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}) .

(1)

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.

ii

 0  0

Solution

 0  0

The function

d_{D} (x) = min_{y \in D} \frac{1}{2} ∥ y - x ∥^{2}

(1)

is the Moreau envelope of the convex indicator

I_{D}

, and is therefore convex. It is nonnegative and vanishes exactly on

D

. Since

C \cap D \neq = \emptyset

, the minimum of

d_{D}

over

C

is zero, and every minimizer belongs to both

C

and

D

iii

 0  0

Solution

 0  0

The squared distance to a convex set satisfies

\nabla d_{D} (x) = x - P_{D} x .

(1)

The map

I - P_{D}

is firmly nonexpansive because

P_{D}

is firmly nonexpansive, so

∥\nabla d_{D} (x) - \nabla d_{D} (y) ∥ \leq ∥ x - y ∥.

(2)

Thus

d_{D}

is one-smooth in the Euclidean norm.

iv

 0  0

Solution

 0  0

Projected gradient descent with unit step is

x_{k + 1} = P_{C} [x_{k} - (x_{k} - P_{D} x_{k})] = P_{C} (P_{D} x_{k}) .

(1)

Thus this instance reduces to alternating Euclidean projections.

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

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