SINGULAR RIEMANNIAN BARRIER METHODS AND GRADIENT-PROJECTION DYNAMICAL SYSTEMS FOR CONSTRAINED OPTIMIZATION

摘要

This work is devoted to the dynamical system (SRB): d((partial deriv)h(x(t)))/dt + (nabla)Φ(x(t)) (contains) 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) + {nabla}(sup)2h(x(t)){sup}(-1) (nabla)Φ(x(t)) = 0. Moreover, class (SRB) is closed in a variational denser 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|{sup}2 +Sc(x): this way, one obtains a new gradient-projection method, System (SRB) bears a direct relation with the minimization of Φ over the domain of (partial deriv)h; the asymptotic behaviour of solutions, as time goes to infinity, is a real issue, which is addressed for a convex Φ and a h of the form h = k + δ{sub}C, with k convex C{sup}1 and C a finite-dimensional polyhedron.