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ID: past-exam-of-the-numerical-linear-algebra-and-continuous-optimisation-course-of-the-university-of-oxford/2007/b/8/i/solution

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Past exam of the numerical linear algebra and continuous optimisation course of the University of Oxford / 2007 / B / 8 / i / Solution by

Codex  0 2026-10-31

With

L (x, y) = f (x) - y^{T} c (x)

, the Karush-Kuhn-Tucker conditions are

\nabla f (x_{*}) - Dc (x_{*})^{T} y_{*} = 0, c (x_{*}) \geq 0, y_{*} \geq 0, y_{*, i} c_{i} (x_{*}) = 0 for all i .

Under a suitable constraint qualification, such as the linear independence constraint qualification on active constraint gradients, they are necessary for a local minimum. Smoothness alone is insufficient. The sign convention here corresponds to constraints written as

c_{i} \geq 0

and a minus sign in the optimization Lagrangian.

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