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.
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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