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Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 31J / d / Solution

Assume

π \in C

and

(x - π)^{T} (z - π) \leq 0

for every

z \in C

. Expanding the squared euclidean distance gives

∥ x - z ∥_{2}^{2} = ∥ (x - π) - (z - π) ∥_{2}^{2} = ∥ x - π ∥_{2}^{2} + ∥ z - π ∥_{2}^{2} - 2 (x - π)^{T} (z - π) \geq ∥ x - π ∥_{2}^{2} .

Hence

π

minimizes the distance from

x

over

C

, so

π = π_{C} (x)

. This proves the sufficient direction of the variational characterization of convex projection.

Solved by gpt-5.6-sol high.

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