On the geometry of saddle point algorithms

There has been great deal of innovative work in recent years relating discrete algorithms to continuous flows. Of particular interest are flows which are gradient flows or Hamiltonian flows. Hamiltonian flows do not have asymptotically stable equilibria, but a restriction of the system to a certain set of variables may have such an equilibrium. In nonlinear optimization and game theory there is an interest in systems with saddle point equilibria. The authors show that certain flows with such equilibria can be both Hamiltonian and gradient and discuss the relationship of such flows with the gradient method for finding saddle points in nonlinear optimization problems. These results are compared with gradient flows associated with the Toda lattice