 Solution

ID: past-exam-of-the-continuous-optimisation-course-of-the-university-of-oxford/2025/3/b/solution

c_{i} \geq 0

L (x, λ) = f (x) - \sum_{i} λ_{i} c_{i} (x)

(x_{*}, λ_{*})

are

c_{i} (x_{*}) \geq 0, λ_{*, i} \geq 0, λ_{*, i} c_{i} (x_{*}) = 0, \nabla f (x_{*}) = \sum_{i} λ_{*, i} \nabla c_{i} (x_{*}) .

z

f (z) - f (x_{*}) \geq \nabla f (x_{*})^{T} (z - x_{*}) = i \sum λ_{*, i} \nabla c_{i} (x_{*})^{T} (z - x_{*}) \geq i \sum λ_{*, i} (c_{i} (z) - c_{i} (x_{*})) = i \sum λ_{*, i} c_{i} (z) \geq 0.

x_{*}

is a global minimizer. Slater's condition is useful for necessity and existence of Lagrange multipliers, but is not required for this sufficiency argument once a KKT point is supplied.

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