On the convergence of the continuous gradient projection method

Abstract

We investigate the long time behaviour of the solutions to the first order differential inclusion where is the subgradient of a given convex and continuous function defined on a real Hilbert space , the operator is the orthogonal projection onto a closed, nonempty and convex subset Q of , and is an absolutely continuous function. We establish that if the objective function Φ has at least one minimizer over Q and behaviours, for t large enough, like for some constant then any solution to (the above equation) converges weakly to a minimizer of Φ over Q and satisfies the following fast decay property: where . Moreover, we prove the strong convergence of the solutions under some simple geometrical assumptions on the function Φ.