Convergence rate estimates for the gradient differential inclusion
Let f : H --> R boolean OR {infinity} be a proper, lower semi-continuous, convex function in a Hilbert space H. The gradient differential inclusion is x'(t) is an element of -partial derivative f (x( t)), x( 0) = x, where x is an element of dom( f) and where partial derivative f (x(t)) is the subdifferential of f at the point x( t). If f is Gateaux differentiable, the inclusion is the differential equation x' (t) = - f' (x( t)), which is the continuous version of the steepest descent method for minimizing f on H. Even if f is not differentiable, the inclusion has a unique solution {x( t): t > 0} which converges weakly to a minimizer of f if such a minimizer exists. In general, the inclusion can be interpreted as the the continuous version of the proximal point method for minimization f on H. There is a remarkable similarity between the behavior of the inclusion and its discrete counterparts as well in the methods used in both cases. As a simple consequence of our previous results on the proximal point method, we prove the convergence rate estimate f (x( t)) - f (u) <= (1/2t)|| u - x||(2) (1/2t)|| u - x( t)||(2) - (t/2)|| partial derivative f(0)( x( t))||(2), where partial derivative f(0)(x( t)) is the least norm element of partial derivative f (x( t)). If f has a minimizer x*, this implies f ( x( t)) - f ( x*) = O(1/t), a result due to Brezis. If x(t) converges strongly to x*, we give a better estimate f (x( t)) - f (x*) <= 1/(integral(o)(t)parallel to x(s) - x*parallel to(-2) ds) = o(1/t).
subdifferential global convergence rate nonlinear semigroups strong convergence