LAUSR

The following problem is considered. We are given a vector space that can be the vector space $$\mathbb {R}^m$$Rm or the vector space of symmetric $$m \times m$$m×m matrices $$\mathbb {S}^m.$$Sm. There are two sets of vectors $$\{a_i, 1 \le i \le r\}$${ai,1≤i≤r} and $$\{b_j, 1 \le j \le q\}$${bj,1≤j≤q} in that vector space. Let K be some convex cone in the corresponding space. Let $$a_i \ge _K b_j, \forall i,j,$$ai≥Kbj,∀i,j, where $$a_i \ge _K b_j$$ai≥Kbj mean that $$a_i-b_j \in K.$$ai-bj∈K. Let $$\mathcal {A}_{\le }=\{x | a_i \ge _K x, \forall i, 1 \le i \le r \},$$A≤={x|ai≥Kx,∀i,1≤i≤r}, where $$a_i \ge _K x$$ai≥Kx mean that $$a_i-x \in K.$$ai-x∈K. Further let $$\mathcal {B}_{\ge }=\{y | y \ge _K b_j, \forall j, 1 \le j \le q \}.$$B≥={y|y≥Kbj,∀j,1≤j≤q}. In this work we study the question of finding and upperbounding the distance from the set $$\mathcal {A}_{\le }$$A≤ to the set $$\mathcal {B}_{\ge }$$B≥ in the case of cones $$\mathbb {R}_+^m, \mathbb {L}^m, \mathbb {S}_+^m$$R+m,Lm,S+m.