On the convergence of the continuous gradient projection method. (2nd September 2019) - May, Ramzi
Abstract : We investigate the long time behaviour of the solutions x ( t ) to the first order differential inclusion x ′ ( t ) + x ( t ) ∈ P Q ( x ( t ) − λ ( t ) ∂ Φ ( x ( t ) ) ), where ∂ Φ is the subgradient of a given convex and continuous function defined on a real Hilbert space H, the operator P Q : H → Q is the orthogonal projection onto a closed, nonempty and convex subset Q of H, and λ : [ 0, + ∞ [ → ] 0, + ∞ [ is an absolutely continuous function. We establish that if the objective function Φ has at least one minimizer over Q and λ ( t ) behaviours, for t large enough, like t θ for some constant θ > − 1 then any solution x ( t ) to (the above equation) converges weakly to a minimizer of Φ over Q and satisfies the following fast decay property: Φ ( x ( t ) ) − Φ ∗ = o 1 t θ + 1 a s t → + ∞, where Φ ∗ = min x ∈ Q Φ ( x ) . Moreover, we prove the strong convergence of the solutions x ( t ) under some simple geometrical assumptions on the function Φ.