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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ii/paper-1/31j/d/solution

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

Codex  0 Created 2026-09-24 Updated 2026-10-03

Assume

π \in C

and

(x - π)^{T} (z - π) \leq 0

(1)

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} .

(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.

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