Let $J$ be a strongly convex and differentiable functional defined on a closed and convex subset $K\subset H$ where $H$ is an Hilbert space. Moreover, $J’$ is $C$- Lipschitz.
I would like to prove that the sequence defined by $u_{n+1}= P_{K}(u_n - \mu J’(u_n))$, where $u_0$ is any vector in $H$ and $0<\mu<\frac{2\alpha}{C^2}$, converges to the solution of $\inf_{v\in K}J(v)$.
Proof : First by the hypothesis of the theorem, the problem has a solution and it is unique, denote $u$ this minimizer. The idea would be to consider $v_n = u_n - u$ and show that the sequence tends to $0_{H}$.
The central observation is that from the necessary condition (Euler-Lagrange) we know that
$$ \forall v\in K,\quad \langle J’(u), v-u\rangle \geq 0 \implies \langle u - (u -\mu J’(u)), v-u\rangle \geq 0 $$
Since $\mu>0$, but this is exactly the variational characterization of the projection of $ u -\mu J’(u)$ on $ K$ (at a sign near). So $u = P_{K}(u -\mu J’(u))$.
Using that we have
$$ \lVert v_{n+1}\rVert^{2} = \langle v_{n+1}, v_{n+1} \rangle = \langle P_{K}(u_n - \mu J’(u_n)) - u, P_{K}(u_n - \mu J’(u_n)) - u\rangle = \langle P_{K}(u_n - u -\mu(J’(u_n) - J’(u)), P_{K}(u_n - u -\mu(J’(u_n) - J’(u))\rangle\leq \lVert u_n - u-\mu(J’(u_n)-J’(u))\rvert^{2} = \langle u_n-u, u_n - u\rangle -2\mu\langle J’(u_n) -J’(u), u_n - u\rangle + \mu^{2}\lvert J’(u_n) -J’(u)\rvert^{2} \leq \lVert v_n \rVert^{2} -2\mu\alpha\lVert v_n\rVert^{2} + \mu^{2}C^2\lVert v_n\rVert^{2} = (1-2\mu\alpha + C^{2}\mu^{2})\lVert v_n \rVert^{2} $$
Where in the last inequality I used the fact that $J’$ is $C$-lipschitz and the fact that J being strongly convex (or $\alpha$ convex to be more explicit on why this $\alpha>0$ ) we have that for all $x,y$
$$ \langle J’(x) - J’(y), x-y\rangle \geq\alpha\lVert x-y\rVert^{2} $$
Thus we have for all $n$
$$ \lVert v_{n+1}\rVert\leq p\lVert v_n\rVert $$
Where $0<p<1$ and we conclude that the sequence $v_n$ tends to $0_{H}$ and so $u_n$ tends to $u$ which ends the proof.
I would like to know if it is correct please.
Thank you !