K.: Extension of Karmarkar's algorithm onto convex quadratically constrained quadratic problems. Math. Prog. 72(3), 273–289 (1996)

    Google Scholar 

  13. Nesterov, Y.E., Todd, M.: Self-scaled barriers and interior-point methods for convex programming. Math. Operat. Res. 22, 1–42 (1997)

    Google Scholar 

  14. Schmieta, S., Alizadeh, F.: Extensions of primal-dual interior point algorithms to symmetric cones. Report RRR 13-99, RUTCOR, Rutgers University, 640 Bartholomew Road, Piscataway, NJ 08854-8003, 1999

  15. Sturm, J.F.: Similarity and other spectral relations for symmetric cones. Linear Algebra and its Appl. 312(1–3), 135–154 (2000)

    Google Scholar 

  16. Tsuchiya, T.: A convergence analysis of the scaling-invariant primal-dual path-following algorithms for second-order cone programming. Optim. Methods and Softw. 11&12, 141–182 (1999)

    Google Scholar 

  17. Wright, S.: Primal–dual Interior Point Algorithms. SIAM Publications, Philadelphia, Pennsylvania, USA, 1997

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Leonid Faybusovich.

Additional information

Key words. polynomial-time primal-dual interior-point methods – JB-algebras – infinite-dimensional problems – optimal control problems

This author was supported in part by DMS98-03191 and DMS01-02698.

This author was supported in part by the Grant-in-Aid for Scientific Research (C) 11680463 of the Ministry of Education, Culture, Sports, Science and Technology of Japan.

Mathematics Subject Classification (1991): 90C51, 90C48, 34H05, 49N05

Rights and permissions

Reprints and permissions

About this article

Cite this article

Faybusovich, L., Tsuchiya, T. Primal-dual algorithms and infinite-dimensional Jordan algebras of finite rank. Math. Program., Ser. B 97, 471–493 (2003). https://doi.org/10.1007/s10107-003-0424-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-003-0424-4

Keywords

Navigation