New inertial factors of the Krasnosel’skiı̆-Mann iteration

Abstract

We consider inertial iterative schemes for approximating a fixed point of any given non-expansive operator in real Hilbert spaces. We provide new conditions on the inertial factors that ensure weak convergence and depend only on the iteration coefficients. For the special case of the Douglas-Rachford splitting, the conditions boil down to a sufficiently small upper bound on the sequence of inertial factors. Rudimentary numerical results indicate practical usefulness of the proposal.

Appendix

Below we give a short proof of Lemma 3.2 for completeness. Note that we no longer follow [7] to introduce the factor ρ_k there.

Proof

Rewrite (2) as

$$ \begin{array}{@{}rcl@{}} y_{k}&=&x_{k}+t_{k}(x_{k}-x_{k-1}), \end{array} $$

(29)

$$ \begin{array}{@{}rcl@{}} x_{k+1}&=&(1-\alpha_{k}) y_{k}+\alpha_{k} T(y_{k}), k=0,1,.... \end{array} $$

(30)

Since z is a fixed point of T, i.e., T(z) = z, it follows from (30) that

$$ x_{k+1}-z=\alpha_{k} (T(y_{k})-T(z))+(1-\alpha_{k}) (y_{k}-z). $$

Combing this with the following well-known identity [21]

$$ \|\alpha u+(1-\alpha)v\|^{2}=\alpha\|u\|^{2}+(1-\alpha)\|v\|^{2} -\alpha(1-\alpha)\|u-v\|^{2}, \forall u, v\in \mathcal{H} $$

for all real number α yields

$$ \begin{array}{@{}rcl@{}} &&\|x_{k+1}-z\|^{2}\\ &=&\|\alpha_{k} (T(y_{k})-T(z))+(1-\alpha_{k}) (y_{k}-z)\|^{2}\\ &=&\alpha_{k}\|T(y_{k})-T(z)\|^{2}+(1-\alpha_{k})\|y_{k}-z\|^{2}-\alpha_{k}(1-\alpha_{k}) \|T(y_{k})-y_{k}\|^{2}\\ &\leq&\alpha_{k}\|y_{k}-z\|^{2}+(1-\alpha_{k})\|y_{k}-z\|^{2}-\alpha_{k}(1-\alpha_{k}) \|T(y_{k})-y_{k}\|^{2}\\ &=&\|y_{k}-z\|^{2}-\alpha_{k}(1-\alpha_{k}) \|T(y_{k})-y_{k}\|^{2}, \end{array} $$

(31)

where the inequality follows from that T is non-expansive.

In view of the identity above and (29), we have

$$ \begin{array}{@{}rcl@{}} \|y_{k}-z\|^{2}&=&\|(1+t_{k})(x_{k}-z)-t_{k}(x_{k-1}-z)\|^{2}\\ &=&(1+t_{k})\|x_{k}-z\|^{2}-t_{k}\|x_{k-1}-z\|^{2}+(1+t_{k})t_{k} \|x_{k}-x_{k-1}\|^{2}. \end{array} $$

On the other hand, it is easy to see that

$$ \begin{array}{@{}rcl@{}} &&\|T(y_{k})-y_{k}\|^{2}\\ &=& \frac{1}{{\alpha_{k}^{2}}}\|(x_{k+1}-x_{k})-t_{k}(x_{k}-x_{k-1})\|^{2}\\ &=& \frac{1}{{\alpha_{k}^{2}}}\|x_{k+1}-x_{k}\|^{2}+\frac{{t_{k}^{2}}}{{\alpha_{k}^{2}}}\|x_{k}-x_{k-1}\|^{2} -2\frac{t_{k}}{{\alpha_{k}^{2}}}\langle x_{k+1}-x_{k}, x_{k}-x_{k-1}\rangle\\ &\geq& \frac{1}{{\alpha_{k}^{2}}}\|x_{k+1}-x_{k}\|^{2}+\frac{{t_{k}^{2}}}{{\alpha_{k}^{2}}}\|x_{k}-x_{k-1}\|^{2} -\frac{t_{k}}{{\alpha_{k}^{2}}}\left( \|x_{k+1}-x_{k}\|^{2}+\|x_{k}-x_{k-1}\|^{2}\right)\\ &=& \frac{1}{{\alpha_{k}^{2}}}(1-t_{k})\|x_{k+1}-x_{k}\|^{2} +\frac{1}{{\alpha_{k}^{2}}}({t_{k}^{2}}-t_{k})\|x_{k}-x_{k-1}\|^{2}. \end{array} $$

(32)

Finally, making use of these two relations to bound (31) yields the desired results. □