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On the convergence of the continuous gradient projection method. (English) Zbl 1430.90462
Summary: We investigate the long time behaviour of the solutions $$x(t)$$ to the first order differential inclusion $x'(t)+x(t) \in P_Q (x(t)-\lambda (t)\partial\Phi (x(t))),$ where $$\partial\Phi$$ is the subgradient of a given convex and continuous function defined on a real Hilbert space $$\mathcal{H}$$, the operator $$P_Q:\mathcal{H}\to Q$$ is the orthogonal projection onto a closed, nonempty and convex subset $$Q$$ of $$\mathcal{H}$$, and $$\lambda :[0,+\infty [\to ]0,+\infty [$$ is an absolutely continuous function. We establish that if the objective function $$\Phi$$ has at least one minimizer over $$Q$$ and $$\lambda (t)$$ behaviours, for $$t$$ large enough, like $$t^\theta$$ for some constant $$\theta >-1$$ then any solution $$x(t)$$ to (the above equation) converges weakly to a minimizer of $$\Phi$$ over $$Q$$ and satisfies the following fast decay property: $\Phi (x(t))-\Phi^* =o \left(\frac{1}{t^{\theta+1}}\right)\text{ as }\to +\infty,$ where $$\Phi^* =\min_{x\in Q} \Phi (x)$$. Moreover, we prove the strong convergence of the solutions $$x(t)$$ under some simple geometrical assumptions on the function $$\Phi$$.
##### MSC:
 90C25 Convex programming 90C30 Nonlinear programming
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