 
Summary: HESSIAN RIEMANNIAN GRADIENT FLOWS
IN CONVEX PROGRAMMING
FELIPE ALVAREZ, J´ER^OME BOLTE, AND OLIVIER BRAHIC§
SIAM J. CONTROL OPTIM. c 2004 Society for Industrial and Applied Mathematics
Vol. 43, No. 2, pp. 477501
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 Bregmantype distances. Then, the general evolution problem is introduced,
and global convergence is established under quasiconvexity 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, continuoustime model of Bregmantype proximal algorithms,
geodesics for some adequate metrics, and projections of qtrajectories of some Lagrange equations
and completely integrable Hamiltonian systems.
Key words. gradient flow, Hessian Riemannian metric, Legendretype convex function, ex
istence, global convergence, Bregman distance, Lyapunov functional, quasiconvex minimization,
