 
Summary: Convergence to the optimal value for barrier methods
combined with Hessian Riemannian gradient flows and
generalized proximal algorithms
Felipe Alvarez & Julio L´opez
Abstract
We consider the problem minxRn {f(x)  Ax = b, x C, gj(x) 0, j = 1, . . . , s}, where
b Rm
, A Rm×n
is a full rank matrix, C is the closure of a nonempty, open and convex
subset C of Rn
, and gj(·), j = 1, . . . , s, are nonlinear convex functions. Our strategy consists
firstly in to introduce a barriertype penalty for the constraints gj(x) 0, then endowing
{x Rn
 Ax = b, x C} with the Riemannian structure induced by the Hessian of an
essentially smooth convex function h such that C = int(dom h), and finally considering the
flow generated by the Riemannian penalty gradient vector field. Under minimal hypotheses,
we investigate the wellposedness of the resulting ODE and we prove that the value of the
objective function along the trajectories, which are strictly feasible, converges to the optimal
value. Moreover, the value convergence is extended to the sequences generated by an implicit
discretization scheme which corresponds to the coupling of an inexact generalized proximal
