Hessian Riemannian gradient flows in convex programming *

Felipe Alvarez (1) , Jérôme Bolte (2) , Olivier Brahic

1 DIM - Departamento de Ingeniería Matemática [Santiago]
2 GREMAQ - Groupe de recherche en économie mathématique et quantitative

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 ˙ q-trajectories of some Lagrange equations and completely integrable Hamiltonian systems.

Mots clés: Gradient flow; Hessian Riemannian metric; Legendre type convex function; ex-istence; global convergence; Bregman distance; Liapounov functional; quasi-convex mini-mization; convex and linear programming; Legendre transform coordinates; Lagrange and Hamilton equations AMS classification: 34G20; 34A12; 34D05; 90C25

Domaines: Optimisation et contrôle [math.OC]

Felipe Alvarez, Jérôme Bolte, Olivier Brahic. Hessian Riemannian gradient flows in convex programming *. SIAM Journal on Control and Optimization, 2004, 43 (2), pp.477 - 501. ⟨10.1137/S0363012902419977⟩. ⟨hal-01928141⟩