Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 35 / 1 / a

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- Solution a

Solution

Suppose the non-vertical supporting hyperplane of a value function has slope

λ

. Every

x \in X

then satisfies

f (x) \geq ϕ (h (x)) \geq ϕ (b) + λ^{T} (h (x) - b),

so

L_{b} (x, λ) \geq ϕ (b)

. Taking the infimum gives

in f_{X} L_{b} \geq ϕ (b)

. Conversely, feasible points approaching the infimum of

f

have

L_{b} = f

, so

in f_{X} L_{b} \leq ϕ (b)

even if no primal minimizer exists. Therefore

X in f L_{b} (x, λ) = ϕ (b),

which is the Strong Lagrangian property.

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