Determining the interset distance

The following problem is considered. We are given a vector space that can be the vector space ℝ^m or the vector space of symmetric m × m matrices 𝕊^m. There are two sets of vectors {a_i, 1 ≤ i ≤ r} and {b_j, 1 ≤ j ≤ q} in that vector space. Let K be some convex cone in the corresponding space. Let a_i ≥ _K b_j, ∀ i,j, where a_i ≥ _K b_j mean that a_i-b_j ∈ K. Let 𝒜_≤={x | a_i ≥ _K x, ∀ i, 1 ≤ i ≤ r }, where a_i ≥ _K x mean that a_i-x ∈ K. Further let ℬ_≥={y | y ≥ _K b_j, ∀ j, 1 ≤ j ≤ q }. In this work we study the question of finding and upperbounding the distance from the set 𝒜_≤ to the set ℬ_≥ in the case of cones ℝ_+^m, 𝕃^m, 𝕊_+^m .