MA208- SLIDES 6 – CONVEXITY EXAM QUESTIONS AND ANSWERS 100% CORRECT

Definition: For x, y∈ℝⁿ the line segment that joins x and y denoted by [x, y] is:

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{λx + (1-λ)y | λ∈[0, 1]}




Theorem: Weak separation for convex sets


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Let S, T⊆ℝⁿ be non-empty convex sets with S∩T=∅. Then ∃α∈ℝ,
c∈ℝⁿ, c≠0 s.t. ∀s∈S, ∀t∈T: c⋅s≤α and α≤c⋅t




Definition: Given f:ℝⁿ→ℝ, f is convex if:

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dom f is a convex set and ∀x, y∈dom f: ∀λ∈[0, 1]: f(λx + (1-
λ)y)≤λf(x) + (1-λ)f(y)




Theorem: Separation for convex sets


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Let C⊆ℝⁿ be a non-empty closed convex set. ∀v∈ℝⁿ s.t. v∉C: ∃c∈ℝⁿ
s.t. ∀x∈C: c⋅x > c⋅v




Definition: Given f:ℝⁿ→ℝ, f is affine if:

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∃α∈ℝⁿ, β∈ℝ s.t.

f(x) = αx + β = α₁x₁ + α₂x₂ +... + αₙxₙ + β

All affine functions are convex (and concave)




Definition: A convex combination of z₁, z₂, ..., zₖ∈ℝⁿ is:

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