Euclidean Jordan algebras and generalized affine-scaling vector fields

L. Faybusovich

We describe the phase portrait of generalized affine- scaling vector fields for optimization problems involving symmetric cones. A Poisson structure on the complexification of a real Euclidean Jordan algebra is introduced. Nonconstrained affine-scaling vector fields are proved to be Hamiltonian with respect to this Poisson structure. Constained affine-scaling vector fields are obtained as a symplectic reduction of unconstrained ones. It is proved that constrained affine-scaling vector fields are completely integrable Hamiltonian vector fields and action-angle variables are constructed for them.

Research report, University of Notre Dame, January, 1998