# Determining the interset distance

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## Abstract

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

## Keywords

Conic optimization Quadratic optimization Semidefinite optimization## Notes

### Acknowledgements

Maksim Barketau is partially supported by the Belarusian Republican Foundation for Fundamental Research through project \(\Phi \hbox {15CO-043}\).

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