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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-326/2/i/solution

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Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 326 / 2 / i / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-06

For a proper convex function

E

on a real Hilbert space, its subdifferential at

y

is

\partial E (y) = {p : E (x) \geq E (y) + ⟨ p, x - y ⟩ for every x} .

In a Banach space, interpret the pairing with an element of the dual space. For a chosen subgradient

p \in \partial E (y)

, the Bregman distance is

D_{E}^{p} (x, y) = E (x) - E (y) - ⟨ p, x - y ⟩ \geq 0.

If also

q \in \partial E (x)

, its symmetric Bregman distance is

D_{E}^{p} (x, y) + D_{E}^{q} (y, x) = ⟨ q - p, x - y ⟩ .

The subgradient inequality proves nonnegativity. These quantities depend on the chosen subgradients and generally are not metrics: they need not distinguish different points or obey the triangle inequality.

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