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Picard iterates of a firmly nonexpansive map converge in finite dimensions

Statement

Let $T : E \to E$ be firmly nonexpansive on $E = R^{n}$ , suppose $Fix (T) = {p : T (p) = p}$ is nonempty, and start at any $w_{0}$ . Then the Picard sequence $w_{k + 1} = T (w_{k})$ converges to a fixed point of $T$ .

Facts & Assumptions

Given: The map $T$ , a fixed point $p$ , and $w_{0} \in E$ .

[A1]

The map is firmly nonexpansive, its fixed-point set is nonempty, and the start is in $E$ .

[F1]

Cauchy-Schwarz $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥_{2} ∥ y ∥_{2}$ with its equality case, the triangle inequality for $∥ \cdot ∥_{2}$ , the parallelogram law and polarisation: the Euclidean norm is a norm and satisfies Cauchy–Schwarz.

[F2]

For $n \geq 1$ every bounded sequence in $R^{n}$ has a convergent subsequence: for $n \geq 1$ , every bounded sequence in $R^{n}$ has a convergent subsequence.

[F3]

The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ : $R^{0}$ consists of the single zero vector.

Proof

1.1A1givenF1F3algebra

If $n = 0$ , [F3] gives $E = {0}$ and the nonempty fixed-point set forces $T (0) = 0$ ; since $w_{0} = 0$ , every iterate equals the fixed point. It remains to consider $n \geq 1$ . Firm nonexpansiveness applied to $w_{k}$ and a fixed point $p$ gives $∥ w_{k + 1} - p ∥_{2}^{2} \leq ⟨ w_{k + 1} - p, w_{k} - p ⟩$ . Expanding $w_{k} - p = (w_{k + 1} - p) + (w_{k} - w_{k + 1})$ and rearranging yields $∥ w_{k + 1} - p ∥_{2}^{2} + ∥ w_{k + 1} - w_{k} ∥_{2}^{2} \leq ∥ w_{k} - p ∥_{2}^{2} .$ Summing this inequality shows that the distances to $p$ are nonincreasing, the sequence is bounded, and $\sum_{k} ∥ w_{k + 1} - w_{k} ∥_{2}^{2} < \infty$ , so the increments tend to zero.

2.1A1F1F2step 1.1algebra

By [F2], some subsequence $w_{k_{j}}$ converges to $\overset{ˉ}{w}$ . Firm nonexpansiveness and [F1] imply $T$ is 1-Lipschitz, hence continuous. Since $w_{k_{j} + 1} - w_{k_{j}} \to 0$ , continuity gives $T (\overset{ˉ}{w}) = \overset{ˉ}{w}$ .

3.1A1step 1.1step 2.1algebra∎

Apply the inequality of step 1.1 with the fixed point $\overset{ˉ}{w}$ in place of $p$ . The distances $∥ w_{k} - \overset{ˉ}{w} ∥_{2}$ are nonincreasing, and along the subsequence $k_{j}$ they tend to zero. A nonincreasing nonnegative sequence with a subsequence tending to zero tends to zero; thus $w_{k} \to \overset{ˉ}{w}$ .

Depends on

Used by

Douglas–Rachford shadows converge to a zero of the operator sum Theorem

Dependency tree · two levels

40 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jonathan Eckstein and Dimitri P. Bertsekas, On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators, §§2–3 (standard reference, not scraped)