Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 339 / 3 / c / i

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- Solution i

Solution

Take

f = I_{C}

and

h = I_{D}

. Their proximal maps are the projections

P_{C}, P_{D}

, so

w_{k + 1} = w_{k} - P_{D} w_{k} + P_{C} (2 P_{D} w_{k} - w_{k}) .

The Douglas--Rachford shadow sequence

P_{D} w_{k}

converges in finite dimensions to a point of

C \cap D

when the intersection is nonempty.

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