Singular Riemannian Barrier methods and gradient-projection dynamical systems for constrained optimization

This work is devoted to the dynamical system (SRB): d(partial derivativeh(x(t)))/dt + delPhi(x(t)) epsilon 0, with h a proper lower semicontinuous convex function. Existence and uniqueness of solutions are examined. Systems (SRB) include the class of gradient systems with respect to a Hessian Riemannian metric induced by a convex Legendre function h: x(t) + del(2)h(x(t))(-1)delPhi(x(t)) = 0. Moreover, class (SRB) is closed in a variational sense: links are made with regularized Lotka-Volterra systems and the limit equations obtained by letting the barrier parameter go to 0. Of particular interest is the case h(x) = (1 /2)\x\(2) + delta(C)(x): this way, one obtains a new gradient-projection method. System (SRB) bears a direct relation with the minimization of Phi over the domain of partial derivativeh; the asymptotic behaviour of solutions, as time goes to infinity, is a real issue, which is addressed for a convex Phi and a h of the form h = k + delta(C), with k convex C-1 and C a finite-dimensional polyhedron.