  Motivated by a constrained minimization problem, it is studied the gradient
flows with respect to Hessian Riemannian metrics induced by convex functions of
Legendre type. The first result characterizes Hessian Riemannian structures on
convex sets as those metrics that have a specific integration property with
respect to variational inequalities, giving a new motivation for the
introduction of Bregman-type distances. Then, the general evolution problem is
introduced and a differential inclusion reformulation is given. A general
existence result is proved and global convergence is established under
quasi-convexity conditions, with interesting refinements in the case of convex
minimization. Some explicit examples of these gradient flows are discussed.
Dual trajectories are identified and sufficient conditions for dual convergence
are examined for a convex program with positivity and equality constraints.
Some convergence rate results are established. In the case of a linear
objective function, several optimality characterizations of the orbits are
given: optimal path of viscosity methods, continuous-time model of Bregman-type
proximal algorithms, geodesics for some adequate metrics and projections of
$\dot q$-trajectories of some Lagrange equations and completely integrable
Hamiltonian systems.
