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

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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.

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Authors and Affiliations

  1. Department of Mathematics, University of Notre Dame Notre Dame, 255, Hurley Building, IN, 46556, USA

    Leonid Faybusovich

  2. The Institute of Statistical Mathematics, 4-6-7, Minami-Azabu, Minato-Ku, Tokyo, 106-8569, Japan

    Takashi Tsuchiya

Authors
  1. Leonid Faybusovich
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  2. Takashi Tsuchiya
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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

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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

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

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