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

Leonid Faybusovich (leonid.faybusovich.1***at***nd.edu)
Takashi Tsuchiya (tsuchiya***at***sun312.ism.ac.jp)

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 size of this algebraic system is $m+1$ by $m+1$, where $m$ is a number of quadratic performance criteria.

Keywords: Interior-point algorithms, primal-dual algorithms, second-order cone programming, infinite-dimensional problems, control theory

Category 1: Linear, Cone and Semidefinite Programming

Category 2: Applications -- Science and Engineering (Control Applications )

Citation: Research Memorandum No. 821, The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106-8569 Japan, November 2001 (Revised: December 2001 and November 2002), To appear in Mathematical Programming.

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Entry Submitted: 03/31/2003
Entry Accepted: 03/31/2003
Entry Last Modified: 03/31/2003

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