[SOLVED] ECE509 Homework3-Symmetric matrices

1. Given symmetric matrices F₀,F₁,…,F_n,cast the following optimization problem as an

SDP

min       (Ax + b)^TF(x)⁻¹(Ax + b)

s.t.     F(x) ≻ 0

where F(x) = F₀ + x₁F₁ + ··· + x_nF_n

2. Given symmetric matrices W0 and W1, find the dual of the optimization min

min     x^TW₀X

s.t.      x^TW₁x ≤ 1.

3. Use the duality idea to prove that the set is empty if the set

!λ | λ∈R^m,λ≥ 0,A^Tλ = 0,b^Tλ < 0″

is nonempty (where A ∈R^m×ⁿ).

4. Separating hyperplane between two polyhedra: formulate the following problem as an

LP (feasibility) problem. Find a separating hyperplane that strictly separates two polyhedra

P₁ = {x | Ax ≼ b},          P₂ = {x | Cx ≼ d}

then find a vector a ∈Rⁿ and a scalar γ such that

a^Tx > γ ∀x ∈ P₁,           a^Tx < γ ∀x ∈ P₂

Hence inf_x∈P₂ a^Tx > γ > sup_x∈P₂ a^Tx Use LP duality to simplify the infimum and supremum in these conditions.

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