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ID: past-exam-of-the-numerical-linear-algebra-and-continuous-optimisation-course-of-the-university-of-oxford/2009/b/6/a/solution

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Past exam of the numerical linear algebra and continuous optimisation course of the University of Oxford / 2009 / B / 6 / a / Solution by

Codex  0 2026-10-31

The contingent tangent cone consists of feasible first-order velocities:

T_{Ω} (x) = {d : \exists t_{n} ↓ 0, x_{n} \in Ω, (x_{n} - x) / t_{n} \to d} .

(1)

x_{*}

is a local minimizer, then

\nabla f (x_{*})^{T} d \geq 0 for every d \in T_{Ω} (x_{*})

. Indeed,

x_{n} \to x_{*}

along the defining sequence, so eventually

f (x_{n}) - f (x_{*}) \geq 0

. Differentiability gives

\frac{f ( x _{n} ) - f ( x _{*} )}{t _{n}} = \nabla f (x_{*})^{T} \frac{x _{n} - x _{*}}{t _{n}} + o (1) ⟶ \nabla f (x_{*})^{T} d .

(2)

This proves the inequality. Using feasible limiting velocities avoids assuming that every tangent direction is realized by a straight segment staying in the feasible set.

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