Papers:
- Using orthogonally structured positive bases for constructing positive k-spanning sets with cosine measure guarantees (Link!)
- Joint work with Clément W. Royer, Gabriel Jarry-Bolduc and Warren Hare. Published in Linear Algebra and its applications.
- A characterization of positive spanning sets with ties to strongly edge-connected digraphs (Link!)
- Joint work with Clément W. Royer and Denis Cornaz. Published in Discrete Applied Mathematics.
Positive spanning sets (PSSs) can be defined as sets properly approximating any given direction in the space. Optimization methods based on PSSs typically favor those with the best cosine measure. However, there is no easy way to compute this measure for an arbitrary set.
This paper shows how the cosine measure of some specific positive spanning sets (called OSPB) can be computed in polynomial time. Using OSPBs, it also shows how to create resilient PSSs whose cosine measure remains large enough after removing some elements.
Despite certain classes of positive spanning sets being well understood, a complete characterization of PSSs remains elusive. In this paper, we explore a relatively understudied relationship between positive spanning sets and strongly edge-connected digraphs, in that the former can be viewed as a generalization of the latter. We leverage this connection to define a decomposition structure for positive spanning sets inspired by the ear decomposition from digraph theory.
Conferences:
- Derivative Free Optimization Symposium (Kelowna, 2022).
- Siam Conference on Optimization (Seattle, 2023). Slides
- Optimization (Aveiro, 2023).
- Siam Conference on Applied Linear Algebra (Paris, 2024).
- Derivative Free Optimization Symposium (Padova, 2024). Slides