 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2022/discrete-mathematics-continuous-mathematics-and-probability-resit/5/c/i/solution

Put

q (x) = x^{T} A x

and

t (x) = x^{T} b

\nabla h (x) = 2 t (x) A x + q (x) b .

Differentiate each term. The derivative of

2 t A x

2 (A x) b^{T} + 2 t A

, while that of

q b

2 b (A x)^{T}

. Hence

H_{h} (x) = 2 (x^{T} b) A + 2 (A x) b^{T} + 2 b (A x)^{T} .

The last two terms are transposes of each other, so this Hessian matrix is symmetric, as expected.

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