[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

! | R^m, 0,A^T = 0,b^T < 0

is nonempty (where A R^mn).

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 such that

a^Tx > x P₁, a^Tx < x P₂

Hence inf_xP₂ a^Tx > > sup_xP₂ a^Tx Use LP duality to simplify the infimum and supremum in these conditions.

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