We study the asymptotic behavior at infinity of solutions of a second order evolution equation with linear damping and convex potential. The differential system is defined in a real Hilbert space. It is proved that if the potential is bounded from below, then the solution trajectories are minimizing for it and converge weakly towards a minimizer of Φ if one exists; this convergence is strong when Φ is even or when the optimal set has a nonempty interior. We introduce a second order proximal-like iterative algorithm for the minimization of a convex function. It is defined by an implicit discretization of the continuous evolution problem and is valid for any closed proper convex function. We find conditions on some parameters of the algorithm in order to have a convergence result similar to the continuous case.

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Abstract. In view of solving theoretically constrained minimization problems, we investigate the properties of the gradient flows with respect to Hessian Riemannian metrics induced by Legendre functions. The first result characterizes Hessian Riemannian structures on convex sets as 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 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 ofq-trajectories of some Lagrange equations and completely integrable Hamiltonian systems.

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